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* Accepted manuscript by Journal of Structural Engineering, ASCE 1
http://dx.doi.org/10.1061/(ASCE)ST.1943-541X.0001897 2
Simplified Numerical Modeling of Axially Loaded Circular 3
Concrete-Filled Steel Stub Columns 4
5
Utsab Katwal1; Zhong Tao, M.ASCE2; Md Kamrul Hassan, A.M.ASCE3; and 6
Wen-Da Wang4 7
8
Abstract: Behavior of concrete-filled steel tubular (CFST) columns can be predicted accurately 9
using detailed finite element (FE) modeling, but such models are tedious to build and impractical 10
for frame analysis. In contrast, computationally efficient fiber beam element (FBE) models can 11
achieve the balance between accuracy and simplicity, and can be utilized for advanced analysis of 12
structural systems. In FBE models, however, the material models themselves have to account for the 13
interaction between the steel tube and core concrete. Therefore, the accuracy of a FBE model 14
depends mainly on the input material models. Although there are a few FBE models available in the 15
literature for CFST columns, these models may not be suitable for some cases, especially when 16
considering the rapid development and application of high strength materials and/or thin-walled 17
steel tubes. In this paper, versatile, computationally simple yet accurate steel and concrete models 18
are proposed based on detailed FE modeling results of circular CFST stub columns under axial 19
compression. The material models are then implemented in FBE modeling in ABAQUS, and the 20
prediction accuracy is verified by a wide range of test data. 21
Keywords: Concrete-filled steel tubes; Confined concrete; Finite element modeling; Fiber beam 22
model; Simplified simulation. 23
1PhD Candidate, Centre for Infrastructure Engineering, Western Sydney University, Penrith, NSW 2751, Australia. 24
E-mail: u.katwal@westernsydney.edu.au 25
2Professor, Centre for Infrastructure Engineering, Western Sydney University, Penrith, NSW 2751, Australia 26
(corresponding author). E-mail: z.tao@westernsydney.edu.au 27
3Postdoctoral Research Fellow, Centre for Infrastructure Engineering, Western Sydney University, Penrith, NSW 28
2751, Australia. E-mail: k.hassan@westernsydney.edu.au 29
4Professor, School of Civil Engineering, Lanzhou University of Technology, Lanzhou 730050, China. E-mail: 30
wangwd@lut.cn 31
2
Introduction 32
Steel-concrete composite structures consisting of concrete-filled steel tubular (CFST) columns have 33
been widely used in modern construction because they offer many structural as well as economic 34
benefits (Han et al. 2014a). Detailed three-dimensional (3D) finite element (FE) models can be 35
developed to precisely predict the behavior of composite structures, but such models are tedious to 36
build and impractical for the analysis of large structural systems or for routine design. This is 37
mainly due to the complexity in modeling, convergence issues and long computational time. In this 38
context, to achieve the balance between efficiency and accuracy in simulating CFST columns, fiber 39
beam element (FBE) models can be utilized because of their simplicity in simulation and high 40
computational efficiency. The FBE models are suitable for use in advanced analysis of composite 41
frames. However, the main challenge is in developing proper material models which themselves 42
have to account for the interaction between the steel tube and core concrete. 43
There are a few steel and concrete stress ()strain models available in the literature 44
developed for FBE modeling of circular CFST columns. The material models proposed by Sushanta 45
et al. (2001), Sakino et al. (2004), Han et al. (2005), Hatzigeorgiou (2008a,b), Liang (2008), Liang 46
and Fragomeni (2009) and Denavit and Hajjar (2012) are empirical and primarily based on 47
experimental data. The difficulty in utilizing experimental data to develop uniaxial material models 48
is that the contributions from the steel or concrete are generally not directly measured and 49
assumptions are required to extract individual responses (Denavit and Hajjar 2012). The normal 50
practice is to assume an elastic-plastic response with or without strain-hardening for steel. Then 51 curve of the concrete is derived from the experimental data by deducting the contribution 52
from the steel. After that, empirical concrete model can be developed based on regression analysis. 53
Although empirical models may give reasonable predictions, they cannot reflect the actual 54
interaction between the steel tube and core concrete since the effects of local buckling and concrete 55
confinement have not been properly considered in the steel model. Furthermore, the accuracy of the 56
empirical models depends on the quality of input information, and the validity is restricted to the 57
3
test data range for optimizing the model parameters. 58
A more scientific way to develop models for FBE analysis is based on 3D FE modeling, 59
provided that the detailed model has been rigorously validated (Shams and Saadeghvaziri 1999; Varma et 60
al. 2005). Lai and Varma (2016) recently conducted 3D FE analysis of circular CFST columns, and they 61
found that the axial stressstrain curve of steel has an initial ascending branch followed by a descending 62
branch. The post-peak response of the steel is mainly due to the tensile hoop stresses developing in the 63
steel tube to confine the concrete infill as it reaches its compressive peak stress. The confined concrete, 64
however, may demonstrate strain-softening or strain-hardening behavior depending on the confinement 65
level. For simplicity, however, Lai and Varma (2016) only proposed idealized elastic-perfectly plastic 66
models for both the steel and concrete. Ideally, steel and concrete models should be proposed to represent 67
the actual material responses. 68
In recent years, high strength steel and concrete materials are increasingly used in structures. 69
For example, steel yield stress of 590 MPa and concrete unconfined strength ( of 150 MPa 70
have been used in CFST columns in Abeno Harukas, Japan (Liew et al. 2014). Xiong et al. (2017) 71
presented test results of CFST columns filled with concrete of close to 200 MPa. With the 72
advancements in high strength materials, thin-walled tubes are more likely to be used in composite 73
columns. To embrace the development of materials, there is a strong need to develop versatile, 74
computationally simple yet accurate steel and concrete models for the analysis of CFST columns. 75
A 3D FE model developed by Tao et al. (2013b) will be utilized in this paper to generate 76
uniaxial effective stressstrain curves for the steel tube and core concrete to cover a wide range of 77
parameters. Based on the numerical data, regression analysis will be conducted to propose 78
comprehensive stressstrain models of steel and concrete for FBE modeling. 79
Numerical Modeling 80
3D Finite Element Model 81
In this paper, the effective stress ()strain relationships of steel and concrete in circular CFST 82
columns will be developed based on the 3D FE model developed by Tao et al. (2013b), which has 83
4
been verified by a total of 142 full-range loaddeformation curves of circular CFST stub columns 84
collected from the literature. A typical 3D FE model built in ABAQUS is shown in Fig. 1(a), where 85
the steel tube and concrete were simulated by four-node shell elements with reduced integration 86
(S4R) and 8-node brick elements with 3 translational degrees of freedom at each node (C3D8R), 87
respectively. The mesh size of each discretized element was taken as D/15, where D is the overall 88
diameter of the circular tube. In general, the detailed model contains over 4000 elements for a 89
typical stub column. Since the nonlinear analysis could be conducted within a reasonable 90
computational time, no efforts were made to improve the computational efficiency using a half or 91
quarter symmetry model. Surface-to-surface contact option available in ABAQUS was used to 92
model the interaction between the steel tube and concrete. It should be noted that the steel model 93
used by Tao et al. (2013b) is only valid when the yield stress is 800 MPa or less. This paper 94
aims to extend the validity range of to 960 MPa. Based on the coupon test results reported by 95
Shi et al. (2012), Qiang et al. (2013) and Shi et al. (2015), the ultimate strength is taken as 96
1.05 for steel with a between 800 MPa and 960 MPa. More details about the FE modeling can 97
be found in Tao et al. (2013b). 98
Fiber Beam Element Model 99
The FBE model is an advanced tool in which the member is divided into a number of longitudinal 100
fiber elements as shown in Fig. 1(b) and (c). The geometric characteristics to be defined for a fiber 101
are its area and location with respect to the cross-section (Taucer et al. 1991). In FBE modeling of 102
CFST columns, the interaction between the steel tube and concrete core needs to be specifically 103
considered in the input material models to get accurate results. Due to the computational efficiency, 104
FBE models have been commonly used in advanced analysis of frames, where the design process 105
can be simplified as the system strength can be directly assessed from the analysis without the need 106
for calculating the effective length factor or checking the specification of beam-column interaction 107
equations (Zhang and Rasmussen 2013). In particular, FBE models are very suitable for analyzing 108
structures subjected to extreme events, such as fire, blast, seismic, and other abnormal events. 109
5
Assumptions Used in FBE Modeling 110
In conducting FBE modeling of CFST stub columns, the following assumptions are adopted: 111
(a) A plane section remains plane during deformation; 112
(b) Perfect bond exists between the steel tube and concrete infill; 113
(c) The longitudinal stress of any fiber is only decided by the strain at that point; 114
(d) The effects of concrete creep and shrinkage are not considered; and 115
(e) Steel fracture is also not considered since it typically does not occur until very late in 116
loading histories (Denavit and Hajjar 2012). 117
Therefore, the results of the FBE modelling are only valid before the steel reaches its tensile 118
strain, which is sufficient for the needs of normal analysis. 119
Procedure for FBE Modeling 120
To generate the FBE model in ABAQUS, the steel tube and concrete core were sub-divided into a 121
finite number of longitudinal fibers, as shown in Fig 1(c). The concrete was simulated using a 122
2-node linear beam element (B31), which is a first-order, three-dimensional Timoshenko element. 123
By changing the number of the material integration points, the number of concrete fibers could be 124
changed accordingly. For the steel tube, the steel fibers were directly defined as material integration 125
points using *rebar option available in the keywords platform in ABAQUS (Wang et al. 2013). The 126
steel and concrete material models were defined through a UMAT subroutine which was developed 127
using the platform of Intel FORTRAN and implemented in ABAQUS through Microsoft Visual 128
Studio. 129
Sensitivity analysis was conducted to find out the influence of section discretization on the 130
simulation accuracy. For CFST stub columns under axial compression, it is found that the effect of 131
the mesh size and section discretization has no obvious influence on the predicted axial load 132 strain curves. Identical specimens H-58-1 and H-58-2 and specimen CU-070 tested by 133
Sakino and Hayashi (1991) and Huang et al. (2002) are taken as examples as shown in Fig. 2(a) and 134
(b), respectively. Almost the same predictions were obtained using different mesh sizes and different 135
6
numbers of fiber elements. This is understandable since the whole stub column is under axial 136
compression. Similar behavior was observed by Patel et al. (2014) for axially loaded concrete-filled 137
stainless steel short columns. However, section discretization is a prerequisite for fiber element 138
modeling (Patel et al. 2014). In this study, the steel tube was divided into 16 longitudinal fiber 139
elements, and 17 longitudinal material integration points were specified for the core concrete. 140
Meanwhile, the CFST column was divided into 10 elements along its length. In specifying the 141
boundary conditions, only vertical displacement at the top end was allowed whereas all other 142
translational degrees of freedom were restrained for both ends of the CFST column. 143
Development of Material Models for FBE Modeling 144
For a CFST column under axial compression, interaction can be developed between the steel tube 145
and concrete. Thus, the concrete can have increased compressive strength and ductility due to the 146
confinement of the steel tube. Meanwhile, tensile hoop stresses will be developed in the steel tube, 147
which reduces its load-carrying capacity in the axial direction (Lai and Varma 2016). During the 148
loading process, the confining stresses change with increasing axial deformation. Furthermore, local 149
buckling of the steel tube might occur during the process, which affects the interaction between the 150
steel tube and concrete. The combined influence of all these factors is very complex and should be 151
properly considered when proposing material models. 152
To develop simplified material models for the steel tube and concrete in CFST columns, the 153
detailed FE model developed by Tao et al. (2013b) was used to analyze 210 circular CFST stub 154
columns using various parameter combinations. Six yield stress levels were chosen for the steel ( 155
= 186, 300, 500, 650, 800, 960 MPa), whereas five cylinder compressive strength levels were 156
chosen for the concrete ( = 20, 50, 100, 150, 200 MPa). The diameter-to-thickness ratio varied at 157
seven levels (/ = 10, 33, 52, 75, 100, 150, 220, where t is the thickness of the steel tube). In the 158
analysis, the length-to-diameter ratio (/) was kept constant at 3, so the columns can be classified 159
as stub columns (Tao et al. 2013b). From the simulation, axial stresses of all elements are extracted 160
from the middle section of the CFST column, respectively. Then the stresses of all steel elements 161
7
are “averaged” to obtain the effective stress for the steel as a function of the axial strain . A 162
similar procedure is adopted to obtain the effective uniaxial relationship for the concrete. 163
Since the averaged curves have already incorporated the influence of the interaction 164
between the steel tube and concrete, these curves can be directly used in FBE modeling. 165
Using the numerical data generated from the 3D FE modeling, regression analysis is conducted in 166
the following subsections to propose suitable steel and concrete material models to represent the 167
effective uniaxial relationships. 168
Steel Material Model 169
Characteristics of the Stress-Strain Curves for Steel 170
In 3D FE modeling, normally only a single
relationship is required as input for steel, and Tao 171
et al. (2013b) used an elastic-plastic model with strain-hardening which is also adopted in the present 172
study. Typical CFST stub columns with different confinement factors / were 173
analyzed using the 3D FE modeling, and the obtained effective
curves of steel are compared in 174
Fig. 3(a). Obviously, these effective
curves are quite different from the input
curve. 175
This is owing to the development of hoop stresses in the steel tube and possible local buckling of the 176
steel tube. This observation highlights the need to develop a proper effective
model for steel in 177
FBE modeling. 178
In general, the effective
curves of different columns coincide with each other very well in 179
the elastic stage. This can be explained by the weak interaction between the steel tube and concrete in 180
the beginning (Han et al. 2014a; Chacon 2015). But after reaching the peak stress, the curves differ 181
significantly from each other, as shown in Fig. 3(a). This is due to the concrete dilation, which 182
strengthens the interaction between the steel and concrete components. The increasing interaction 183
leads to the fast increase of the hoop stress and decrease of the axial stress in the steel tube. More 184
discussion of this phenomenon is given by Liew and Xiong (2012). As can also be seen from Fig. 3(a), 185
the descending speed of a column with a smaller is faster than that of the column with a larger . 186
The former also has lower residual strength. This is because the concrete dilates faster when the 187
8
confinement is less significant for the column with a smaller . After reaching a critical point 188
[
,
), where
and
are the critical strain and stress respectively], the axial stress 189
increases again because of the strain hardening effect of steel considered in the input model. 190
There are a few simplified steel models available in the literature for the fiber modeling 191
of the steel tubes in CFST columns. Elastic-perfectly plastic models with yield stress 192
reduction factors of 0.89 and 0.9 were proposed by Sakino et al. (2004) and Lai and Varma (2016), 193
respectively. An idealized linear-rounded-linear model with strain hardening was proposed 194
by Liang and Fragomeni (2009) for normal steel, and the rounded part of the curve was replaced 195
with a straight line for high strength steel. Only Denavit and Hajjar (2012) presented a steel model 196
with a softening branch after yielding for circular CFST columns. But a constant slope has been 197
adopted for the descending branch. There is no existing model that can capture all the 198
characteristics of the effective stress-strain curves presented in Fig. 3(a) for steel. Therefore, a new 199
model is proposed in the following subsection to fill this research gap. 200
Proposed Steel Stress-Strain Relationship 201
The steel
model used by Tao et al. (2013b) in 3D FE modeling was originally proposed by 202
Tao et al. (2013a) based on statistical analysis of a wide range of
curves of steel. Since that 203
model cannot be directly used in FBE modeling as discussed in the last subsection, suitable 204
modifications should be made to capture the interaction between the steel tube and core concrete. In 205
the present study, the model proposed by Tao et al. (2013a) is revised and expressed by Eq. (1) for 206
FBE modeling. 207
0
∙
∙
(1)
208
in whichis the first peak stress of steel in the CFST column; / is the strain 209
corresponding to ; is the Young’s modulus of steel, which can be taken as 200 GPa if the value 210
9
was not reported in a test; and are the strain softening and hardening exponents, respectively; 211
and is the effective stress of steel corresponding to the ultimate strain . Eq. (2) was 212
proposed by Tao et al. (2013a) to determine of steel. It should be noted that the upper limit for 213 is 800 MPa in the original equation. A linear equation as presented in Eq. (2) is proposed to 214
determine when is between 800 and 960 MPa based on ten coupon test results of S960 steel 215
reported by Shi et al. (2012), Qiang et al. (2013) and Shi et al. (2015). 216
100300MPa
1000.15300300800MPa
250.1800800960MPa (2)
217
where is the yield strain of steel, taking as /. A schematic view of the simplified 218
curves with high, medium and low -values is shown in Fig. 4. As can be seen, six parameters 219
(,
,
,,,and ) are required to define the
relationship of steel. Regression analysis 220
was conducted to derive equations for these parameters using the numerical data generated from the 221
3D FE modeling. 222
First Peak Stress
223
The ratio of /is an indication of the initial intensity of the interaction between the steel tube 224
and concrete. The stronger the interaction, the higher the hoop stresses developed in the steel tube 225
and the lower the /ratio. Based on parametric analysis, it is found that the ratio of /is 226
mainly affected by / and / ratio, where is the strain at peak stress of the 227
corresponding unconfined concrete. can be determined by Eq. (3) proposed by De Nicolo et al. 228
(1994): 229 0.000760.6264.3310 (3) 230
where c
fis expressed in MPa. 231
The ratio of /decreases with increasing /ratio. This is due to the fact that a smaller 232 /ratio represents a relatively slower initiation of the concrete dilation, leading to a weaker 233
10
initial interaction. Meanwhile, /decreases with an increase in/ ratio. When / 234
decreases, the concrete is under increased confinement. However, the ratio of the hoop tensile stress 235
to the yield stress of the steel tube decreases, leading to increased /ratio. Based on regression 236
analysis, Eq. (4) is proposed to determine /, andthe prediction accuracy is demonstrated in Fig. 237
5. The coefficient of determination is 0.81, indicating a reasonably good fitting. 238
1.020.01∙
.
. 1 (4) 239
Critical Stress
and Critical Strain
240
By analyzing the numerical data obtained from FE modeling, it is found that the ratio of the critical 241
stress
to the yield stress is mainly determined by . When increases to about 0.6, 242
/ increases almost linearly to 0.6. After that,
/ increases slowly with increasing . Eq. 243
(5) is developed to determine
, and the predictions from this equation are compared with the data 244
obtained from FE modeling in Fig. 6. The value of is found to be 0.99 for the proposed 245
equation, which indicates an excellent correlation between the predictions and the numerical data. 246
∙...
0and (5) 247
The critical strain
is also mainly dependent on . As shown in Fig. 3,
increases with 248
increasing. When increases, the confinement to concrete is stronger, leading to a slower 249
concrete dilation. Thus, the strain-hardening of the steel is delayed. When is smaller than 0.5, 250
increases almost linearly with the increase of . After that, the increase in
becomes slower. 251
Regression analysis indicates that
may be expressed as a function of only. However, if other 252
terms, such as , and / are introduced as additional terms, a better model can be produced 253
for
as shown in Fig. 7, where the value of is 0.96. Eq. (6) is proposed to predict
: 254
280.07
.0.13.
∙..and (6)255
Stress
256
Steel under uniaxial tension can reach its tensile strength corresponding to the ultimate strain 257
11
. However, for the steel tube of the CFST column, the obtained effective stress at is 258
smaller than because the steel tube has to resist the additional hoop stress in the lateral direction. 259
It is found that factors affecting
also have similar influence on . Therefore, similar to Eq. (6), 260
Eq. (7) is proposed to determine , which has very good agreement (=0.92) with the data 261
obtained from FE modeling as shown in Fig. 8. 262
6.80.013.
.1.3.
∙..
(7) 263
Strain Softening Exponent and Strain Hardening Exponent 264
Strain softening exponent is calibrated from the FE modeling results. It is found that a constant 265
value of 1.5 can be reasonably used to represent , as shown in Fig. 9. Although there is some 266
variation found, this constant value of 1.5 suggested for is acceptable since it only slightly 267
affects the softening branch of the curve. 268
The strain hardening exponent is proposed as shown in Eq. (8), which is modified from an 269
equation originally proposed by Tao et al. (2013a). The modification is made by simply replacing 270
the relevant parameters with
, and
respectively. 271
(8) 272
in which is the initial modulus of elasticity at the onset of strain-hardening, and can be taken as 273 0.02. 274
The proposed steel model can accurately predict the effective
curve of steel obtained from 275
3D FE modeling, as shown in Fig. 10(a), where the
model input into ABAQUS is designated 276
as “3D FE input” and the obtained effective
curve is shown as “3D FE output”. In this 277
example, the specimen 4LN tested by Tomii et al. (1977) has been used. 278
Concrete Material Model 279
Characteristics of the Stress-Strain Curves for Concrete 280
It is well-documented in the literature that the confinement provided by the steel tube can increase 281
12
the concrete strength and ductility (Han et al. 2014b; Liew and Xiong 2012). However, the concrete 282
confinement is of passive nature and very difficult to quantify. The confinement factor is a 283
comprehensive parameter, which can reasonably reflect the intensity of the concrete confinement 284
(Han et al. 2014b). Based on 3D FE modeling, Fig. 3(b) depicts the effective curves of 285
concrete for CFST columns with different -values. When the confinement is strong, there is a 286
significant improvement in strength and ductility, and no softening branch is available. On the other 287
hand, the improvement in concrete strength and ductility is relatively limited if the confinement is 288
weak. 289
Lai and Varma (2016) proposed an elastic-perfectly plastic
model for concrete of 290
circular CFST columns. However, it cannot be used for weakly-confined concrete with a 291
strain-softening branch. In contrast, other empirical concrete models proposed by Susantha et al. 292
(2001), Sakino et al. (2004), and Liang and Fragomeni (2009) normally have a strain-softening 293
response after reaching its peak stress. Since their steel models did not properly consider the 294
strength reduction resulting from the interaction between the steel tube and concrete, the strength 295
reduction of steel has to be incorporated into the concrete models. Thus, these empirical concrete 296
models cannot reflect the actual concrete response shown in Fig. 3(b). Meanwhile, these 297
existing empirical models are normally only validated by test results of normal CFST columns. 298
With the development of high-strength steel and concrete, there is a need to develop a more 299
versatile concrete model to cover a wider range of parameters. 300
Proposed Concrete Stress-Strain Relationship 301
Samani and Attard (2012) proposed a
model for confined concrete, which had been verified 302
by extensive test results. A single expression was used in that model to represent both the ascending 303
and descending branches. The model proposed by Samani and Attard (2012) is revised in the 304
present study to represent the effective curve of concrete confined by the steel tube, which is 305
expressed by Eq. (9): 306
13
∙∙
∙∙∙
1or1andσ
1andσ (9) 307
where/
;
and
are the confined concrete strength and the corresponding strain; 308
is the residual stress of concrete, as shown in Fig. 11; and A and B are coefficients to determine the 309
shape of the curve. 310
Fig. 11 shows the effective curves for weakly-confined concrete and strongly-confined 311
concrete. To define the full-range curves, five parameters including
,
, , , and , are 312
required. Based on the numerical data generated from the 3D FE modelling, regression analysis was 313
conducted to derive suitable equations for these parameters as follows. 314
Confined Concrete Strength
and Corresponding Ultimate Strain
315
The parameter
directly reflects the concrete strength increase due to the confinement effect. 316
Parametric analysis indicates that
depends mainly on . The ratio of
/increases with 317
increasing . To further improve the prediction accuracy, other terms including , and / 318
ratio are introduced into Eq. (10) to determine
. As shown in Fig. 12, excellent prediction 319
accuracy (=0.97) has been obtained between
calculated from the proposed equation and that320
obtained from FE modeling. 321
10.2∙
.0.90.25∙
.∙1and3 (10)322
in which is in MPa. 323
The strain
corresponding to
partially reflects the deformation capacity and ductility of 324
a CFST column.Based on the same procedure of regression analysis, Wang et al. (2017) proposed 325
Eq. (11) to predict
and this equation is directly adopted in this paper. It should be noted that the 326
maximum value of
is limited to 0.01 by Wang et al. (2017) for design purposes. This limitation 327
is removed in this study. 328
300010.4∙..0.733785.8
.με (11) 329
14
Residual Concrete Strength 330
To analyze structures with large deformation, it is necessary to define the residual strength for 331
the confined concrete. Parametrical analysis indicates that the ratio of /
is mainly affected by 332 / ,and .It is found that /
decreases with increasing / or, and increases with an 333
increase in .Regression analysis is conducted and Eq. (12) is proposed to predict /
, which is 334
a function of / , , and . The correlation (=0.97) between the proposed equation and the 335
simulation results is very close, as shown in Fig. 13. 336
3.5∙
∙...
.
(12)337
in which is in MPa. 338
Coefficients and 339
The coefficient determines the shape of the ascending part, and Samani and Attard (2012) 340
suggested an equation of
/
for it, where is the modulus of elasticity of unconfined 341
concrete. can be taken as 4700 according to ACI 318 (2011), where is in MPa. 342
However, the proposed equation for by Samani and Attard (2012) is for actively-confined 343
concrete, which cannot be directly used for CFST columns. The concrete inside a CFST column is 344
passively-confined, and the direct use of the coefficient leads to the underestimation of the 345
initial stiffness of the CFST column. Therefore, a correction factor ranging from 1 to 1.3 is 346
introduced into Eq. (13) to determine the coefficient : 347
(13) 348
Parametric analysis indicates that has a strong correlation with . Based on numerical tests, 349
suitable values of are determined for CFST columns with different -values. Then a 350
regression analysis is conducted and Eq. (14) is proposed accordingly to determine as follows: 351 10.25∙../ (14) 352
The coefficient controls the shape of the descending part. The smaller the coefficient , the 353
15
steeper the descending curve. The coefficient B increases with increasing or decreasing. The 354
value of normally ranges from 0.75 to 2. For normal strength concrete with reasonably good 355
confinement, will be positive. However, becomes negative for weakly-confined concrete or 356
high-strength concrete. Based on numerical tests, suitable values of are determined for CFST 357
columns with different combinations of and . Based on regression analysis, Eq. (15) is 358
proposed to determine the coefficient as follows: 359 2.152.05
0.00760.75(15) 360
The proposed concrete model can accurately predict the effective
curve of concrete obtained 361
from 3D FE modeling, as shown in Fig. 10(b). As demonstrated by this example, all characteristics 362
of the effective
curve of concrete have been well captured by the proposed model. 363
Verification of the Fiber Beam Element Modeling 364
The axial load axial strain curves of 150 circular CFST stub columns collected from 22 365
sources are used to develop the proposed FBE model. The majority of the test data was collected by 366
Tao et al. (2013b) for developing the 3D FE model. In addition, some newly reported test data is 367
collected and assembled in the database, as summarized in Table 1. It would be worth noting that 368
the majority of the test data is from extensively cited references. The ranges of parameters for the 369
test specimens are: = 186-853 MPa; = 18-193 MPa; = 60-450 mm; and / =17-221 370
and /=1.8-4.8. As can be observed in Table 1, these parameters cover sufficiently wide practical 371
ranges. 372
The predicted ultimate strengths from the FBE modeling are firstly compared with the 373
measured ultimate strengths . Following the definition in Tao et al. (2013b), the ultimate 374
strength in this paper is defined as the peak load if the curve has softening branch and the 375
strain corresponding to the peak load is less than 0.01; otherwise it is defined as the load at a strain 376
of 0.01. The comparison of / with respect to for all 150 columns is presented in Fig. 377
14(a), where the mean value and standard deviation are found to be 0.985 and 0.066 378
16
respectively. Meanwhile, the ultimate strengths are also predicted using the 3D FE 379
modelling. The comparison of/ with respect to is presented in Fig. 14(b), where the 380
obtained and are 0.992 and 0.064 respectively. As can be seen, comparable results are 381
obtained from the FBE modeling and the detailed modeling in terms of ultimate strength. In general, 382
the predictions from the FBE modeling are slightly more conservative than the 3D FE predictions, 383
but have similar precision. 384
The effects of concrete strength and steel yield stress on the prediction accuracy of ultimate 385
strength are shown in Fig. 15(a) and (b) respectively. In this paper, concrete with less than 60 386
MPa is considered as normal strength concrete (NSC). If is between 60 MPa and 120 MPa, the 387
concrete is referred to as high-strength concrete (HSC). Concrete with higher than 120 MPa is 388
considered as ultra-high strength concrete (UHSC). Similarly, steel with less than 460 MPa is 389
considered as normal strength steel (NSS). Otherwise, it is grouped into high strength steel (HSS). 390
The comparison demonstrated in Fig. 15 indicates that the prediction accuracy of the ultimate strength 391
using the FBE modeling is not obviously affected by or. 392
In predicting the curves of normal CFST columns, the predictions from the FBE 393
modeling also agree with the test results and the 3D FE modeling very well. This can be seen from 394
the comparisons shown in Fig. 10(c) and Fig. 16, where specimens 4LN and 3HN tested by Tomii et 395
al., (1977) and C2 tested by Schneider (1988) are taken as examples. The load carried by the steel 396
tube or concrete can be determined by simply multiplying the effective stress of the steel or concrete 397
by the cross-sectional area of the corresponding component. When the load is mainly carried by the 398
steel tube (C2) or it is almost evenly shared by the steel tube and concrete, the composite column 399
usually does not have a post-peak softening response. In contrast, it usually demonstrates a softening 400
response after the peak load if the concrete carries the majority of the load. 401
The FBE model can also be successfully used to predict curves of CFST columns with 402
HSC. This is demonstrated in Fig. 17 for specimens S12CS80A and C-100-3D tested by O’Shea 403
and Bridge (1998) and de Oliveira et al. (2009), respectively. The predicted curves agree 404
17
very well with the experimental curves and those predicted by the 3D FE modeling. The FBE model 405
is also used to simulate CFST columns with UHSC concrete. Compared with NSC or HSC, UHSC 406
is more brittle under compression and demonstrates an almost linear curve even with 407
confinement from the steel tube. Accordingly, a steep drop in the loadaxial shortening curves was 408
observed for UHSC filled tubes right after the peak load (Liew et al. 2014). This feature has been 409
considered when proposing the concrete model for FBE modeling. Therefore, the steep drop in 410 curves for columns with UHSC has been successfully captured in the FBE modeling, which 411
can be seen in Fig. 18 for the two typical specimens C15 and C14 reported by Xiong et al. (2017). 412
However, the 3D FE model does not accurately capture this behavior. 413
There are only a few loaddeformation curves reported in the literature for circular CFST stub 414
columns with HSS steel. For these columns, very good predictions by the FBE modeling are also 415
obtained as shown in Fig. 19, where specimen 049C36_30 tested by Lee et al. (2011) and specimen 416
CC8-A-8 tested by Sakino et al. (2004) are taken as examples. 417
It should be noted that the proposed FBE model can also be used for concrete-filled thin-walled 418
tubes, or stocky CFST columns with small / ratios. The prediction accuracy can be observed for 419
specimen S12CS80A with a tube thickness 1.13 mm and specimen S2-2-4 with a tube thickness of 420
10 mm as shown in Fig. 17(a) and Fig. 18(b), respectively. 421
It should also be noted that the proposed equations in this paper can be directly utilized to 422
calculate the loaddeformation curves of circular CFST stub columns using simple spreadsheet 423
software. This can help design engineers to conduct preliminary design of CFST columns. 424
Concluding Remarks 425
This paper developed a fiber beam element model for axially loaded circular concrete-filled steel 426
stub columns to achieve a high computational efficiency. Effective steel and concrete stressstrain 427
models were proposed based on detailed finite element modeling. The proposed stressstrain curves 428
for steel has implicitly considered the interaction between the steel tube and concrete, possible local 429
buckling of the steel tube, and strain-hardening of the steel material. Meanwhile, the concrete model 430
18
has considered the increase in strength and ductility resulting from the concrete confinement. 431
The proposed material models were implemented in the simplified fiber beam element 432
modeling, and the predictions were verified by 3D FE modeling and a large amount of test data 433
collected from the literature. The simplified numerical model proposed in this paper covers a wide 434
range of parameters: diameter-to-thickness ratio (/ = 10-220); yield stress ( = 186-960 MPa) 435
and concrete cylinder compressive strength ( = 20-200 MPa). The strength increase or 436
degradation of a CFST column after reaching its ultimate strength can be automatically captured in 437
the simulation. 438
It should be noted that the current research is only limited to axially loaded circular CFST stub 439
columns. Further research is required to propose a similar model for rectangular columns due to the 440
difference in section stability and concrete confinement between the two types of columns. Further 441
research is also required to check if the proposed steel and concrete models can be used in the 442
simulation of slender CFST columns or columns under eccentric loading. The efficiency of the fiber 443
beam element modeling has been well demonstrated by Wu et al. (2006) in analyzing the Second 444
Saikai Arch Bridge with a main span of 240 m. Further research should be conducted to incorporate 445
the proposed material models in FBE modeling of composite frames with CFST columns. 446
Acknowledgements 447
This study is supported by Western Sydney University under the International Postgraduate 448
Research Scholarship scheme. This support is gratefully acknowledged. 449
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22
Table 1. Summary of Test Data for Circular CFST Stub Columns. 564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
Number of specimens D (mm) (mm) / L/D
(MPa)
(MPa) Source
7 76-153 1.7-4.1 1.6-2.6 30-48 2.0 363-605 21-34 Gardener and Jacobson (1967)
1 169 2.6 0.563 65 1.8 317 37 Gardener (1968)
23 150 2.0-4.3 0.65-1.6 35-75 3.0 280-336 18-29 Tomii et al. (1977)
12 174-179 3.0-9.0 0.4-3.1 20-58 2.0 248-283 22-46 Sakino and Hayashi (1991)
2 190 1.15 0.05 165 3.5 202.8 110.3 O’Shea and Bridge (1994)
10 165-190 0.9-2.8 0.05-0.54 59-221 3.5 186-363 41-80 O’Shea and Bridge (1998)
2 141 3.0-6.5 0.92-2.8 22-47 4.3 285-313 24-28 Schneider (1998)
12 108-133 1.0-4.7
0.07-0.71 24-125 3.5 232-358 92-106 Tan et al. (1999)
6 102-319 3.2-10.3 0.92-2.5 31-32 3.0 334-452 23-52 Yamamoto et al. (2000)
3 200-300 2.0-5.0 0.34-1.1 40-150 3.0 266-342 27-31 Huang et al. (2002)
2 100-200 3.0 0.385 33-67 3.0 304 50 Han and Yao (2004)
7 114-115 3.8-5.0 0.58-2.0 23-30 2.6 343-365 26-95 Giakoumelis and Lam (2004)
5 108-450 3.0-6.5 0.64-3.22 17-52 3.0 308-853 41-85 Sakino et al. (2004)
22 60-250 1.9-2 0.12-0.52 30-134 3.0 282-404 75-80 Han et al. (2005)
8 89-113 2.7-2.9 1.3-1.7 33-39 3-3.8 360 28-33 Gupta et al. (2007)
2 165 2.7 0.36-0.50 61 3.1 350 48-67 Yu et al. (2007)
4 114.3 3.35 0.35-1.13 34 3.0 287 33-106 de Oliveira et al. (2009)
1 360 6 1.11 60 4.8 498 31.5 Lee et al. (2011)
16 114-219 3.6-10 0.19-1.5 18-44 2.2-2.7 300-428 54-193 Xiong D. X. (2012)
2 76.2 3-3.3 0.34-0.42 23-25 3.9 278-316 145 Guler et al. (2013)
2 114.3 4.0-5.9 0.42-0.67 19-28 3.5 306-314 115 Guler et al. (2014)
1 160 3.8 0.83 42 3.0 409 51 Han et al. (2014)
23
587
588
589
Fig. 1. 590
591
592
593
594
(a) Influence of mesh size (b) Number of fiber elements 595
Fig. 2. 596
597
598
599
600
0
700
1400
2100
0 10000 20000 30000 40000
Test (Sakino and Hayashi 1991)
FE model
Specimens
H-58-1 and H-58-2
Mesh size 3.6 mm
Mesh size 18 mm
Mesh size 36 mm
0
1000
2000
3000
4000
0 10000 20000 30000 40000
Test (Huang et al. 2002)
FE model
Specimen CU-070
8 fiber elements
16 fiber elements 64 fiber elements
Integration points
(b) FBE model
(a) Solid FE model
Steel tube
Column load
Render view of FBE
model
Concrete
Steel fibers as *rebar elements
Cross-section discretization
Core concrete as B31 element
Fiber elements
Concrete
Steel tube
(c) Discretization of the steel tube and concrete core
=280 mm, =4 mm, =840 mm
=174 mm, =3 mm, =360 mm
=265.8 MPa, =45.7 MPa, =0.42 =272.6 MPa, =31.15 MPa, =0.52
Axial load (kN)
Axial strain μ Axial strain μ
Axial load (kN)
24
601
602
603
(a) Steel curves
(b) Concrete curves 604
Fig. 3. 605
606
607
608
Fig. 4. 609
610
611
612
Fig. 5. 613
0
0.4
0.8
1.2
1.6
2
2.4
0 25000 50000 75000 100000
FE input ξ=3.4
ξ=1.5 ξ=0.65
ξ=0.35 ξ=0.15
FE input
ξc=1.50
ξc=0.35 ξc=3.40
ξc=0.65
ξc=0.15
0
0.5
1
1.5
2
2.5
3
3.5
0 10000 20000 30000 40000
ξ=4.5 ξ=2.0
ξ=0.85 ξ=0.45
ξ=0.20
ξc=3.40
ξc=0.65
ξc=0.15 ξc=1.50
ξc=0.35
Typical σ-ε curve in 3D FE modelling
Proposed σ-ε curves for fiber models
Low
High
ε
_
y
^′
0.6
0.8
1
1.2
0 0.05 0.1 0.15
2=0.81
Eq. (4)
,
/
,
,
,
0.01/./.
Stress σ/
Stress
σ
/
=300 MPa, =50 MPa, =220 mm =300 MPa, =50 MPa, =220 mm
Strain μ Strain μ
Strain
Stress
/
25
614
Fig. 6.
615
616
617
618
Fig. 7. 619
620
621
622
Fig. 8. 623
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
3D FE output
Proposed equation
2=0.99
Eq. (5)
0
0.005
0.01
0.015
0.02
0.025
0 0.005 0.01 0.015 0.02 0.025
2=0.96
Eq. (6)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.5 1 1.5
2=0.92
Eq. (7)
280.0712.0.13./..
6.8-0.0133.5.1.3../.
.../
/
/
26
624
Fig. 9. 625
626
627
628
629
630
(a) Comparison of steel models (b) Comparison of concrete models 631
632
(c) Comparison of predicted and measured curves 633
634
Fig. 10. 635
636
637
0
0.5
1
1.5
2
2.5
3
01234
ξ
c
3D FE output
Proposed value
ψ=1.5
0
100
200
300
400
500
0 25000 50000 75000 100000
3D FE input (Tao et al. 2013)
3D FE output
Proposed equation
0
20
40
60
0 10000 20000 30000 40000
3D FE input (Tao et al. 2013)
3D FE output
Proposed equation
0
500
1000
1500
0 10000 20000 30000 40000
Test (Tomii et al. 1977)
3D FE model
Fiber beam model
Load carried by
steel tube
Load carried by
concrete
Axial load (kN)
Stress (MPa)
Strain μ
Stress (MPa)
Strain μ
Axial strain μ
=150 mm, =4.3 mm, =-450 mm
=279.6 MPa, =18.03 MPa, =1.94
27
638
Fig. 11. 639
640
641
642
Fig. 12.643
644
645
646
647
648
Fig. 13.649
650
651
652
Tao et al. (2013) for FE modelling
Proposed σ-ε curves for fiber models
Low confined
Highly confined
0
0.5
1
1.5
2
2.5
3
0 0.25 0.5 0.75 1 1.25
R2=0.97
Eq. (10)
0
0.2
0.4
0.6
0.8
1
1.2
00.511.5
2=0.97
Eq. (12)
0.2
/
.0.90.25/.
,
3.5
./.0.2.
Stress
Strain
/
/
28
653
(a) Comparison between and (b) Comparison between and 654
Fig. 14. 655
656
(a) Concrete strength (b) Steel yield stress 657
658
Fig. 15. 659
660
661
(a) Specimen 3HN (b) Specimen C2 662
Fig. 16. 663
664
0
0.2
0.4
0.6
0.8
1
1.2
1.4
00.511.522.533.5
10%
10%
Mean ()=0.985
Standarad deviation ()=0.067
0
0.2
0.4
0.6
0.8
1
1.2
1.4
00.511.522.533.5
10%
10%
=0.992, =0.064
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 30 60 90 120 150 180 210
10%
10%
UHSCHSC
NSC
0
0.2
0.4
0.6
0.8
1
1.2
1.4
180 460 740 1020
10%
10%
HSSNSS
0
450
900
1350
1800
0 10000 20000 30000
Test (Tomii et al. 1977)
3D FE model
Fiber beam model
Load carried by steel tube
Load carried by concrete
0
500
1000
1500
2000
0 10000 20000 30000 40000
Test (Schneider 1998)
FE model
Fiber beam model
Load carried
by concrete
Load carried
by steel tube
Axial load (kN)
Axial strain μ
Axial load (kN)
Axial strain μ
Steel yield stress
(MPa)
Confinement factor
/
Confinement factor
/
/
/
Concrete strength
(MPa)
=150 mm, =3.2 mm, =450 mm
=287.43 MPa, =28.71 MPa, =0.91
=141.4 mm, =6.5 mm, =602 mm
=313 MPa, =23.8 MPa, =2.79
29
665
(a) Specimen S12CS80A (b) Specimen C-100-3D 666
Fig. 17. 667
668
(a) Specimen C15 (b) Specimen C14 669
Fig. 18. 670
671
672
(a) Specimen 049C36_30 (b) Specimen CC8-A-8 673
Fig. 19. 674
675
0
800
1600
2400
0 10000 20000 30000 40000
Test (O'Shea and Bridge 1998)
FE model
Fiber beam model
Load carried by concrete
Load carried by steel tube
0
700
1400
2100
0 10000 20000 30000
Test (de Oliveira et al. 2009)
3D FE model
Fiber beam model
Load carried
by concrete Load carried
by steel tube
0
3000
6000
9000
0 10000 20000 30000 40000
Test (Xiong et al. 2017)
3D FE model
Fiber beam model
Load carried
by concrete
Load carried by
steel tubes
0
4000
8000
12000
0 20000 40000 60000
Test (Xiong et al. 2017)
FE model
Fiber beam model
Load carried
by concrete
Load carried by
steel tubes
0
2000
4000
6000
8000
10000
0 10000 20000 30000
Test (Lee et al. 2011)
3D FE model
Fiber beam model
Load carried
by concrete
Load carried
by steel tube
0
1000
2000
3000
4000
0 10000 20000 30000 40000
Test (Sakino et al. 2004)
3D FE model
Fiber beam model
Load carried
by concrete
Load carried
by steel tube
Axial load (kN)
Axial strain μ
Axial load (kN)
Axial strain μ
Axial load (kN)
Axial strain μ
Axial load (kN)
Axial strain μ
Axial load (kN)
Axial strain μ
Axial load (kN)
Axial strain μ
=108 mm, =6.47 mm, =324 mm
=854 MPa, =77 MPa, =3.22
=114.3 mm, =3.35 mm,
=342.9 mm,=287.3 MPa,
=105.5 MPa, =0.35
=219.1 mm, =6.3 mm, =600 mm
=300 MPa, =163 MPa, =0.23 =219.1 mm, =10 mm, =600 mm
=381 MPa, =193.3 MPa, =0.41
=360 mm, =6 mm, =1760 mm
=498 MPa, =31.5 MPa, =1.11
=190 mm, =1.13 mm,
=662.5 mm,=185.7 MPa,
=80.2 MPa, =0.06
30
Captions for Figures 676
Fig. 1. Typical sketch of FE and FBE models for circular CFST columns 677
Fig. 2. Influence of mesh size and number of fiber elements of steel tube 678
Fig. 3. Effective curves of steel and concrete 679
Fig. 4. Proposed steel curves for FBE modelling 680
Fig. 5. Verification of proposed equation of681
Fig. 6. Verification of proposed equation of
682
Fig. 7. Verification of proposed equation of
683
Fig. 8. Verification of proposed equation of 684
Fig. 9. Verification of proposed value of 685
Fig. 10.Validation of steel and concrete material models 686
Fig. 11. Proposed curves of confined concrete 687
Fig. 12. Verification of proposed equation of
688
Fig. 13.Verification of proposed equation of 689
Fig. 14.Comparison between Nue with Nuc and NuFE with respect to confinement factor 690
Fig. 15.Comparison between Nuc and Nue with respect to material strength 691
Fig. 16.Comparison between predicted and measured curves for columns with normal 692
materials 693
Fig. 17.Comparison between predicted and measured curves for columns with HSC 694
Fig. 18.Comparison between predicted and measured curves for columns with UHSC 695
Fig. 19.Comparison between predicted and measured curves for columns with high 696
strength steel 697
