• No results found

Behaviour of reinforced and concrete-encased composite columns subjected to biaxial bending and axial load

N/A
N/A
Protected

Academic year: 2021

Share "Behaviour of reinforced and concrete-encased composite columns subjected to biaxial bending and axial load"

Copied!
12
0
0

Loading.... (view fulltext now)

Full text

(1)

Building and Environment 43 (2008) 1109–1120

Behaviour of reinforced and concrete-encased composite columns

subjected to biaxial bending and axial load

Cengiz Dundar



, Serkan Tokgoz, A. Kamil Tanrikulu, Tarik Baran

Civil Engineering Department, Cukurova University, 01330 Adana, Turkey

Received 3 February 2006; received in revised form 26 January 2007; accepted 2 February 2007

Abstract

An experimental investigation of the behaviour of reinforced concrete columns and a theoretical procedure for analysis of both short and slender reinforced and composite columns of arbitrarily shaped cross section subjected to biaxial bending and axial load are presented. In the proposed procedure, nonlinear stress–strain relations are assumed for concrete, reinforcing steel and structural steel materials. The compression zone of the concrete section and the entire section of the structural steel are divided into adequate number of segments in order to use various stress–strain models for the analysis. The slenderness effect of the member is taken into account by using the Moment Magnification Method. The proposed procedure was compared with test results of 12 square and three L-shaped reinforced concrete columns subjected to short-term axial load and biaxial bending, and also some experimental results available in the literature for composite columns compared with the theoretical results obtained by the proposed procedure and a good degree of accuracy was obtained.

r2007 Elsevier Ltd. All rights reserved.

Keywords: Reinforced concrete column; Composite column; Biaxial loading; Ultimate strength; Stress–strain models

1. Introduction

Reinforced and concrete-encased composite columns of arbitrarily shaped cross section subjected to biaxial bending and axial load are commonly used in many structures, such as buildings and bridges. A composite column is a combination of concrete, structural steel and reinforcing steel to provide an adequate load carrying capacity of the member. Thus, such composite members can provide rigidity, usable floor areas and cost economy for mid-to-high buildings. Many experimental and analy-tical studies have been carried out on reinforced and

composite members in the past years. Furlong [1] has

carried out analytical and experimental studies on re-inforced concrete columns using well-known rectangular stress block for the concrete compression zone in the

analysis. Brondum-Nielsen [2]has proposed a method of calculating the ultimate strength capacity of cracked polygonal concrete sections using rectangular stress block in the concrete compression zone of the section under biaxial bending. Hsu [3,4] has presented theoretical and experimental results for L-shaped and channel shaped

reinforced concrete sections. Dundar [5] has studied

reinforced concrete box sections under biaxial bending

and axial load. Rangan [6] has presented a method to

calculate the strength of reinforced concrete slender columns including creep deflection due to sustained load as an additional eccentricity and the method compared

with ACI 318-Building Code Method [7]. Dundar and

Sahin [8] have researched arbitrarily shaped reinforced concrete sections subjected to biaxial bending and axial load using Whitney’s stress block [9] in the compression zone of the concrete section. Rodriguez and Ochoa[10]and

Fafitis [11] have suggested numerical methods for the

computation of the failure surface for reinforced concrete sections of arbitrary shape. Hong [12] has proposed a simple approach for estimating the strength of slender www.elsevier.com/locate/buildenv

0360-1323/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.buildenv.2007.02.010

Corresponding author. Tel.: +90 322 338 6762; fax: +90 322 338 6702. E-mail addresses:dundar@cu.edu.tr (C. Dundar),stokgoz@cu.edu.tr (S. Tokgoz),akt@cu.edu.tr (A.K. Tanrikulu),tbaran@cu.edu.tr (T. Baran).

(2)

reinforced concrete columns with arbitrarily shaped cross section using nonlinear stress–strain relationship for the materials. Saatcioglu and Razvi [13] have presented an experimental research to investigate the behaviour of high strength concrete columns confined by rectilinear reinfor-cement under concentric compression. Furlong et al. [14]

have examined several design procedures for ultimate strength analysis of reinforced concrete columns and compared with many short and slender experimental columns under short-term axial load and biaxial bending. Morino et al. [15]have presented a series of experimental results on short and slender square composite columns. Roik and Bergmann[16]have proposed a simplified design method based on the strength interaction curve and reported test results for short and slender composite columns with unsymmetrical square and rectangular

shaped cross sections. Virdi and Dowling [17] have

presented a numerical method and test results for square composite columns to predict the ultimate strength capacity of the composite column members under biaxial bending and axial load. Mirza[18]has examined the effects of variables, such as the confinement effect, the ratio of structural steel to gross area, the compressive strength of concrete, the yield strength of steel and the slenderness ratio, on the ultimate strength of composite columns. Mirza and Skrabek [19] have carried out a statistical analysis on the variability of ultimate strength capacity of

slender composite beam-columns. Munoz[20] and Munoz

and Hsu [21,22] have presented an experimental and a

theoretical study based on the finite differences method, including confinement effect for the concrete, on square and rectangular cross sections of short and slender composite columns under biaxial load. Weng and Yen

[23] have investigated the differences between the ACI[7]

and AISC [24] approaches for the design of

concrete-encased composite columns. Lachance[25], Chen et al.[26]

and Sfakianakis [27] have proposed a numerical analysis method for short composite columns of arbitrarily shaped cross section.

Confinement provided by lateral ties increases the ultimate strength capacity and ductility of reinforced concrete columns under combined biaxial bending and axial load. Strength and ductility gain in concrete are obtained by many confinement parameters e.g., the compressive strength of concrete, longitudinal reinforce-ment, type and the yield strength of lateral ties, tie spacing, etc. Because of such parameters, determination of mechanical behaviour of confined concrete is not as easy as unconfined concrete. Some researchers for instance, Kent and Park [28], Sheikh and Uzumeri[29], Saatcioglu

and Razvi [30], Chung et al. [31] have presented a

stress–strain relationship to describe the confined concrete behaviour.

The main objective of this paper is to present an iterative computing procedure for the rapid design and ultimate strength analysis of arbitrarily shaped both short and slender reinforced, and concrete-encased composite

mem-bers having arbitrarily located reinforcing steel bars and structural steel elements subjected to biaxial bending and axial load. For this aim a computer program has been developed which considers various confined or unconfined concrete stress–strain models for the concrete compression zone for both short and slender reinforced and composite columns. Thus, results of the ultimate strength analysis, with various concrete models, can be compared with each other.

In the experimental part of the study, 12 square and three L-shaped short and slender reinforced concrete columns were tested to determine the ultimate strength capacity, load-deflection behaviour, load-axial strain be-haviour and confinement effect of column members. The test results were compared with the theoretical results obtained by the developed computer program which uses various stress–strain models for the confined concrete or unconfined concrete in the compression zone of the member.

Finally, the theoretical results obtained using the proposed procedure were also compared with the test results available in the literature for short and slender composite columns.

2. Experimental program

An experimental investigation of the behaviour of reinforced concrete columns under short-term axial load and biaxial bending is presented. The primary objective of this investigation was to examine the ultimate strength capacity and load-deflection behaviour of short and slender reinforced concrete columns and to compare the test results of ultimate strength capacities of specimens with the results obtained by the proposed theoretical procedure using various stress–strain models for the materials. For this reason, reinforced concrete specimens were designed with different length, dimension and cross section with different diameter and arrangement of longitudinal and lateral reinforcements.

2.1. Test specimens

The experimental program includes 15 reinforced con-crete columns. Five specimens are short square tied columns (C1–C5), seven specimens are slender square tied columns (C11–C14, C21–C23) and the other three are L-shaped section slender tied columns (LC1–LC3). The cross section details and dimensions of each specimen are shown inFig. 1.

The reinforced concrete column specimens were cast horizontally inside a formwork in Structural Laboratory at Cukurova University, Adana. Maximum 20 mm diameter local aggregate and Normal Portland Cement were used in all concrete batches. Three standard cylinder specimens (150 mm in diameter by 300 mm long) were cast from each column specimen concrete mix and cured under the same condition as the column specimen in the Structural

(3)

Laboratory. The concrete cylinder specimens were tested at the same day of that column specimen in order to determine the mean compressive strength of concrete and to attain a stress–strain relationship for each column specimen to use in the analysis. A typical concrete stress–strain relationship obtained experimentally is given inFig. 2.

The longitudinal reinforcement consisted of 6 and 8 mm diameter of deformed bars with the yield strength of 630 and 550 MPa, respectively. Lateral reinforcements were arranged using 6 and 6.5 mm diameter of deformed reinforcing bars with the yield strength of 630 MPa for the specimens. The lateral ties were bent into 135hooks at the ends. The overall length ðLÞ, the mean compressive strength of the concrete ðfcÞ, the eccentricities of the applied load (ex and ey) and the lateral reinforcement

arrangements of the specimens (diameter+ and spacing s) are presented inTable 1.

2.2. Test procedure

The specimens were tested with pinned conditions at both ends using 400 kN capacity HI-TECH MAGNUS hydraulic testing machine in the Structural Laboratory at Cukurova University. A photograph of test setup for reinforced concrete column specimen is shown inFig. 3.

For the application of the biaxially eccentric load to the column specimen in the vertical position, the heavily reinforced brackets were designed for both ends of the specimen to prevent local failures. The point of the applied biaxially eccentric axial load was sensitively marked up on

17.5 17.5 100 mm 17.5 17.5 100 mm 6 mm (C1−C5) 8 mm (C11−C14) 25 25 150 mm 25 25 150 mm 8 mm 150 mm 150 mm 100 100 x y G 17.5 6 mm 17.5 (C1−C5, C11−C14) (C21−C23) LC1−LC3

Fig. 1. Reinforced concrete column specimen cross sections.

0 10 20 30 40 50 0 0.002 0.0035 Stress-Strain Relation (LC3) Stress (MPa) Strain 0.0005 0.001 0.0015 0.0025 0.003 0.004

Fig. 2. Experimental concrete stress–strain relationship for specimen LC3.

Table 1

Specimen details of reinforced concrete columns

Specimen no. L (mm) fc(MPa) ex(mm) ey(mm) f=s (mm/cm)

C1 870 19.18 25 25 6/12.5 C2 870 31.54 25 25 6/15 C3 870 28.13 25 25 6/10 C4 870 26.92 30 30 6/8 C5 870 25.02 30 30 6/10 C11 1300 32.27 35 35 6.5/10.5 C12 1300 47.86 40 40 6.5/10.5 C13 1300 33.10 35 35 6.5/10.5 C14 1300 29.87 45 45 6.5/12.5 C21 1300 31.70 40 40 6.5/10.5 C22 1300 40.76 50 50 6.5/10.5 C23 1300 34.32 50 50 6.5/10.5 LC1 1300 35.12 36.25 36.25 6/10 LC2 1300 32.77 41.25 41.25 6/11 LC3 1300 44.88 46.25 46.25 6/13

(4)

the surface of each bracket of the specimen. Two different types of load cells were accommodated at both ends to obtain the biaxially applied axial load. Linear variable transducers were located at the middle height of each column specimen to measure the lateral deflection of the member into two principal directions. In addition, the axial displacement measurements of the most heavily com-pressed fiber were recorded by using displacement

trans-ducer in order to compute axial strain for the specimens C1–C5, C11–C14. The load cells and the transducers were calibrated before they were used in the test. A schematic diagram of test setup and instrumentation are presented in

Fig. 4.

At load controlled tests, the monotonically increasing compressive biaxially axial load was applied to the column specimen at a rate of 1 kN/s. In the meantime, short-term axial load, lateral and axial displacements of the column specimen were recorded. The test was continued until a significant drop in load resistance occurred.

2.3. Test results

In the experimental investigation, all column specimens behaved in a similar manner until crushing of concrete occurred. It was observed that the tensile crack in the tension zone are and the concrete crash in the compression zone are located nearly in the middle height of the column specimens as shown inFig. 5.

Experimental load-deflection diagrams for the specimens C5, C14, C23 and LC3 are presented inFig. 6(a)–(d) for x and y directions. As can be seen in the diagrams, the column specimens behaved in a ductile manner in both sides. The specimens deflected until reaching the peak load and tensile crack observed on the convex side of the

Fig. 3. Photograph of test setup for reinforced concrete column specimen.

8 channel data logger Personal computer Steel plate Hydraulic pump Laboratory floor Load cell L/2 L/2 Displacement transducers Signal cables Steel ball HI-TECH MAGNUS Load cell 8 channel data logger

(5)

specimens at that stage. After that, a sudden drop occurred in load resistance and lateral displacements increased. This indicated that the lateral reinforcements played a signifi-cant role in column ductility and confinement.

Load-axial strain relation for the specimen C14 is given inFig. 7. As shown from the figure the specimen behaved in a ductile manner until crushing of concrete. Because of

the confinement provided by lateral ties, the maximum strain value for the column specimen shown in the figure exceeded the maximum compressive strain value obtained experimentally from the uniaxially tested cylinder specimen.

The experimental results of the specimens are evaluated theoretically in Section 4.

Fig. 5. Reinforced concrete column after testing.

0 20 40 60 80 100 0 5 Load (kN) X Axis Y Axis 0 20 40 60 80 0 4 Load (kN) Load-Deflection (C5) Deflection (mm) Load-Deflection (C14) Deflection (mm) 6 8 10 12 2 2.5 7.5 10 12.5 X Axis Y Axis X Axis Y Axis X Axis Y Axis 0 50 100 150 200 250 0 4 8 Load (kN) 0 50 100 150 200 0 4 Load (kN) 2 Deflection (mm) 6 10 Load-Deflection (LC3) 2 6 8 Deflection (mm) 10 12 Load-Deflection (C23)

(6)

3. Analysis method 3.1. Assumptions

The proposed method is based on the following assumptions:

1. Plane sections remain plane after deformations (Bernoulli’s assumption).

2. Arbitrary monotonic stress–strain relationships for each of the three materials (i.e., concrete, structural steel and the reinforcing bars) may be assumed.

3. The longitudinal reinforcing bars are identical in diameter and are subjected to the same amount of strain as the adjacent concrete.

4. The effect of creep and the tensile strength of concrete and any direct tension stresses due to shrinkage etc. are neglected.

5. Shear deformation is neglected. 3.2. Stress–strain models for the materials

In the proposed procedure, the idealized stress–strain relationships for the concrete and the steel materials are expressed in the following generalized form: s ¼ f ðÞ

(Fig. 8). In this study, various stress–strain models available in literature have been used in the compression zone of the concrete section. The mathematical expressions of these models are presented inTable 2.

InTable 2, the parameters used in Kent and Park model

[28] and Saatcioglu and Razvi model [30] are defined as follows:

In Kent and Park model for confined and unconfined concrete: c20¼1:6ð50uþ50hÞ 0:6coc, 50u¼ 3 þ 0:285fc 142fc1000Xco ðfctaken as MPaÞ, coc¼Kco; fcc¼Kfc; K ¼ 1 þ rsfywk fc ; rs¼ AoLs sbkhk , 50h¼0:75rs bk s  1=2 ; Zu¼ 0:5 50uco , Zc¼ 0:5 50uþ50hcoc ,

where rs, Ao, Ls, bk, and hk are volumetric ratio, cross

sectional area, length, small and large edge of lateral ties, respectively; s is the centre-to-centre spacing between lateral ties; K is the strength enhancement factor and fywk is the yield strength of lateral ties.

In Saatcioglu and Razvi model for confined concrete: c20¼ 0:8 0:15ð85ccÞ þcc; 85 ¼260rccþ0:0038, cc¼coð1 þ 5KÞ, K ¼k1fle fc ; k1¼6:7ðfleÞ0:17; fcc¼fcþk1fle,

where 85, ccand co are the strain at 85% strength, is the

strain corresponding to peak stress of confined concrete and the strain corresponding to peak stress of unconfined concrete, respectively; fle, fc and fcc are the equivalent uniform lateral pressure, unconfined strength of concrete 0 10 20 30 40 50 60 70 0 Load (kN)

Load-Axial Strain Relation (C14)

Axial Strain

0.002

0.001 0.003 0.004 0.005 0.006

Fig. 7. Load-axial strain relation for the specimen C14.

Concrete Steel Strain Stress εcu εco fc Strain Stress fy1 ε1 εn fyn

a

b

(7)

and confined strength of concrete, respectively; r is the volumetric ratio of the section[30].

3.3. Formulation of the problem

An arbitrarily shaped concrete cross section with any arrangement of reinforcement and structural steel sub-jected to biaxially eccentric load ðNÞ is shown inFig. 9.

The coordinates (xN, yN) of the axial load N with respect

to x–y axis system are obtained as follows:

xN ¼xpcex, ð1Þ

yN ¼ypcey, ð2Þ

in which exand eyare the eccentricities of the axial load N

with respect to the parallel axis system through the plastic centroid (PC) of the arbitrarily shaped composite cross section. xpc and ypc indicate the coordinates of the plastic

centroid of the arbitrarily shaped composite cross section and given as follows[16]:

xpc¼ Acxcfc=gcþAsxsfy=gsþAtxtft=gt Acfc=gcþAsfy=gsþAtft=gt , ð3Þ ypc ¼Acycfc=gcþAsysfy=gsþAtytft=gt Acfc=gcþAsfy=gsþAtft=gt , ð4Þ

where Ac, As; and At are the total areas of concrete,

reinforcing bars and the structural steel, respectively; fc, fy; and ft are the strength of concrete, reinforcing bars and structural steel, respectively; xc, yc, xs, ys, xt, yt are the

centroid coordinates of the x–y axis system for concrete, reinforcing bars and structural steel, respectively; and gc, gs, and gt are the partial safety factors of concrete,

reinforcing bars and structural steel, respectively.

In the analysis, the plastic centre is assumed as the geometric centre of the section for reinforced concrete members.

The strain distribution is linear across the concrete section according to Bernoulli’s assumption. Therefore, the strain at any point (xi, yi) in the cross section is

indicated by i¼cu yi cþ xi a   1 h i , (5)

where a and c are the horizontal and the vertical distances between the origin of the x–y axis system and the neutral axis. cu denotes the strain at the location of the maximum

compressive stress.

In the analysis, any stress–strain model can be used for the compression zone of the concrete and the steel materials. For this reason, the compression zone of the arbitrarily shaped cross section and entire section of the structural steel are divided into adequate number of parallel segments to the neutral axis. The geometric properties (i.e., the area and the coordinates of the centroid) of each segment are computed according to the

procedure given in Dundar and Sahin [8]. The stress

resultants of the arbitrarily shaped concrete member and the structural steel are calculated at the centroid of each segment by using Eq. (5) and assumed stress–strain model for each material. Hence, using a sufficient number of segments for the compression zone of the concrete member provide more compatible compressive stress distribution with the assumed stress–strain model. The optimum number of segments is computed iteratively until conver-gence factor satisfy in the analysis. The same approach is also applied to the structural steel element in compression and tension zone for the composite column analysis. 3.4. Equilibrium equations

The equilibrium equations for an arbitrarily shaped composite column member subjected to biaxial bending and axial load can be written with respect to the x0–y0axis

Table 2

Mathematical expressions for the concrete stress–strain relations

Model 0pcpco coocpcu co, cu Hognestad[32] sc¼fc 2c co  c co  2 " # sc¼fc 1  0:15 cco cuco     0.002, 0.0038 CEC[33] sc¼fc 2c co  c co  2 " # s c¼fc 0.002, 0.0035

Kent and ParkðuÞunconfined[28]

sc¼fc 2c co  c co  2 " # s c¼fc½1  ZuðccoÞ 0.002, 0.004

Kent and ParkðcÞconfined[28]

sc¼fcc 2c coc  c coc  2 " # s c¼fcc½1  ZcðccocÞX0:2fcc 0.002, c20

Saatcioglu and Razvi[30]

sc¼fcc 2c cc  c cc  2 " #1=1þ2K pfcc sc¼fccþ fccf85 cc85   ccc ð ÞX0:2fcc 0.002, c20

(8)

system with its origin at the coordinates of the biaxially eccentric load (xN, yN) and parallel to the x–y axis system

(Fig. 9) as follows: f1¼X t k Acksck Ast m Xm i ssi Xn j AtjstjN ¼ 0, (6) f2¼ Ast m Xm i ðxixNÞssiþ Xn j AtjstjðxtjxNÞ X t k AcksckðxckxNÞ ¼0, ð7Þ f3¼ Ast m Xm i ðyiyNÞssiþ Xn j AtjstjðytjyNÞ X t k AcksckðyckyNÞ ¼0, ð8Þ

where Ast is the total area of the reinforcing bars within

the cross-section; sck is the concrete compressive stress at

the centroid of the kth segment; Ack and (xck, yck) indicate

the area and the centroid coordinates of kth concrete segment, respectively; t is the number of segment of the concrete in compression zone; ssi is the stress of ith

reinforcing bar; xi and yi are the coordinates of the ith

reinforcing bar; m is the total number of reinforcing bars; Atjand (xtj, ytj) are the area and the centroid coordinates of

the jth structural steel segment, respectively; stj is the

structural steel stress at the centroid of the jth segment; n is the total number of segment of the structural steel.

In the design procedure, the algebraic equations (Eqs. (6)–(8)) have three unknowns ða; c; AstÞ. These

equations are solved by a numerical procedure [8] based on the Newton–Raphson iterative method.

For the ultimate strength analysis, the biaxially eccentric ultimate load Nucan be determined by solving Eq. (7) and

Eq. (8) for (a, c) by using the aforementioned procedure and substituting them in Eq. (6), resulting in

Nu¼ Xt k Acksck Ast m Xm i ssi Xn j Atjstj. (9) 3.5. Slenderness effect

The slenderness effect of reinforced or composite column is considered by using the Moment Magnification Method (ACI 318-99[7]) as follows:

The moment magnification factor d is expressed as

d ¼ Cm

1  1:33Nu=Ncr

X1:0, (10)

in which Cm is the end effect factor (Cm¼1:0 for the pin

ended column), taken as follows: Cm¼0:6 þ 0:4

Mu1

Mu2

X0:4; Mu1pMu2 (11)

with Mu1 and Mu2 are the end moments of the column.

σc εcu εsi Asi x' y' x y N Neutral axis c a Typical segment Opening Structural steel Reinforcing bar PC xN yN ex ey xpc ypc

(9)

Ncr is the elastic buckling load of column:

Ncr¼

p2EI

ðkLÞ2, (12)

where ðkLÞ is the effective length and EI is the effective flexural rigidity of the column section. The effective flexural rigidity of the section plays an important role on the computation of the ultimate strength capacity of slender reinforced and composite columns and may be determined as follows:

For reinforced concrete columns (ACI 318-99[7]): EI ¼0:2EcIg 1 þ bd þEsIs (13) or EI ¼0:4EcIg 1 þ bd (14) where Ec and Es are the modulus of elasticity of the

concrete and the steel materials, respectively; Ig is the

moment of inertia of gross concrete section of the column;

Is is the moment of inertia of reinforcement about

centroidal axis of member cross section; bdis the sustained load factor (bd¼0 for short-term axial load).

For composite columns[16]:

EI ¼ EceIceþEsIsþEtIt, (15)

where

Ece ¼600fc. (16)

In Eq. (15), Ice, Is and It are the moments of inertia of

the uncracked concrete, the reinforcement and the struc-tural steel section, calculated with respect to the elastic centroidal axis, respectively. Es and Et are the modulus of

elasticity of reinforcement and structural steel, respectively. For the arbitrarily shaped composite cross section, the coordinates of the elastic centre may be expressed with respect to the x–y axis system as[16]:

xec¼ AcEcxcþAsEsxsþAtEtxt AcEcþAsEsþAtEt , ð17Þ yec¼AcEcycþAsEsysþAtEtyt AcEcþAsEsþAtEt , ð18Þ

in which Ecis the modulus of elasticity of the concrete (TS

500[34]), given as Ec¼3250 ffiffiffiffiffi fc p þ14 000 ðMPaÞ. (19)

For biaxial bending, ACI 318-99 [7] recommends that the moment magnification factors shall be computed for each axis separately and multiplied by the corresponding moments as follows:

Mux¼dxNuey; Muy¼dyNuex. (20)

To reach the ultimate strength value ðNuÞ of a slender

column, dx and dy are computed for each iteration until

satisfying the equilibrium equations.

The flowchart of the developed computer program based on the aforementioned procedure is illustrated inFig. 10.

As shown in the flowchart, at the beginning of the analysis, the initial values for the location of the neutral axis (a, c) and the number of segment ðtÞ are read from the data file, then Nuor Astis computed using an iterative procedure. To

optimise the number of segment ðtÞ, it is increased by one until the following criteria is satisfied:

BiBi1 Bi p, (21)

where Biis Nuor Astvalue which is obtained at the ith step,

 is the convergence factor.

Due to space limitation, the listing of developed computer program is not given in the paper. A personal computer version and the manual of the program can be obtained free of charge from the authors upon request.

4. Evaluation of test results

4.1. Computer analysis of reinforced concrete columns tested by the authors

The reinforced concrete column specimens (C1–C5, C11–C14, C21–C23, LC1–LC3) were tested with pinned conditions at both ends under short-term axial load and biaxial bending. These specimens were also analysed for

Read Input Data: Material properties, Column geometrical properties Initial parameters (a, c, t)

Calculate cross section properties (Area, Geometric centre coordinates)

Iterative calculation for Nu or Ast

Calculate segment stresses Calculate reinforcement bar stresses Calculate structural steel stresses

Satisfying equilibrium equations? Derive new values

for neutral axis parameters (a, c) No Yes ≤ε Bi Bi - Bi-1 Yes Store results to output file Increase number of segment (t = t +1) No

(10)

the ultimate strength capacities using the computer program.

In the ultimate strength analysis, various stress–strain models, Hognestad (HOG)[32], CEC[33], Kent and Park unconfined [28] ðK&PðuÞÞ, Kent and Park confined [28]

ðK&PðcÞÞ, Saatcioglu and Razvi[30](S&R), Whitney Stress Block [9] (WSB) and experimental stress–strain relation-ships obtained from the cylinder specimens of the columns by the authors (EXP), were used for the concrete compression zone in order to compute the theoretical ultimate strength capacity and to compare with the experimental results of the column specimens. Experimen-tal results and obtained theoretical capacities according to

each stress–strain model are presented in Table 3,

comparative ratios of the theoretical load to the experi-mental test load for the specimens are given inTable 4.

A good degree of accuracy has been obtained between the theoretical results according to each concrete stress– strain models and the experimental results. The maximum permissible strain assumed in the K&PðcÞ model for the concrete compression zone is affected by confinement provided by lateral ties. Most of the calculated loads by using this model are higher than the test results and the results of the other models. The mean ratios of the comparative results indicate that the shape of the concrete stress–strain relationship has little effect on the ultimate strength capacity of the column members as seen in Table 4. On the other hand, the maximum permissible strain plays the most important role on the ultimate strength capacity.

4.2. Computer analysis of composite columns tested by Munoz and Hsu

Munoz and Hsu [21] presented one short (MC1) and

three slender composite columns (MC2–MC4) with pinned

conditions at both ends. The cross section of 63:5  63:5 mm2 had four reinforcing steel bars at each corner with a total area of 126:68 mm2. An I-shaped structural steel of 25.4 mm in width, 25.4 mm in depth and 2.39 mm flange and web thickness was situated at the centre of the

composite column section as shown in Fig. 11. The

columns were subjected to biaxial bending and axial load. The stress and the strain values for a selected number of points described the piecewise linear stress–strain relation-ship for the steel bars and the structural steel materials[20]. The overall length, the compressive strength of concrete for each specimen and the eccentricities of the applied loads are presented inTable 5.

These composite columns are solved by the developed computer program for the ultimate strength analysis using various stress distribution models (HOG, CEC, K&PðuÞ, WSB) in the concrete compression zone of the cross section. The obtained theoretical results for the maximum load capacity as well as the test results are presented in

Table 6for comparison.

Table 3

Ultimate strength capacities of reinforced concrete columns Column no. Ntest (kN) (A) Nu(theoretical) HOG (B) CEC (C) K&PðuÞ (D) K&PðcÞ (E) S&R (F) WSB (G) EXP (H) C1 89 90.45 88.95 89.75 104.44 88.18 79.66 — C2 121 127.98 126.78 118.76 130.29 110.15 114.04 119.85 C3 125 117.83 116.51 111.45 127.00 105.63 104.71 109.45 C4 99 95.21 93.57 91.25 107.14 87.76 83.67 94.46 C5 94 90.47 88.83 87.70 100.42 83.78 79.55 89.46 C11 104 90.53 88.81 86.86 92.54 82.38 80.71 83.45 C12 95 99.24 91.39 93.74 98.96 87.63 82.38 88.31 C13 98 91.51 90.00 87.58 94.73 83.22 81.81 83.23 C14 58 63.46 63.02 61.38 67.88 59.51 60.57 64.10 C21 238 236.45 233.41 219.86 224.17 197.44 211.89 205.46 C22 199 208.82 208.83 188.86 205.71 183.00 187.88 204.78 C23 192 189.46 189.38 176.12 188.41 167.15 176.35 175.26 LC1 196 187.47 179.96 190.02 256.35 — 170.68 194.54 LC2 182 160.30 153.38 163.21 215.52 — 153.23 158.53 LC3 178 166.28 158.53 169.61 215.40 — 149.91 164.47 Table 4

Comparative results for the reinforced concrete column specimens Column no. Ratio

B/A C/A D/A E/A F/A G/A H/A

C1 1.016 0.999 1.008 1.173 0.991 0.895 — C2 1.058 1.048 0.981 1.077 0.910 0.942 0.990 C3 0.943 0.932 0.892 1.016 0.845 0.838 0.876 C4 0.962 0.945 0.922 1.082 0.886 0.845 0.954 C5 0.962 0.945 0.933 1.068 0.891 0.846 0.952 C11 0.870 0.854 0.835 0.890 0.792 0.776 0.802 C12 1.045 0.962 0.987 1.041 0.922 0.867 0.930 C13 0.934 0.918 0.894 0.966 0.849 0.835 0.849 C14 1.094 1.086 1.058 1.170 1.026 1.044 1.105 C21 0.993 0.981 0.924 0.942 0.830 0.890 0.863 C22 1.049 1.049 0.949 1.033 0.920 0.944 1.029 C23 0.987 0.986 0.917 0.981 0.871 0.918 0.913 LC1 0.956 0.918 0.969 1.308 — 0.871 0.992 LC2 0.881 0.843 0.896 1.184 — 0.842 0.871 LC3 0.934 0.891 0.953 1.210 — 0.842 0.924 Mean ratio 0.979 0.957 0.941 1.076 0.894 0.880 0.932 14.3 14.3 63.5 mm 63.5 mm 14.3 14.3 19.05 19.05 25.4 19.05 19.05 25.4

(11)

As can be seen from Table 6 the results obtained by using WSB model are below the test results, since maximum permissible strain ðcuÞ in concrete plays an

important role to achieve the ultimate strength capacity of the composite column section. Comparative results by using the other stress–strain models show an excellent degree of accuracy with the test results reported by Munoz and Hsu[21].

4.3. Computer analysis of composite columns tested by Morino et al.

Morino et al. [15] tested pin ended short and slender composite columns under biaxial bending and axial load. The dimensions of the composite column specimen as well as the steel arrangements are shown inFig. 12. The overall length and the compressive strength of the composite column specimens named A4, B4, C4 are 960, 2400, 3600 mm and 21.10, 23.37, 23.30 MPa, respectively. The rectangular shaped lateral ties of 4 mm in diameter are spaced at 150 mm. The yield strength of the reinforced and the structural steel are 413.70 and 344.75 MPa, respec-tively, the modulus of elasticity of the steel is taken as 200 kN=mm2. The authors solved the column specimens

with the developed computer program using various stress distribution models, including confinement effect for the concrete compression zone of the section, to demonstrate the accuracy and validity of the proposed procedure. Experimental and theoretical results are pre-sented inTable 7.

The ultimate strength results of the composite columns compare well with the test results reported by Morino[15]. Most of the computed loads obtained by using the K&PðcÞ

model are higher than the test results and the other

theoretical results, since the maximum permissible concrete strain and the concrete confinement provided by lateral ties play an important role in enhancing the ultimate strength capacity and ductility of short and slender composite column members.

5. Conclusions

An experimental investigation of the behaviour of reinforced concrete columns and an iterative numerical procedure for the strength analysis and design of short and slender reinforced concrete and concrete-encased compo-site columns of arbitrarily shaped cross section under biaxial bending and axial load by using various stress– strain models, including confinement have been presented in this paper. The computational procedure takes into account the nonlinear behaviour of the materials (i.e., concrete, structural steel and reinforcing bars) and includes the second order effects due to the additional eccentricity

Table 5 Specimen details of MC1–MC4 Column L (mm) fc(MPa) ex(mm) ey(mm) MC1 812.8 36.77 38.10 38.10 MC2 1219.2 30.97 31.75 31.75 MC3 1219.2 25.83 25.40 25.40 MC4 1219.2 27.51 38.10 38.10 Table 6

Comparative results for the composite column specimens MC1–MC4 Column Ntest (kN) (A) Nu(theoretical) (kN) Ratio HOG (B) CEC (C) K&PðuÞ (D) WSB (E)

B/A C/A D/A E/A MC1 28.17 29.01 28.19 27.65 25.29 1.029 1.001 0.981 0.897 MC2 26.48 27.81 27.02 27.15 24.41 1.050 1.020 1.025 0.922 MC3 29.06 30.17 29.31 29.92 26.55 1.038 1.008 1.029 0.913 MC4 22.03 22.19 21.54 21.88 19.43 1.007 0.978 0.993 0.882 Mean ratio 1.031 1.002 1.007 0.903 160 mm 160 mm 6 mm 4 mm H-100×100×6×8 mm 30 30 100 30 30 100 19 19 19 19

Fig. 12. Composite column cross section. Table 7

Load carrying capacities and test results of composite columns Column ex (mm) ey (mm) Ntest (kN) Nu(theoretical) (kN)

HOG CEC K&PðcÞ S&R

A4-00 40 0 499.91 488.37 497.82 492.69 473.82 A4-30 34.64 20 513.44 488.37 482.16 511.88 485.92 A4-45 28.28 28.28 518.83 507.22 500.10 541.82 511.98 A4-60 20 34.64 524.25 544.40 537.35 581.47 552.39 A4-90 0 40 740.44 635.28 645.67 584.12 479.29 B4-00 40 0 371.04 396.32 383.38 409.22 371.57 B4-30 34.64 20 392.62 410.89 406.42 422.39 405.46 B4-45 28.28 28.28 389.64 428.81 423.07 447.12 428.58 B4-60 20 34.64 436.41 461.39 455.33 482.60 464.06 B4-90 0 40 503.56 559.18 564.19 513.29 491.49 C4-00 40 0 274.61 313.95 311.97 293.34 311.03 C4-30 34.64 20 283.55 308.83 306.48 314.64 306.54 C4-45 28.28 28.28 304.47 322.10 319.04 331.83 323.38 C4-60 20 34.64 340.29 346.69 343.07 357.39 350.42 C4-90 0 40 411.94 435.75 429.75 416.51 410.14

(12)

of the applied axial load by the Moment Magnification Method.

The capability and the reliability of the proposed procedure and its computer program have been tested by means of comparisons with the experimental results reported by the authors of this study and other researchers. The theoretical and experimental results show that the compressive strength of concrete and its corresponding compressive strain are the most effective parameters on the ultimate strength capacity of column members. However, the shape of the concrete stress–strain relationship has little effect on the ultimate strength capacity of the column members.

Consequently, the ratio of computed axial load to experimental load for different cross section of reinforced and composite column members indicate that the results obtained by the proposed procedure have been in good agreement with the test results of columns subjected to biaxial bending and axial load.

References

[1] Furlong RW. Concrete columns under biaxially eccentric thrust. ACI Journal October 1979:1093–118.

[2] Brondum-Nielsen T. Ultimate flexural capacity of fully prestressed, partially prestressed, arbitrary concrete sections under symmetric bending. ACI Journal 1986;83:29–35.

[3] Hsu CTT. Biaxially loaded L-shaped reinforced concrete columns. Journal of Structural Engineering ASCE 1985;111(12):2576–95. [4] Hsu CTT. Channel-shaped reinforced concrete compression members

under biaxial bending. ACI Structural Journal 1987;84:201–11. [5] Dundar C. Concrete box sections under biaxial bending and axial

load. Journal of Structural Engineering 1990;116:860–5.

[6] Rangan BV. Strength of reinforced concrete slender columns. ACI Structural Journal 1990;87(1):32–8.

[7] Building code requirements for structural concrete (ACI 318-99). Detroit (MI): American Concrete Institute (ACI); 1999.

[8] Dundar C, Sahin B. Arbitrarily shaped reinforced concrete members subjected to biaxial bending and axial load. Computers and Structures 1993;49:643–62.

[9] Whitney CS. Plastic theory of reinforced concrete design. Transac-tions, ASCE 1940;107:251–60.

[10] Rodriguez JA, Aristizabal-Ochoa JD. Biaxial interaction diagrams for short RC columns of any cross section. Journal of Structural Engineering 1999;125(6):672–83.

[11] Fafitis A. Interaction surfaces of reinforced-concrete sections in biaxial bending. Journal of Structural Engineering 2001;127(7):840–6. [12] Hong HP. Strength of slender reinforced concrete columns under biaxial bending. Journal of Structural Engineering 2001;127(7): 758–62.

[13] Saatcioglu M, Razvi SR. High-strength concrete columns with square sections under concentric compression. Journal of Structural Engineering 1998;124(12):1438–47.

[14] Furlong RW, Hsu CTT, Mirza SA. Analysis and design of concrete columns for biaxial bending-overview. ACI Structural Journal 2004;101(3):413–23.

[15] Morino S, Matsui C, Watanabe H. Strength of biaxially loaded SRC columns. In: Proceedings of the US/Japan joint seminar on composite and mixed construction. New York, NY: ASCE; 1984. p. 185–94.

[16] Roik K, Bergmann R. Design method for composite columns with unsymmetrical cross-sections. Journal of Constructional Steel Research 1990;15:153–68.

[17] Virdi KS, Dowling PJ. The ultimate strength of composite columns in biaxial bending. In: Proceedings of the institution of civil engineers, Part 2; 1973. p. 251–72.

[18] Mirza SA. Parametric study of composite column strength varia-bility. Journal of Constructional Steel Research 1989;14:121–37. [19] Mirza SA, Skrabek W. Statistical analysis of slender composite

beam-column strength. Journal of Structural Engineering 1992;118(1): 1312–31.

[20] Munoz PR, Behavior of biaxially loaded concrete-encased composite columns. PhD thesis, New Jersey Institute of Technology; 1994. [21] Munoz PR, Hsu CT. Behavior of biaxially loaded concrete-encased

composite columns. Journal of Structural Engineering 1997;123(9): 1163–71.

[22] Munoz PR, Hsu CT. Biaxially loaded concrete-encased composite columns: design equation. Journal of Structural Engineering 1997; 123(12):1576–85.

[23] Weng CC, Yen SI. Comparisons of concrete-encased composite column strength provisions of ACI code and AISC specifications. Engineering Structures 2002;24:59–72.

[24] Load and resistance factor design specification for structural steel buildings. 2nd ed. Chicago, IL: American Institute of Steel Construction (AISC); 1993.

[25] Lachance L. Ultimate strength of biaxially loaded composite sections. Journal of Structural Division, ASCE 1982;108:2313–29.

[26] Chen SF, Teng JG, Chan SL. Design of biaxially loaded short composite columns of arbitrary section. Journal of Structural Engineering 2001;127(6):678–85.

[27] Sfakianakis MG. Biaxial bending with axial force of reinforced, composite and repaired concrete sections of arbitrary shape by fiber model and computer graphics. Advances in Engineering Software 2002;33:227–42.

[28] Kent DC, Park R. Flexural members with confined concrete. Journal of Structural Division ASCE 1971;97(7):1969–90.

[29] Sheikh SA, Uzumeri SM. Analytical model for concrete confinement in tied columns. Journal of Structural Division ASCE 1982;108(12): 2703–22.

[30] Saatcioglu M, Razvi SR. Strength and ductility of confined concrete. Journal of Structural Engineering 1992;118(6):1590–607.

[31] Chung HS, Yang KH, Lee YH, Eun HC. Stress–strain curve of laterally confined concrete. Engineering Structures 2002;24: 1153–63.

[32] Hognestad E, Hanson NW, McHenry D. Concrete stress distribution in ultimate stress design. ACI Journal 1955;27(4):455–79.

[33] Commission of the European Communities (CEC). Design of composite steel and concrete structures. Brussels: Eurocode 4; 1984. [34] Building code requirements for reinforced concrete (TS 500). Ankara:

References

Related documents

We considered that all operators performed the task intuitively, like conventional laparoscopic surgery, without a sense of discomfort due to the relationship between the

Results: Top-ranked barriers included the lack of bridge funding accessible to UK CTUs to support the design of ADs, limited practical implementation knowledge, preference

Source: Consumer Spending data are derived from the 2016 and 2017 Consumer Expenditure Surveys, Bureau of Labor

different aspects of Parkland such as Study Abroad, Athletics, Student Life, Archives, Student Club Presidents and the Dean of Students... Surveys at

The following research questions guided the study: (i) to expand understanding of leadership by exploring leaders’ media data (i.e., interviews, speeches and team announcements,

Reactive disperse dyes containing monochlorotriazine/sulphatoethylsulphone reactive moieties were synthesized by condensing aniline with cyanuric chloride and further

Care of the patient with naso-gastric tube - check for placement and patency, maintain suction, check bowel sounds, perform nasal care, and

Our study in the short rainy season showed that cabbage production in Adola areas and the cultivation of Royal and Monarch varieties evaluated during the short rainy season