Modelling and experimental studies of the stress-strain state of compressed concrete filled steel tube elements of a continuous section

Modern building structures must meet the requirements of efficiency and resource-saving. The main direction for implementing these requirements is to reduce the consumption of steel (14–16%) and save cement (10–12%). These tasks can be solved through the rational combination of concrete and steel when they work together and the use of high-strength materials. One embodiment of this task is the use of Concrete Filled Steel Tube (CFST) structures. The purpose of this study is to identify the possibility of using the Drucker-Prager model by comparing the results of test studies on short compressed Concrete Filled Steel Tube (CFST) elements with different strength and deformation characteristics of the concrete core to the results from finite element analysis (FEA) modelling of corresponding CFST elements. Additionally, the behaviour of a steel pipe without a concrete core was investigated. The results show that the FEA method used in this work is sufficiently accurate for studying the behaviours of short CFST elements. The modelling technique adopted in the study made it possible to consider the redistribution of stresses in the concrete and pipe dynamically. The stress distribution patterns inside the concrete during both linear and non-linear deformations of CFST elements, as well as the characteristics of the interaction between the concrete and pipe in the contact zone, have been revealed.


The problem formulation and its relationship with scientific and practical tasks
The construction of modern buildings and structures requires the use of compressed elements with a high bearing capacity.One of the solutions to this problem is the use of Concrete Filled Steel Tube (CFST) elements.However, the wide distribution of this type of structural element is their significant metal consumption.In compressed CFST elements, the load is taken by both the pipe shell and the concrete core [1,2].Increasing the bearing capacity of the core helps to reduce steel consumption while maintaining the bearing capacity of the CFST element [3].In the limiting state, CFST does not collapse in the conventional sense.Because the destroyed concrete reinforced by the shell is included in the joint work.That is, the combination of one structural 1254 (2023) 012005 IOP Publishing doi:10.1088/1755-1315/1254/1/012005 2 element of several components from different structural materials leads to the formation of a special state of interaction between them.Therefore, the use of a rationally selected core will lead to a significant reduction in the need for metal and cost savings during the construction of the structure.Thus, the study of the behaviour of solid section elements compressed by CFST will allow choosing of the most effective design solutions for such structures.Modern FEA methods for simulating the behaviour of structures allow you to get significant savings in time and resources when checking the correctness of the design decisions made.However, to ensure the similarity of mathematical and physical models, it is necessary to use modern models of the behaviour of materials, in particular, concrete.

Research and publications analysis
At present, buildings and structures designed and built using CFST are represented almost all over the world.At the moment, the robot TB under static loading has been widely studied.
In work [4] stress-strain state CFST was estimated using the complex variable apparatus.The resulting solution is exactly within the boundaries of the formulated prerequisites.The key of which is the continuous joint work of individual components (pipe and concrete core) at all stages of loading until the limit state is reached.An analysis of the comparison results revealed that satisfactory agreement between theoretical and experimental studies was only for samples with a core of ordinary heavy concrete of low and medium ranges of strength classes.Thus, this approach limits the possibility of extending the obtained solution to CFST elements with other component materials, for example, high-strength concretes.In addition, the results obtained cannot be extended to other design solutions other than a solid two-component CFST element.For example, elements with voids in a concrete core, reinforced cores, etc.In the work [5] the method of estimating the stress-strain state based on the mechanical model of a composite body is considered.This approach makes it possible to take into account the loss of contact interaction between individual CFST components.This adequately describes the operation of a CFST element with a wide range of physical and mechanical characteristics of the components.In addition, the general methodological approach allows us to consider any design solutions for the CFST element, for example, to take into account additional bar reinforcement, etc.But a significant drawback is the laborious process of forming a system of resolving equations for the skin of such design solutions.At the same time, the issue of ambiguity in the conduct of calculations for such structures hinders the wider introduction of this kind of combination of steel and concrete, including taking into account the strengthening of their cores.The aim of the research was to formulate the prerequisites for performing an analytical calculation of compressed CFST elements with a high-strength core, as well as to evaluate the effectiveness of structural CFST elements with hardened cores compared to steel elements.
Due to many advantages that draw attention to the presented designs [6,7], previous studies were aimed at clarifying the mechanism of development of the stress-strain state of CFST elements with a hardened core for a more reliable and specific assessment of their characteristics [8][9][10][11][12][13][14][15].
Previous CFST studies using FEA [16][17][18][19][20][21][22] have shown that it is possible to accurately model the behaviour of this type of structural element.However, most of the considered models worked only at the stage of linear deformations of the concrete core.The study of the nonlinear behaviour of concrete was hampered by the lack of software implementation of nonlinear models of concrete behaviour, such as the Drucker-Prager model.

Formulation of the problem
Combining the geometric parameters and mechanical properties of the materials of the CFST components provides a variety of design options.Each of them has its own advantages and disadvantages.This creates uncertainty in the final decision.The choice can be made only by having an idea about the work under a load of a particular design solution.The developed methods for designing CFST structures, which are based on exact mathematical solutions, are cumbersome and limited in application.For adequate modelling of the stress-strain state of complex structural elements, it is not enough to imagine a model in which individual elements have excellent mechanical characteristics.The transition between different stress-strain state CFST structures throughout their life cycle is caused by a change in the magnitude and direction of external force factors.Therefore, the search for an adequate mechanism for assessing the stress-strain state of CFST structures is relevant.
The purpose of this study is to identify the possibility of using the Drucker-Prager model by comparing the results of test studies of short CFST elements with different strength and deformation characteristics of the concrete core with the results of FEA modelling of the corresponding CFST elements.

Methodology for conducting test studies
The purpose of this study is to identify the possibility of using the Drucker-Prager model by comparing the results of test studies of short CFST elements with different strength and deformation characteristics of the concrete core with the results of FEA modelling of the corresponding CFST elements.
Samples were tested after 28 days of natural hardening.Samples were tested in a GSM-250 press.The samples were loaded through the hinges along the longitudinal physical axis, the position of which was established by the trial loading method.The loading steps were 0.1 of the expected breaking force at the beginning of the test and 0.05 before reaching the ultimate limit state.
Longitudinal deformations during the testing of samples were measured using dial gauges with a division value of 0.01 mm.Simultaneously, the deformations in the compressed elements were measured using electric strain gauges.The AID-2M device was used to take the indicators.Each electric strain gauge was connected to an AID-1 measuring device, through which deformations were recorded.
The mechanical characteristics of the metal were determined by testing empty pipes for compression.Tensile strength was determined on standard samples-strips cut from the walls of pipes.The test results are given in table 1.The characteristics of concrete were determined

Modelling method adopted in the work
The modelling of the operation of CFST elements was carried out considering the nonlinear function of the components.The calculation was carried out considering large deformations.By the task, the CFST elements have a symmetrical tubular section and a central load case, to reduce the size of the model and reduce computational costs, the analysis was carried out considering the modelling of cyclic symmetry.In this case, the behaviour of one symmetrical sector (1/4 of the tubular component) was calculated, and the results obtained were used to build the response behaviour of the full component (as a post-processing step).
The simulation was carried out using the non-commercial software ANSYS Student by the finite element method in the Workbench environment.The non-linear calculation of CFST elements was carried out considering the concrete work model -Drucker-Prager and taking into account the multilinear isotropic hardening of steel.The joint work of the steel pipe and the concrete core in the model was provided by frictional contact.Following studies [18], the friction coefficient was taken equal to 0.5 in the simulation.[23].
Three models of CFST elements were analysed, which had the same external diameter of 119 mm but differed in the strength of the concrete core.In addition, the behaviour of a steel pipe without a concrete core was investigated.

Features of concrete modelling
The Drucker-Prager model was used to simulate the behaviour of concrete, which describes the nonlinear behaviour of concrete and is based on the theory of plastic flows.The calculation of the parameters of the Drucker-Prager model was carried out according to the methods proposed in the works [24][25][26][27][28][29].The validity of the accepted parameters was tested in the work [30].
The parameters given in table 3 were used in the calculations of the behaviour of concrete.

Features of steel modelling
Considering large plastic deformations, the steel pipe material model was developed considering multilinear isotropic hardening.Non-linear steel curves are found in Figure 1, and steel structural information data are presented in table 4.

CFST FEA model
When preparing the mathematical model, the parametric design was applied, which allows changing the model when its geometry changes.CAD models were created in SpaceClaim.Models were transferred to ANSYS Workbench via a direct interface.The contact between the concrete core and the steel pipe was assumed in the calculations as Frictional, with a friction coefficient of 0.5.The connection between the base plates and the pipe was set as Bonded and between the base plates and the pipe was Frictionless.
All elements were modelled using SOLID186 hex elements.The SOLID186 element is defined by 20 nodes having three degrees of freedom per node.The element maintains ductility, hyperelasticity, creep, large deflections, and large stresses.The size of the concrete element  was taken equal to 15.2 mm, for the pipe -5.5 mm.The size of the elements of the base plates was assumed to be 32 mm (figure 2).The boundary conditions of the lower support plate were modelled as Fixed Support (figure 3).To improve the convergence of the matrix, the elements were loaded not through the application of a load, but through deformations.Therefore, the boundary conditions of the upper base plate were modelled as Displacement along the Y-axis.The displacement was carried out step by step in equal shares.In total, during 10 steps, the plate was displaced by 30 mm.The CFST behaviour was studied at each loading step.

Results comparison of simulation and natural experiments
The dependences of deformations of CFST elements on the load were obtained because of test and FEA studies.The results are shown in figure 4 and figure 5.A comparison of the two groups of results shows that the simulation method used in the work is sufficiently accurate to study the behaviour of short CFST.Particularly good correlation is obtained in the region of linear deformations CFST.This allows us to conclude that the proposed method can be used to predict the strength characteristics of CFST elements.The proposed method also made it possible to "look inside" CFST elements and obtain data that is difficult to obtain as a result of laboratory tests.The stress distribution patterns inside the CFST components at different loading levels are obtained.For example, figure 6 shows the distribution of equivalent stresses for sample CFST-2 at a load of 871 kN.It has been observed that the stresses within the volume of the concrete core are distributed unevenly.The modelling technique utilized in this study enables consideration of the redistribution of stresses within both the concrete core and pipe dynamically.Within the region of linear deformations, the stress distribution of the stress element at the centre of the concrete core and pipe did not  differ.However, upon transitioning to the zone of non-linear deformations, the stresses in the longitudinal axis of the concrete core were found to be higher than those in the contact zone with the pipe.
Figure 7 shows the information on the Sliding Distance between the concrete core and the pipe for sample CFST-3 at a load of 2339 kN.Displacement of concrete relative to the pipe was observed in the edge zones adjacent to where the elements were fixed.The maximum

Conclusion
In this work, a comparison of the results of test studies of short CFST elements with different strength and deformation characteristics of the concrete core with the results of FEA modelling of corresponding CFST elements was carried out.This study's conclusions can be summarized as follows: − The Drucker-Prager model, for which the parameters were calculated based on existing methods and verified for their adequacy, was used to model concrete.− A model with multilinear isotropic hardening, for which a nonlinear stress-strain curve was set, was used to model steel.− Parametric design, contact conditions, boundary conditions and step loading were used in the modelling.− The study showed sufficient accuracy and adequacy of the proposed method for studying the behaviour of short CFST elements.− The high degree of convergence between modelling and test results provides the possibility of using the Drucker-Prager model to model the behaviour of compressed CFST elements with different characteristics of strength and deformation of the concrete core.− The application of the FEA method allows for a deeper study of the mechanism of interaction between the concrete core and the steel pipe.− The proposed method made it possible to reveal the features of stress and displacement distribution in the components of CFST elements, which were detected by modelling and which are difficult to obtain by laboratory studies.In particular, unevenness of stresses in the concrete core, displacement of concrete relative to the pipe in the edge zones and appearance of a gap between concrete and pipe in the transitional regime were observed.
− The features of stress distribution inside the concrete core both in linear and nonlinear deformations of CFST elements, as well as the features of interaction between concrete and pipe in the contact zone, were revealed.

Figure 1 .
Figure 1.The curve of isotropic hardening of pipe material.

Figure 4 .
Figure 4.The longitudinal strain of a metal pipe and CFST elements.

Figure 5 .
Figure 5.The transverse strain of a metal pipe and CFST elements.

Figure 7 .
Figure 7. Sliding Distance between concrete core and pipe for sample CFST-3 at 2339 kN.

Table 1 .
Physical and mechanical characteristics of steel pipes.The longitudinal and transverse deformations were measured with the help of electric strain gauges.Longitudinal strains were also measured with dial gauges.According to the results of the test, graphs of the dependence of deformations on stress σ − ϵ were plotted, the prismatic strength was determined, and Young's modulus and Poisson's ratio of concrete were calculated.The cubic strength of concrete was determined by testing cubes with an edge of 150 mm.The deformation -strength characteristics of concrete are given in table 2.

Table 3 .
Concrete parameters of CFST FEA models.

Table 4 .
Steel characteristics in CFST FEA models.