Strength Calculation of Short Concretefilled Steel Tube Columns
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Abstract
The aim of this work is to propose a technique to calculate the strength of short concretefilled steel tube columns under the shortterm action of a compressive load, based on the phenomenological approach and the theoretical positions of reinforced concrete mechanics. The main dependencies that allow the realization of the deformation calculation model in practice are considered. A distinctive feature of the proposed approach is the method of the multipoint construction of deformation diagrams for a concrete core and steel shell. In this case, two main factors are taken into account. First, the steel shell and the concrete core work under conditions of a complex stress state. Since the proposed dependencies to determine the strength and the ultimate relative strain of volumetrically compressed concrete are obtained phenomenologically, they are more versatile than the commonly used empirical formulas. In particular, they can be used for selfstressing, finegrained and other types of concrete. Second, with a stepbystep increase in the relative deformation, the lateral pressure on a concrete core and a steel shell constantly change. Thus, the parametric points of the concrete and steel deformation diagrams also change at each step. This circumstance was not taken into account in earlier calculations. A comparison of the theoretical and experimental results indicates that the practical application of the developed calculation procedure gives a reliable and fairly stable estimate of the stress–strain state and the strength of concretefilled steel tube columns.
Keywords
concretefilled steel tube columns strength calculation technique deformation model deformation diagram concrete core steel shell1 Introduction
Concretefilled steel tube columns (CFTC) are steelreinforced concrete structures. The proposed methods for the calculation of a CFTC’s bearing capacity stipulated in regulatory instruments in a range of countries, e.g., Australia (Standard Australia AS5100.62004), Brazil (Brazilian Code NBR 8800:2008), Canada (CAN/CSA S1601:2001), People’s Republic of China (China Standard GB 500102010), Russia (CR 266.1325800.2016), and the USA (AINSI/AISC 360:2005), as well as the general European regulations (Eurocode EN 199411:2004), are essentially based upon empirical formulas. They have significant limitations in practical application, since they were obtained either from the results of specific laboratory sample tests (Furlong 1967; Tang et al. 1982; Saatcioglu and Razvi 1992; Shams and Saadeghvaziri 1997; Tsuda et al. 2000; Nishiyama et al. 2002; Johanson 2002; Fujimoto et al. 2004; Han 2007; Tao et al. 2008; Fattah 2012) or by the statistical processing of relevant data (Mander et al. 1988). First, these formulas are valid only for heavy concrete. For instance, in the case of the finegrain concrete columns, they provide excessively high estimates. In compliance with the published results (Karpenko 1996), one can observe a significantly smaller gain in strength for finegrain concrete under volumetric compression. When selfstressing concrete is used, such methods are not acceptable, as they do not take into account the prestress of the concrete core. Furthermore, empirical formulas are generally obtained by the results of research on smallscale samples, and their use often results in significant errors in the calculation of a bearing capacity for CFTCs with large crosssections (500 mm and more). This is confirmed by the results of experiments on CFTCs having section diameters of 630–1020 mm (Fonov et al. 1989). In addition, these methods generally do not allow making consistent calculations of eccentrically compressed CFTCs when they have any differences from a “classical” design. Examples include the presence of a highstrength rod and (or) spiral reinforcement in a concrete core, which can be rather efficiently used in CFTCs. To calculate the stresses in such a reinforcement, if possible, it is necessary to obtain sample the axial and circumferential deformations. The known calculation methods for the strength of compressed CFTCs do not provide such an opportunity (Krishan et al. 2016).
Previous publications (De Oliveira et al. 2009; Dundu 2012) note that the most reliable results are given by the calculations of the CFTC strength performed on the basis of Eurocode 2 (EN 199211 2004). These norms suggest two approaches for calculations. The first approach is simplistic, based on empirical formulas, and acceptable when a number of simplifications are made. The general case of calculation is also considered. For this approach, it is necessary to take into account the geometric and physical nonlinearity of the structure and the processes of crack formation, concrete creep and shrinkage. However, these prescriptions are declarative in nature and are not supported by specific methods or formulas. At that, the indication of the necessity of deformation diagram use for volumetrically compressed concrete proposed in § 3.1.5 of EN 199211 is erroneous, as will be shown below.
Computer calculations also do not allow obtaining reliable results using finite element models (Hu et al. 2005; Liang and Fragomeni 2010). The software packages used for this purpose (ANSIS, ABAQUS, etc.) do not contain the possibility to change the transverse deformation ratio of concrete and steel as the voltage level increases. Without an appropriate modification in the form of a special subprogram, they do not allow the accurate determination of the magnitude of the concrete lateral pressure on a steel shell. This value determines the strength and deformation properties of the concrete and the ratio of the axial and circumferential stresses in the steel.
The purpose of this paper is to propose a calculation technique for the strength of short, noncentrically compressed CFTCs under shortterm load action based on the phenomenological approach and the theoretical positions of reinforced concrete mechanics (Karpenko 1996).
2 Basic Design Provisions

10 mm.

1/Θ,
The calculation of the sample strength, performed using values of the concrete and steel strength characteristics obtained from the experimental results, should be carried out taking into account the understated values of random eccentricities. When samples are centred in the process of testing on a physical axis, this value is recommended to be reduced by 3 times.
The deformation calculation is based on material deformation diagrams. The CFTC power resistance is the work of the concrete and steel shell under the conditions of a volumetric stress state. As the loading level increases, the stress state changes not only quantitatively but also qualitatively. Therefore, the dependencies between the stresses and strains for concrete and steel are not known before the calculation begins. In this regard, the deformation calculation of the CFTC strength is proposed to be conducted in two stages.
In the first stage, diagrams of the concrete core \( \langle\langle \sigma_{cz}  \varepsilon_{cz} \rangle\rangle \) and the steel shell \( \langle\langle \sigma_{pz}  \varepsilon_{pz} \rangle\rangle \) are developed for the axial direction of the element. The diagram set is recommended to be of a multipoint form. It is shown (Karpenko et al. 2013) that this method is the most universal. In our case, it is practically realized during the calculation of the normal crosssection strength of a short, centrally compressed CFTC, as is shown below.
In the second stage, using the known dependencies of the nonlinear deformation model, given, for example, in a set of rules (Krishan 2008), the strength of the centrally compressed CFTC is calculated.
2.1 Development of Material Deformation Diagrams
To develop the diagrams \( \langle\langle \sigma_{cz}  \varepsilon_{cz} \rangle\rangle \; \) and \( \langle\langle \sigma_{pz}  \varepsilon_{pz} \rangle\rangle \), the conditions for the joint operation of the concrete core and the steel shell are considered in the process of the incremental growth of the axial deformations \( \varepsilon_{cz} = \varepsilon_{pz} \) of the centrally compressed column.
The consideration of the inelastic properties of the concrete core is made by the use of variable coefficients of elasticity \( \nu_{cj} \) (\( j = z,r,i \)) and transverse deformation \( \upsilon_{zr} ,\upsilon_{rr} \). Here, the indices z and r correspond to the axial and radial directions, and the index i corresponds to the elasticity coefficient, which determines the concrete deformation modulus in dependence between stress intensity and strain intensity.
The steel shell is represented as an isotropic body. The hypothesis of a single curve by Ilyushin is used to define it (Ilyushin 1948). According to this hypothesis, the connection between the stresses and deformations \( \langle\langle \sigma_{p}  \varepsilon_{p} \rangle\rangle \), obtained with the uniaxial tension of a steel shell, is considered valid for all stressed states. Hence, the current stresses \( \sigma_{p} \) and deformations \( \varepsilon_{p} \) are replaced by the intensity of the current stresses \( \sigma_{pi} \) and the strain intensity \( \varepsilon_{pi} \), respectively, for the sections of the shell under a complex stress state.
Coordinates of diagram characteristic points of the steel shell deformation.
Diagram parameter  Steel grades according to CR 16.13330.2011  

C245 C255  C285  C345 C345К C375  C390  C440  C590 C590К  
\( \overline{\varepsilon }_{el} \)  0.80  0.80  0.80  0.90  0.90  0.90 
\( \overline{\sigma }_{elp} \)  0.92  0.92  0.92  1.00  1.00  1.00 
\( \overline{\varepsilon }_{op} \)  1.70  1.70  1.70  1.70  1.70  1.70 
\( \overline{\sigma }_{yp} \)  1.00  1.00  1.00  1.00  1.00  1.00 
\( \overline{\varepsilon }_{2p} \)  14.0  15.0  16.0  17.0  17.0  18.0 
Here, \( \sigma_{pz} ,\;\sigma_{p\tau } ,\;\sigma_{pr} \) are the normal stresses of a pipe in the axial, tangential and radial directions; \( \varepsilon_{pz} ,\;\varepsilon_{p\tau } ,\;\varepsilon_{pr} \) are the relative deformations of a steel shell in the corresponding directions; \( \nu_{p} \) is the steel elasticity ratio; and \( \upsilon_{p} \) is the coefficient of a steel pipe transverse deformation.
The values of \( \upsilon_{p} \) calculated by the formula (10) are in good agreement with the data provided in the Russian Federation standards (Construction Rules and Regulations (CRaR) 2.05.0685*), but calculations based on the elastic coefficients make it possible to minimize the number of iterations.
At a known value of \( \sigma_{cr} \), the stresses in the steel shell \( \sigma_{p\tau } \) and \( \sigma_{pr} \) are calculated. To do this, they consider the equilibrium condition for the cross section and use the Lame solution for the problem of stress determination in the walls of a thickwalled cylinder (Ilyushin 1948). Then, the stresses \( \sigma_{cz} \) are found from the first equation of the system (2), and the stresses \( \sigma_{pz} \) are found from the first equation of the system (9).
The analysis of formulas (18)–(20) shows that the values of both coordinates of the diagram top concerning the deformation of the concrete core largely depend on the magnitude of the concrete lateral pressure. In the process of the gradual increase of the axial deformation \( \varepsilon_{cz} \), the pressure \( \sigma_{cr} \) does not remain constant. It increases from the values close to zero to a certain limiting value at \( \varepsilon_{cz} = \varepsilon_{cc1} \) that depends on the constructive and geometric parameters of the CFTC and is determined by the means of formula (21). It is assumed that at \( \varepsilon_{cc1} \le \varepsilon_{cz} \le \varepsilon_{cc2} \), the lateral pressure does not change, i.e., \( \sigma_{cr} = const. \) Therefore, the value of stress in a concrete core at the deformation \( \varepsilon_{cc2} \) can be assumed as being equal to \( f_{cc} \).
Intermediate diagrams are used for a more accurate calculation of the elasticity coefficients, transverse strains and, ultimately, the stresses \( \sigma_{cz} \) corresponding to the accepted value of the deformation \( \varepsilon_{cz} \).
Hence, it is obvious that it is impossible to use the deformation diagram of volumetrically compressed concrete in the limiting state recommended by Eurocode 4 (EN 199411 2004). In this case, the calculations give an increased bearing capacity, which is dangerous for practical use.
The practical implementation of the proposed calculation methodology is based on the stepiteration method. The axial deformation \( \varepsilon_{cz} = \varepsilon_{pz} \) increases gradually, and at each step, the stress–strain state of the concrete core and the steel shell are calculated. It is then taken into account that the value of the relative deformation \( \varepsilon_{cc1} \), originally calculated from formula (19), is only approximate, and it is specified during the calculation. Therefore, the iterations continue until the condition when the axial stresses in the concrete \( \sigma_{cz} \) reach its strength f_{cc} and the condition \( \left {\varepsilon_{cc1}^{\left( k \right)}  \varepsilon_{cc1}^{{\left( {k  1} \right)}} } \right < \Delta_{\varepsilon } \) is fulfilled. In this condition k is the approximation number, and \( \Delta_{\varepsilon } \) is the set accuracy of the calculations.
The result of this decision is arrays of numerical data that relate the relative deformations and stresses of the concrete and steel \( \left\{ {\varepsilon_{czi} } \right\} {} \left\{ {\sigma_{czi} } \right\} \) and \( \left\{ {\varepsilon_{pzj} } \right\} {} \left\{ {\sigma_{pzj} } \right\} \). They are then used to develop material deformation diagrams.
2.2 The Second Stage of the Calculation
With the direction of the axis n in the plane of the bending moment \( M_{n} = \sqrt {M_{x}^{2} + M_{y}^{2} } \), the case of compression by a force \( N_{z} \) applied with eccentricity \( e_{0} = {{M_{n} } \mathord{\left/ {\vphantom {{M_{n} } {N_{z} }}} \right. \kern0pt} {N_{z} }} \) is regarded (see Fig. 4).
In the calculation process, the deformation of the most compressed fibre of the concrete core \( \varepsilon_{c\text{max} } \) is increased step by step. At each step, using the Bernoulli hypothesis, a diagram of the relative deformations of the normal cross section is designed, corresponding to the equilibrium condition of the calculated element. To develop such a diagram, it is necessary to find the corresponding value of the relative deformation of the least compressed (stretched) fibre of the steel shell \( \varepsilon_{p\text{min} } \). The search for this value is carried out with a gradual shortening deformation decrease (starting from \( \varepsilon_{c\text{max} } \)) or a buildup of the elongation deformation \( \varepsilon_{p\text{min} } \) (starting from zero).
The normal section of the calculated element is conditionally divided into small sections with areas of concrete \( A_{ci} \) and steel shell \( A_{pk} \). The origin of the coordinate system is combined with the geometric centre of an element crosssection.
Within each small section of the concrete core and the steel shell, the stresses are assumed to be uniformly distributed (averaged). The magnitude of the stresses is determined at the centre of gravity of each section, depending on the values of the corresponding deformations, using the results of the first stage of the calculation.
Then, \( \varepsilon_{c\text{min} } \le 0 \) (elongation deformation) should be taken as \( \varepsilon_{cu} = \varepsilon_{cc2} . \)
In this regard, the criterion for the loss of the loadcarrying capacity of a column in the deformation calculation is the achievement of the maximum compressive force in the process of shortening the relative deformation increase of the most compressed fibre of a normal section.
On the other hand, in the process of a CFTC loading with certain geometric and constructive parameters, its axial deformation can reach excessively large values even before the exhaustion of the strength, and in such conditions, the operation of real structures becomes impossible. In a number of experiments, the axial deformations of CFTC samples reached 5–10%. In such cases, the limiting deformation can become predominant, determining the first limiting state. Therefore, during the determination of the CFTC loadbearing capacity, it is recommended to limit their axial deformations. The maximum permissible values of these deformations can be set by a calculator, depending on a specific design situation for a building or structure.
3 Comparison of Calculated Bearing Capacity with Experimental Data
In compliance with the proposed method, the authors developed an algorithm for the evaluation of the strain–stress state and the calculation of the bearing capacity of CFSTs having circular and annular sections. This algorithm is implemented in the software “CFST 3.3”, registered with the Russian Agency for Patents and Trademarks (certificate of registration No. 2014614664). It provides an opportunity to calculate CFSTs made of heavy or finegrain concrete (including selfstressing concrete) of classes from B15 up to B100 and a sheath made of any steel type, applied today, as well as to take into account the load duration and other norms for the manufacturing conditions and operation.
With the help of this software, the authors determined the theoretical values of the strength of CFTC shortcompressed samples that were previously studied by the famous Russian scientific schools performing concretefilled steel research (Fonov et al. 1989; Storozhenko et al. 1991). In addition, the experimental data of studies carried out at the Nosov Magnitogorsk State Technical University in 2002–2017, both on columns with a prestressed concrete core and on conventional CFTCs (Krishan et al. 2016), are used.
CFTC samples of a circular crosssection, working under compression with random eccentricities (107 samples) and eccentric compression (60 samples), were considered. The relative eccentricity of the longitudinal force for the latter was in the interval \( e_{0} /d = 0.06 {} 0.375 \). Some of the studied structures were made of selfstressing concrete. The conditions for carrying out all of the experiments and the obtained results are detailed in Krishan et al. (2016).
Results of comparison of theoretical and empirical loads to failure.
No  Specimen ID  d, mm  δ, mm  f_{yp}, MPa  f_{c}, MPa  N _{ u} ^{ Exp} , kN  N _{ u} ^{ Th} , kN  N _{ u} ^{ Exp} /N _{ u} ^{ Th} 

1  CC4A2  149  2.96  308  21.6  941  930  0.99 
2  CC4A41  149  2.96  308  35.6  1064  1165  0.91 
3  CC4A42  149  2.96  308  35.6  1080  1165  0.93 
4  CC4A8  149  2.96  308  69.3  1781  1630  1.09 
5  CC4C2  301  2.96  279  21.6  2382  2645  0.90 
6  CC4C41  300  2.96  279  36.1  3277  3511  0.93 
7  CC4C42  300  2.96  279  36.1  3152  3511  0.90 
8  CC4C8  301  2.96  279  72.3  5540  5573  0.99 
9  CC4D2  450  2.96  279  21.6  4415  5010  0.88 
10  CC4D41  450  2.96  279  36.1  6870  6964  0.99 
11  CC4D42  450  2.96  279  36.1  6985  6964  1.00 
12  CC4D8  450  2.96  279  76.6  11,665  11691  0.99 
13  CC6A2  122  4.54  576  21.6  1509  1211  1.24 
14  CC6A41  122  4.54  576  35.6  1657  1450  1.14 
15  CC6A42  122  4.54  576  35.6  1663  1450  1.15 
16  CC6A8  122  4.54  576  69.3  2100  1890  1.11 
17  CC6C2  239  4.54  507  21.6  3035  3026  1.00 
18  CC6C41  238  4.54  507  35.6  3583  3818  0.94 
19  CC6C42  238  4.54  507  35.6  3647  3818  0.95 
20  CC6C8  238  4.54  507  69.3  5578  5210  1.07 
21  CC6D2  361  4.54  525  21.6  5633  5700  0.99 
22  CC6D41  361  4.54  525  36.1  7260  7300  0.99 
23  CC6D42  360  4.54  525  36.1  7045  7300  0.99 
24  CC6D8  360  4.54  525  76.6  11,505  10,812  1.06 
25  CC8A2  108  6.47  853  21.6  2275  2131  1.07 
26  CC8A41  109  6.47  853  35.6  2446  2254  1.09 
27  CC8A42  108  6.47  853  35.6  2402  2250  1.07 
28  CC8A8  108  6.47  853  69.3  2713  2618  1.04 
29  CC8C2  222  6.47  843  21.6  4964  4389  1.02 
30  CC8C41  222  6.47  843  35.6  5638  5207  1.08 
31  CC8C42  222  6.47  843  35.6  5714  5207  1.10 
32  CC8C8  222  6.47  843  69.3  7304  7346  0.99 
33  CC8D2  337  6.47  823  21.6  8475  7765  1.09 
34  CC8D41  337  6.47  823  36.1  9668  9929  0.97 
35  CC8D42  337  6.47  823  36.1  9835  9929  0.99 
36  CC8D8  337  6.47  823  76.6  13,776  13,971  0.99 
Max  450  6.47  853  85.1  13,776  13,971  1.24  
Min  108  2.96  279  25.4  941  930  0.88  
V _{ b}  0.085 

The outer diameter of the outer steel shell − d = 93–1020 mm;

The thickness of the outer steel shell wall − δ = 0.8–13.3 mm;

The yield point of the shell steel − f_{yp} = 240–853 MPa;

The initial prismatic strength of the concrete − f_{c} = 11.7–104 MPa.
4 Discussion of Results
An analysis of these results indicates that the practical application of the developed calculation method gives a reliable and sufficiently stable estimate of the CFTC stress–strain state and strength. The nonlinear deformation model used as the basis of the method has significant advantages in comparison with the method of extreme forces. It allows one to use a singlesystem approach to the calculation of the bearing capacity and the stress–strain state of compressed concretefilled steel tube elements at all stages of their operation. The practical implementation of this model is carried out using iterative calculation performed with the consideration of the complex stress state, as well as the inelastic deformations of the materials and the changes in the transverse deformation ratios in the core and the steel shell as the stress level rises. Such a calculation makes it possible to take into account the specifics of the compressed concretefilled steel tube element operation.
The main dependencies for the determination of the strength and ultimate relative strain of triaxially compressed concrete were obtained phenomenologically on the basis of known positions in solidbody mechanics. Unlike the commonly used empirical formulas, they are more universal. In particular, they can be used for selfstressing, finegrained and other types of concrete.
In addition, it should be noted that the developed method is applicable for the calculation of the bearing capacity both for traditional CFTCs and for those with prestressed concrete. The shape of the CFTC cross section and the nature of their reinforcement can thus be different. These differences are easily taken into account when calculation algorithms for specific problems are developed.
The proposed multipoint method for concrete deformation diagram development is of great importance for improving the calculation accuracy. So far, this diagram has been adopted either for uniaxially compressed concrete (without taking into account the effect of the cage) or for the volumetrically compressed concrete core of the structure under conditions of the limiting equilibrium. Thus, the value of the bearing capacity was underestimated in the first case and inflated in the second case. No new method had previously been developed.
The proposed criterion to achieve the loadcarrying capacity of compressed elements is also important for practical calculations. It indicates that the strength properties of the concrete core cannot be used fully in certain cases. This fact cannot be ignored. In this regard, the calculation by the method of extreme forces does not always reflect the physical essence of the process and can result in large discrepancies from the experiments.
5 Conclusions
The authors proposed a method to determine the strength of short CFTCs. Based upon the known principles of a deformation calculation, it properly takes into account specific CFTC peculiarities. The method uses new dependencies to obtain the strength and ultimate deformation of a concrete core suitable for various concrete types, which makes it universal compared to other methods.
The deformation method of the strength calculation is based upon the numerical deformation diagram construction of a stressed concrete core and sheath. The implemented multipoint method of this diagram construction is significant for increasing the calculation accuracy.
The maximum value of a compression force is taken as a bearing capacity that is reached in the process of an incrementing relative deformation of shortening in the most compressed fibre with a normal section. This criterion allows a significant increase in the accuracy of the deformation calculation compared to the traditionally applied breaking stress method as allows a reliable assessment of the strain–stress state of compressed CFTCs.
Notes
Authors’ contributions
ALK has developed the method to determining the strength of short CFTCs. MAA participated in the comparison of theoretical and experimental data of the work. EPC participated in the development of the method to determining the strength of short CFTCs and drafted the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This article was prepared from the results of the implementation of a scientific project within the state task of the Ministry of Education and Science of the Russian Federation No. 7.3379.2017/4.6.
Competing interests
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