Experimental study and finite element analysis on mechanical performance of the steel tube column filled with aluminium foam under eccentric compression

Abstract

To investigate the compressive deformation mode and flexural mechanical performance of steel tube columns filled with aluminium foam, a series of corresponding experiments under the quasi-static eccentric compressive conditions were performed to obtain the failure deformation diagram, load-deflection curve and load-strain relationship. Based on the experimental study, A finite element model was established to analyze the mechanical performance of steel tube columns filled with aluminum foam under different loading eccentric distances. The results show that the restrained bending deformation was more pronounced in the steel tube column filled with aluminum foam, and this deformation significantly improved the overall mechanical performance as the porosity of the aluminum foam decreased. Compared with the pure steel column, the ultimate loads of the aluminium foam-filled steel tube column with porosities of 90%, 80% and 70% were increased by 87.13%, 93.41%, and 104.11%, respectively. The tensile or compressive strains of the pure steel tube column and that filled with aluminium foam respectively appeared symmetric and asymmetric phenomena because the failure deformation mode of the steel tube column was influenced by filling aluminium foam. Compared with axial compression, the ultimate vertical loads of the aluminium foam-filled steel tube column with loading eccentricities of 30 mm, 50 mm, 100 mm and 150 mm decreased by 48.78%, 60.35%, 80.19% and 88.61%. The steel tube columns filled with aluminium foam under axial compressive action and eccentric load appeared different failure modes, which were respectively the progressive crushing symmetrical deformation and bending deformation. The corresponding failure phenomena of these tubes were observed to transition from local buckling occurring at the loading or fixed end to overall bending deformation along the column length with the gradual increase of eccentricity. Furthermore, the bending moment value of the steel tube column filled with aluminium foam obtained from the experiment under different loading eccentric distances increases with the decrease of load eccentricity.

Keywords

aluminium foam steel tube column mechanical performance eccentric compression vertical load

Introduction

With the continuous rapid growth of the social economy and the forward development of science and technology, the advanced materials in the construction industry are increasingly in demand. A material with excellent mechanical performance, remarkable energy absorption and damping properties is urgently developed. Variations in the local or overall instability and deformation forms of thin-walled steel tubes occur when excessive external loads are applied, resulting in a significant decrease in mechanical properties and energy absorption capacity (Arnold and Altenhof, 2004; Mantena et al., 2021; Rahi, 2018). Aluminium foam inherently possesses multiple excellent functions such as light weight, high energy absorption, sound insulation and noise reduction, buffer and shock absorption and so on (Barzigar et al., 2023). However, aluminium foam with low strength is limited in numerous practical application fields. The composite structure composed of aluminium foam-filled into thin-walled metal tubes by various technical methods is a kind of impact-resistant member with excellent mechanical performance.

In recent years, domestic and foreign researchers have performed plentiful fundamental research on the mechanical performance and energy absorption properties of structures filled with aluminium foam and achieved abundant valuable scientific results (Bhutada et al., 2023; Liu et al., 2022b; Salehi et al., 2021; Xu et al., 2019). The working performance of gradient aluminum foam under impact loading was investigated by Liu et al. (2024), who concluded that gradient aluminum foams exhibit a better energy absorption effect than homogeneous aluminum foam under impact action, with negative gradient aluminum foam demonstrating a 46.9% higher specific energy absorption rate. The ultimate bearing capacity and deformation behavior of aluminium foam-filled steel tubes with different lengths under quasi-static axial load were studied by Liu et al. (2022a), Shao et al. (2019) and Wang and Shao (2023) from the loading test, where the steel tube filled with aluminium foam was observed to show a larger residual bearing capacity after the axial force exceeded the limit load compared with the empty steel tube and to exhibit a steady compression through the whole compressive loading process.

The crashworthiness of functionally graded lattice structures filled by aluminium tubes with different parameters of rod diameter, draft angle and length-to-diameter ratio under impact load was investigated by Baykasoğlu et al. (2024) who concluded that the specific energy absorption value of square tubes with the selection of appropriate lattice filler parameters could be improved by up to 76%. Axial compression tests of aluminium foam filled with steel tubes under different high temperatures were studied by Wang et al. (2021, 2022) to discover that the peak axial compressive capacity of composite members decreases to a certain extent with the increase of fire temperature, and the difference between the yield load and the upper or lower limit fluctuated at the yield platform increases with the increase of aluminium foam density. The crash-over resistance of high thin-walled circular tubes filled with aluminium foam under a quasi-static axial loading test was analyzed by De Lemos Coutinho et al. (2022) who concluded that the energy absorption and peak force of the tubes filled with aluminium foam increased by 37% and 20% respectively when reinforced by CFRP and PVC foam, and the reinforcement material showed complementary effects on the crashworthiness of these tubes. Meanwhile, the finite element software was adopted by Ferdynus and Rogala (2019) and Xu et al. (2011) to simulate the thin pilaster filled with aluminium foam, wherein the thin-walled cylindrical aluminium alloy tube filled with aluminium foam could withstand higher stress than the empty tube. The mechanical performance of steel tubes filled with aluminium foam under the axial and oblique impact action was investigated by Djamaluddin et al. (2015) and Gao et al. (2016) to show that the steel tube filled with aluminium foam possessed excellent crash resistance under axial load and was adopted as the optimal choice of energy absorption in the structure under oblique load. In addition, the crashworthiness performance of thin-walled tubes filled with aluminium foam under compression at different strain rates was analyzed by Rajak et al. (2019) who concluded that the plateau stress and energy absorption increased with the increase of strain rate. With the widespread application of components filled with aluminium foam, the deformation mode and energy absorption of the sandwich panel filled with multiple aluminium foam under both low-velocity and high-velocity impacts were analyzed by Anant Mishra et al. (2024) through numerical simulations to show that the tubular cores as primary energy absorbers accounted for approximately 2/3 of the total energy absorption and possessed higher energy dissipation compared to aluminium foam and flat steel plates.

The columns in the actual structural system generally not only resist the axial force but also bear the bending moment subjected to the external horizontal forces such as earthquake load or wind action, and the mechanical performance of columns as eccentric compressive members should be analyzed in practical engineering. However, the plentiful above researches on structures filled with aluminium foam focus on the quasi-static axial compression experiments for tube structures such as rectangular or circular tubes to analyze the effects of filling metal foam on the mechanical performance and energy absorption capacity. Therefore, the investigation of the deformation behavior of steel tubes filled with foam metal under eccentric loading action is extremely deficient.

Based on the previous research on steel tubes filled with foam metal, the quasi-static eccentric compression experiment was performed to investigate the failure deformation mode and bending mechanical performance of the steel tube filled with aluminium foam in this paper. Meanwhile, the finite element software was utilized to analyze the influence of eccentricity on the mechanical properties including stiffness and bearing capacity of these tubes.

Investigation on eccentric compression experiment

Specimens

Three different types of aluminum foam-filled steel tube columns with respective porosities of 70%, 80% and 90% were customized for comparison with a pure steel tube in the eccentric compression experiment. The cross-section size of the steel tube column with the respective thickness, length and loading eccentricity of 2 mm, 300 mm and 50 mm was 40 mm × 40 mm. The porosity of foam aluminum is calculated using the density method, as illustrated in the following formula (1).

P = (1 - \frac{ρ_{f o a m}}{ρ_{b a s e}}) \times 100 %

(1)

In the formula:P—Foam aluminum porosity; ρ_foam—Foam aluminum density;ρ_base—Base material density.where the base material density refers to the density of pure aluminum.

Aluminium foam was filled into the steel tube columns by a hydraulic press after the steel tube column, aluminium foam, fixed steel plate, loading steel plate and stiffener were independently manufactured. Subsequently, the steel tube column filled with aluminium foam was respectively welded with the fixed steel plate, the loading steel plate and the stiffener to associate a whole. The material of steel tube column and stiffener was adopted as Q235 steel, and Q355 steel was used for other steel plates. The processing dimensions of aluminium foam, steel tube, fixed steel plate, loading steel plate and stiffener are shown in Figure 1, where the thickness of fixed steel plate, loading steel plate and stiffener was respectively 20 mm, 20 mm and 2 mm.

Figure 1.

Processing dimensions of specimens (unit: mm).

Experimental process

The failure deformation mode and compressive bending mechanical performance of the steel tube column filled with aluminium foam were investigated in this paper. The manufactured specimen and overall experimental device are respectively shown in Figure 2 and Figure 3. Both ends of the steel tube filled with aluminium foam were respectively welded by the loading and fixed steel plates to construct the integrality of specimen, and the fixed steel plate of this specimen was immobilized with the reaction frame connecting the ground through the bolt rod. The strain gauge was pasted to the tension and pressure side of the steel tube column at the upper, middle and lower positions. The horizontal displacement meters were installed respectively at the middle and lower parts of the steel tube to monitor the lateral deformation, whereas a vertical displacement meter mounted at the loading steel plate was utilized to measure the applied displacement. Finally, the eccentricity of the loading situation was transformed by moving the position of a jack as plotted in Figure 4.

Figure 2.

Steel tube column filled with aluminium foam.

Figure 3.

Installation layout of experiment.

Figure 4.

Plane view of loading eccentricity (unit: mm).

To ensure the accuracy of the load position in practice, multiple calibrations of the experimental equipment was conducted to verify the accuracy of the load position through preliminary tests. During the experiment, the load position is monitored through a real-time monitoring system. Small deviations in the loading distance are immediately adjusted to ensure the reliability of the experiment.

The specimen was firstly preloaded to 0.5 kN and the corresponding loading time was maintained 1min to ensure the effectiveness and normal operation of the measurement system, therewith the load was unloaded to 0 kN. The loading increments of the steel tube column before and after yielding were respectively 0.5 kN and 0.2 kN during the formal loading. When the maximum load of the steel tube column drops to 85% of the peak load or this specimen has already been damaged, which was unsuitable for further loading, it was considered that the specimen had failed and the loading test was stopped.

Size effect

The similarity ratio refers to the proportional relationship between the model and the actual structure in terms of geometric dimensions, material properties and loads. The similarity ratio is used to infer the performance of the actual structure from the results of model tests. The geometric similarity ratio is plotted in equation (2).

λ = \frac{L}{L_{m}}

(2)

In the formula:λ—Geometric similarity ratio;L—Dimensions for the actual structure;L_m—Dimensions for the test or model.

The load similarity ratio is typically proportional to the square of the geometric similarity ratio in structural mechanics. This is because the load is proportional to the cross-sectional area, and the cross-sectional area is proportional to the square of the geometric dimensions. The load similarity ratio is given by equation (3).

λ_{F} = λ^{2}

(3)

In the formula:λ_F—Load similarity ratio;λ—Geometric similarity ratio.

The load of the actual structure is inferred from the load of the scaled-down model through equation (4).

F_{A} = F_{T s} \cdot λ_{F}

(4)

In the formula:F_A—Load for the actual structure;F_TS—Load for the text or simulation;λ_F—Load similarity ratio.

Experimental result

Experimental phenomenon

The aluminium foam-filled steel tube columns filled with porosities of 70%, 80% and 90% and the pure steel tube were tested under the eccentric compression action with the loading eccentricity of 50 mm to obtain the bending failure and deformation states, as shown in Figure 5.

Figure 5.

Failure deformation of steel tube columns with a loading eccentricity of 50 mm.

When the eccentric loads of pure steel tube column and aluminium foam-filled steel tube with porosities of 90%, 80% and 70% were respectively applied to 18.6 kN, 33.2 kN, 35.5 kN and 36.6 kN during the test process, the slight bending deformation was observed at the middle of the specimen and local buckling occurred at the loading region of stiffeners, which indicated that the applied eccentric loading value caused by the initial tiny bending phenomenon of the specimen gradually increased with the decreased porosity of aluminium foam. The lateral bending deformation of the steel tube column was effectively constrained by filling aluminium foam, and the filler with smaller porosity provided the stronger restricted ability to effectively prevent bending distortion of the specimen. When the eccentric loads of the pure steel tube column and the aluminium foam-filled steel tube column with porosities of 90%, 80% and 70% were further applied to 20.1 kN, 39.6 kN, 39.4 kN and 41.8 kN, the visible bending deformations at the middle of specimens and warping phenomenon at the loading steel plate were obviously observed. Meanwhile, the aluminum foam-filled steel tube columns with different porosities bore similar loads, while the pure steel tube column withstood only approximately 50% of those loads. The generated highly serious bending deformation and the warping phenomenon of the specimens caused the extremely decreased bearing capacity when the pure steel tube and the specimens filled by porosities of 90%, 80% and 70% were respectively further loaded to 21.6 kN, 40.3 kN, 42.2 kN and 45.8 kN. Therefore, the specimens were considered the failure, the eccentric experiments were compulsively stopped and those loads were regarded as the ultimate bearing capacity of specimens.

Load-displacement curve

The displacement and vertical load during the eccentric compression experiment were respectively obtained by the displacement meter at the loading ending plate and the pressure sensor connected with the jack to draw the load-displacement curve, as shown in Figure 6. The obvious torsion of the specimen was undiscovered during the experiment process. Therefore, the effect of rotation angle on the bearing capacity of the specimen was comparatively small and could be ignored.

Figure 6.

Load-displacement curve of specimens with loading eccentricity of 50 mm.

The curve exhibited that the displacement and vertical load of the steel tube columns were linear relation before the loading displacement was 2 mm, and the graphs were extremely clustered to each other. The bearing load of the pure steel tube column tended to gradually stabilize when the displacement exceeded 2 mm, and the curves of the aluminium foam-filled steel tube columns with different porosities appeared obvious bifurcation. The corresponding vertical load values of the specimens increased as the aluminum foam porosity decreased. When the load-displacement curve gradually reached the horizontal section, this indicated that the premature yielding of the pure steel tube column was prevented by the aluminum foam. Additionally, the bearing capacity of the filled specimens was remarkably enhanced as the porosity decreased.

The schematic diagram of the methods for determining yield and ultimate load is shown in Figure 7. The corresponding load and displacement at the final loading stage were respectively adopted as the ultimate load and displacement when there was no falling section of the load-displacement curve. The lower limiting value was controlled to 0.85V_max and the ultimate load was taken by the point corresponding to 0.85V_max when the curve occurred a decline section after entering the plastic stage, as illustrated in Figure 7(a). The load-displacement curves of the specimens do not exhibit a decay stage. The ultimate load for each specimen was taken as the load corresponding to its final loading displacement. A straight line parallel to the initial elastic slope was drawn on the load-displacement relationship of the specimen, intersecting the curve at two points, such that the area under the curve above the line equals the area above the curve below that line. The load value at which the areas on both sides are balanced was the yield load, as plotted in Figure 7(b).

Figure 7.

Determination method for the yield and ultimate load.

The mechanical parameters about the bearing capacity of the steel tube columns including stiffness and ultimate load obtained from the load-displacement curve are shown in Table 1. The initial stiffness was equal to the ratio of yield load to yield displacement. The table was observed that the initial stiffness and yield load of the filled specimen under the experiment with an eccentricity of 50 mm were insusceptible to the porosity of aluminium foam, but the ultimate load was remarkably affected by aluminium foam. The increasing rates of ultimate load for the specimens with porosities of 90%, 80%, and 70% were respectively improved by 87.13%, 93.41% and 104.11% compared with the pure steel tube column. These increases gradually accelerated as the porosity of the aluminum foam decreased, indicating that the bearing capacity of the specimens was enhanced by filling with aluminum foam. Moreover, the increased amplitude became more remarkable as the porosity of the aluminum foam became smaller.

Table 1.

Mechanical parameters of specimens.

Specimens	Porosity of aluminium foam	Initial stiffness/(kN/mm)	Yield load/kN	Yield displacement/mm	Ultimate load/kN	Ultimate displacement/mm
Steel tube column filled with aluminium foam	70%	11.50	32.12	2.79	46.15	9.13
	80%	11.46	32.83	2.86	43.73	9.31
	90%	11.52	32.61	2.83	42.31	11.26
Pure steel tube column	–	10.89	18.63	1.71	22.61	9.13

Load-strain curve

Strain gauges were attached to the upper, middle and lower of the tension and compression sides of the steel tube column, and the schematic position of the arranged strain gauge is illustrated in Figure 8.

Figure 8.

Position of strain gauges.

The strain data was derived from the strain gauges at the upper, middle and lower of the tension and compression side of the specimens to draw the load-strain curves shown in Figure 9. The figure shows that the strains at the tension and compression sides of the pure steel tube column and the aluminum foam-filled steel tube column exhibited completely symmetrical and asymmetrical phenomena, respectively. The strain value in the compression zone of the aluminum foam-filled specimen was larger than that in the tension zone, indicating that the failure deformation mode of the steel tube column was significantly influenced by the aluminum foam filling. The measured maximum strain values of the compression zone and tension zone at the upper of the pure specimen were respectively shown as −4000με and 3800με. The maximum strain values of the filled steel tube column with porosities of 70%, 80% and 90% were −5600με, −4300με and −4100με at the upper compression zone of the steel tube column and 3500με, 3000με and 2654με at the upper tension of the specimen. Meanwhile, the strain values of the aluminium foam-filled steel tube column gradually decreased with the decreasing aluminium foam porosity, which represented that the buckling deformability of the steel tube column was intensely constrained by aluminium foam with smaller porosity.

Figure 9.

Load-strain curves of specimens at different positions.

Load-deflection curve

The lateral deflection was obtained by the displacement meter mounted at the middle of the steel tube column illustrated in Figure 10 to draw the load-displacement curve, as shown in Figure 11. The curve showed that the pure steel tube column had premature yield even when the lateral deflection was slight, while the load-deflection curve trend of the aluminium foam-filled specimen with different porosities that possessed higher bearing capacity than the pure specimen was similar. The bearing load value of the filled steel tube column under the same lateral deflection was enlarged with the gradually decreasing porosity of the aluminum foam, which indicated that the bearing capacity of the specimen was prominently enhanced by filling aluminium foam, and the steel tube column could be extremely constrained with the porosity decrease.

Figure 10.

Position of displacement meters.

Figure 11.

Load-deflection curves of specimens with different porosities of aluminium foam.

Finite element numerical simulation

Modeling process

To investigate the mechanical performance of the aluminium foam-filled steel tube column influenced by the loading eccentricity of 0 mm, 30 mm, 50 mm, 100 mm and 150 mm, the finite element software of ABAQUS was adopted to establish the analytical models shown in Figure 12 and conduct the parametric numerical simulation.

Figure 12.

Model of numerical simulation.

The element types of steel tube columns and other components such as the aluminium foam, fixed steel plate, loaded steel plate and stiffener were respectively adopted as shell units (S4R) and solid units (3D8R) to build a finite element model with a dead-weight attribute. The fixed steel plate was attached with a mechanics characteristic of a rigid body where all degrees of freedom were constrained, while the vertical freedom degree in the loaded steel plate was not constrained. The contact type between the aluminium foam and the steel tube column, the fixed and loaded steel plates were adopted as face-to-face contact. Furthermore, the friction formula of the penalty function with a friction coefficient of 0.3 was adopted as the tangential slip and the relationship of hard contact was employed in the normal contact. The stiffener and steel tube column were respectively contacted with the steel plates by a binding connection to simulate the welding fabrication in the eccentric compression experiment. The type of load applied to the specimen was the loading displacement during the experiment, and the corresponding mechanical parameters of material in each component are shown in Table 2.

Table 2.

Material parameters on each component.

Component	Density/(kg/m³)	Elastic modulus/(MPa)	Yield stress/(MPa)	Poisson’s ratio
Steel tube	7900	206000	235	0.3
Aluminium foam	567	253	3.59	0.2
Fixed and loaded steel plates	7900	206000	355	0.3

Constitutive relation of material

To ensure that the actual loading process during the eccentric compression experiment was accurately simulated using the finite element method, the constitutive relations of different materials were individually and elaborately determined to describe the changing rules of mechanical performance.

Constitutive relation of steel

The Mises yield criterion and hardening rule were generally adopted to calculate the constitutive relation of steel by the mathematical expression represented in equation (5). The stress–strain curve of steel under the uniaxial tensile experiment is illustrated in Figure 13, where the f_p, f_y and f_u are respectively the proportional limit, yield strength and tensile strength. The steel materials of the steel tube, stiffener and steel plate were respectively selected by Q235, Q235 and Q355, and the elastic modulus of steel with the Poisson’s ratio of 0.3 was 2.06 × 10⁵ N/mm².

σ = {\begin{cases} \begin{array}{l} \begin{array}{l} E_{s} ε_{s} & ε_{s} \leq ε_{e} \\ - A {ε_{s}}^{2} + B ε_{s} + C & ε_{e} < ε_{s} \leq ε_{e 1} \end{array} \\ \begin{array}{l} f_{y} & ε_{e 1} < ε_{s} \leq ε_{e 2} \\ f_{y} [1 + 0.6 \frac{ε_{s} - ε_{e 2}}{ε_{e 3} - ε_{e 2}}] & ε_{e 2} < ε_{s} \leq ε_{e 3} \end{array} \end{array} \\ \begin{array}{l} 1.6 f_{y} & ε_{s} > ε_{e 3} \end{array} \end{cases}

(5)

Figure 13.

Stress–strain relationship of steel.

In the formula: $ε_{e} = 0.8 \frac{f_{y}}{E_{s}} 、$ $ε_{e 1} = 1.5 ε_{e} 、$ $ε_{e 2} = 10 ε_{e 1} 、$ $ε_{e 3} = 10 {0 ε}_{e 1}$ $A = 0.2 \frac{f_{y}}{{(ε_{e 1} - ε_{e})}^{2}} 、$ $B = 2 A ε_{e 1} 、$ $C = 0.8 f_{y} + A ε_{e}^{2} - B ε_{e}$

Constitutive relation of aluminium foam

The measured stress–strain curve of aluminium foam with 90% porosity plotted in Figure 14 was utilized to investigate the mechanical performance of steel tubes under the different loading eccentric distances in numerical simulation. The achieved elastic modulus, proportional limit, yield strength and densification strain derived from the mechanical experiment of material are respectively 99.78 MPa, 3.75 MPa, 3.59 MPa and 0.82.

Figure 14.

Stress–strain curve of aluminium foam with porosity of 90%.

Comparison between simulation and experiment

After completing the eccentric loading tests, the finite element software of ABAQUS was adopted to establish numerical models of foam aluminum-filled steel tube columns with different dimensions and the eccentric loading simulations was performed. By comparing the failure modes and load-displacement curves derived from the experiments and numerical simulations, it was found that the failure modes were almost identical and the ultimate load values were considerably approximate. Although there are issues related to size effects, the scaled-down specimens can still reflect the force distribution and failure mode of the actual-sized members. The current numerical analysis will provide significant insights into the basic mechanical characteristics of the practical member. In order to ensure that the actual mechanical performance of a steel tube column filled with aluminium foam under the eccentric compression action was correctly simulated by a finite element method, the numerical model of pure specimen and steel tube column with a cross-section of 40 mm × 40 mm, the thickness of 2 mm and length of 300 mm which was filled by aluminium foam with porosities of 70%、80% and 90% under loading eccentricity of 50 mm was established for analysis.

Comparison of failure phenomena

The destructive models of the pure steel tube column and the steel tube column filled by foam aluminum with a porosity of 90% were selected for simulation and experimental comparison. The pure specimen and the aluminium foam-filled steel tube column with 90% porosity were simulated by using the finite element method at a loading eccentricity of 50 mm to obtain the failure deformation diagram of the specimen shown in Figure 15. It can be seen from Figure 15(a) that the loading end of the pure steel tube column in the numerical simulation mainly exhibited bending deformation failure, with a significant bending amplitude and severe stress concentration on the tensile side. Similarly, the loading end of the pure steel tube column in the experiment also primarily shown bending deformation failure, and the bending amplitude was also considerable. It can be seen from Figure 15(b) that the loading end of the foam aluminum-filled steel tube column in the numerical simulation shown a noticeable bulging phenomenon in the compression zone, with the compression zone severely squeezed and indented at the loading end, and there was also a slight bulging phenomenon on the side. The tension zone at the loading end of the foam aluminum-filled steel tube column also experiences a minor protrusion. Similarly, the foam aluminum-filled steel tube column in the experiment exhibits a significant bulging phenomenon at the loading end, with the compression zone at the loading end being severely squeezed and indented, as well as a slight bulging phenomenon on the side. It was proved that the satisfactory agreement can be found between the experimental and numerically simulated failure deformations. Therefore, the numerical method can comparatively accurately respond to the experimental process under eccentric compression.

Figure 15.

Comparison of failure deformation between experimental and numerical results.

Comparison of load-displacement curve

A comparison of the load-displacement curve for the pure steel tube column and the aluminium foam-filled steel tube column with porosities of 70%、80% and 90% between the experiment and the finite element simulation is illustrated in Figure 16.

Figure 16.

Comparison of load-displacement curves between experimental and numerical results.

This compared figure demonstrated that the calculated numerical result showed the excellent agreement with the measured experimental values at the elastic stage. However, the load-displacement curves of the specimens were slightly different when the filled steel tube column entered the elastoplastic stage with increasing applied load. This discrepancy may be due to the simplified treatment of boundary conditions in the simulation and insufficiently refined mesh division. Nevertheless, the ultimate bearing capacity between the simulation and the experiment was not significantly different. Relative error between simulated and numerical mechanical properties of the specimen is illustrated in Table 3.

Table 3.

Relative error between simulated and numerical mechanical properties of the specimen.

Specimens		Simulation		Experiment		Relative error
Specimens		Ultimate displacement/mm	Ultimate load/kN	Ultimate displacement/mm	Ultimate load/kN	Ultimate displacement/mm	Ultimate load/kN
Pure steel tube column		10.0	21.2	9.0	22.9	11.1%	7.4%
Filled steel tube column	70%	10.0	43.9	9.2	46.3	8.7%	5.2%
	80%	10.0	39.9	9.3	42.6	7.5%	6.3%
	90%	10.0	37.98	11.2	40.9	10.7%	7.1%

It can be seen from the above table that the relative errors of ultimate displacements and loads between the simulated and experimental results were between 5% and 12%, which meet the accuracy requirements of engineering. Therefore, it was further concluded that the mechanical performance of the steel tube column filled with aluminium foam under eccentric load was accurately simulated by adopting the finite element model, and the numerical methodology was commendably applied to the parametric analysis.

Numerical simulation under different eccentricities

Stress nephogram

The aluminium foam-filled steel tube columns were numerically simulated by using finite element software at the different loading eccentricities to obtain the effect of eccentricity on stress nephogram of members, as plotted in Figure 17. This figure described that the failure mode of specimens was remarkably affected by the loading eccentricity, and the failure pattern of the filled steel tube column at axial compression and eccentric compression respectively appeared progressive crushing symmetric and bending deformation mode. Figure 17(a) expressed that the stress of the outer steel tube and aluminium foam was evenly distributed along the whole length direction under axial load, and the stress values at the upper and lower ends of the specimen were lower than other parts due to partial compressive buckling. Figure 17(b) showed that the outer steel tube prominently appeared a stress concentration in the tension and compression zones near the loading end under the applied load eccentricity of 50 mm, and the stress value and distribution range of the outer steel tube and aluminium foam at the tensile and compressive zones were significantly different. The stress value of the outer component at the loading area was greater than that at the fixed region, and the loading end of aluminium foam at the compressive side appeared serious stress concentration with maximum stress distribution. It can be seen from Figure 17(c) that there was no obvious local buckling and stress concentration occurred at the component. Moreover, the lateral bending deformation of the component was relatively larger compared to other members with smaller eccentricity. The distribution area of the greater stress located at the loading end was also wider. These observations indicated that as the loading eccentricity gradually increased to 150 mm, the failure mode of the aluminum foam-filled steel tube column changed from local compressive buckling to overall compression-bending failure.

Figure 17.

Stress nephogram of the aluminium foam-filled steel tube column with different eccentricities (unit: MPa).

Vertical load-displacement curve

Through the finite element numerical simulation of the aluminium foam-filled steel tube column under different loading eccentricities, the vertical force and displacement were respectively obtained by acquiring the calculated results of the coupling reference points located at the bottom of the column and loading point at the beam to draw the vertical load-displacement curve, as shown in Figure 18. The curve illustrates that the bearing capacity of the filled steel tube column is remarkably affected by loading eccentricity. The vertical bearing load of the specimen with an eccentricity of 0 mm which was subjected to the axial force was significantly higher than that of other steel tube columns, and the yield load of the specimen under axial compressive action was about 82.1 kN. The vertical load value of the aluminium foam-filled steel tube columns with an eccentricity of 30 mm decreased sharply, and the yield load was only about 50% compared with the specimen under axial compression. Moreover, the vertical bearing load of the steel tube columns filled with aluminium foam decreased with the gradual increase of eccentricity. The ultimate vertical bearing loads of the filled steel tube columns with eccentricities of 30 mm, 50 mm, 100 mm and 150 mm decreased by 48.78%, 60.35%, 80.19% and 88.61% compared to the specimen with the eccentricity of 0 mm.

Figure 18.

Vertical load-displacement curves of filled steel tube columns with different eccentricities.

The bending moment value was obtained by multiplying the eccentricity by the vertical load to plot the bending moment-displacement curve shown in Figure 19. The curve at the elastic stage was observed that the bending moment of the specimen filled with aluminium foam increased with the gradual decrease of the eccentricity, which described that the vertical force of the filled steel tube column was dramatically effective by the eccentricity. When entering the elastoplastic stage, the varying trend of bending moment of the filled member with the eccentricity of 30 mm, 50 mm and 150 mm was comparatively similar and gradually appeared the obvious platform segment. However, the bending moment of aluminium foam-filled steel tube columns with an eccentricity of 100 mm still slowly enhanced with the gradual increase of displacement. The meaningful achievements in bearing capacity obtained from the numerical analysis for the design of composite columns subjected to axial compression-bending action can be interpreted as follows: the failure mode of the member is largely related to the loading eccentricity, which results in different characteristics of the load-displacement curves. The members under smaller, larger and moderate loading eccentricities are primarily controlled by compressive failure, bending damage and compression-bending coupling failure, respectively.

Figure 19.

Bending moment-displacement curves of filled specimens with different eccentricities.

Load bearing performance of members with different eccentricities

The load performance parameters of steel tube column filled with aluminium foam obtained from the finite element simulation are shown in Table 4, where the ultimate bending moment value of the member was defined as multiplying the ultimate load by eccentricity. The table showed that the bearing capacity of the filled steel tube column was prominently affected by the eccentricity, and the initial stiffness, yield load and ultimate capacity of the member decreased sharply with the increase of load eccentricity. The initial stiffness values of filled steel tube columns with eccentricities of 30 mm, 50 mm, 100 mm and 150 mm were respectively reduced by 75.61%, 87.80%, 95.45% and 97.57%, and the yield loads were decreased by 54.68%, 63.64%, 84.52% and 88.90%, compared to the member under the axial compression. In the calculation of bearing capacity for the eccentric compression member, as illustrated in the following formula (6), the vertical bearing load of the member subjected to small eccentric compression action increased and the corresponding bending moment decreased due to the compression-bending coupling effect. However, this variable situation was opposite at the large eccentric compression region. The strength check of compression-bending steel members subjected to bi-directional bending moments is calculated as follows.

\frac{N}{A_{n}} + \frac{M_{x}}{γ_{x} W_{n x}} + \frac{M_{y}}{γ_{y} W_{n y}} \leq f

(6)

Table 4.

Ultimate bearing capacity of filled steel tube columns.

Loading eccentricity/mm	Initial stiffness/(kN/mm)	Yield load/(kN)	Yield displacement/(mm)	Ultimate load/(kN)	Ultimate displacement/(mm)	Ultimate bending moment/(kN.m)
0	89.48	82.32	0.92	95.78	10.00	0.00
30	21.82	37.31	1.71	49.06	10.00	1.47
50	10.92	29.93	2.74	37.98	10.00	1.90
100	4.07	12.74	3.13	18.97	10.00	1.90
150	2.17	9.14	4.21	10.91	10.00	1.64

In the formula:A_n—Net sectional area;W_nx, W_ny—Net sectional modulus for principal axis x and y;γ_x, γ_y—Plastic development coefficient for sections of principal axis x and y.

Only the axial force and bending moments in the direction of principal axis x and y are considered variables and other parameters are fixed values for this parametric numerical analysis according to the above formula. The lateral bending deformation of aluminium foam-filled steel tube column with an eccentricity of 30 mm to 100 mm mainly occurred in the x-axis direction and the corresponding deformation in the y-axis direction was extremely tiny. Therefore, the bending moment of the member in the x-axis direction was principally expected. However, the filled steel tube column with an eccentricity of 150 mm caused significant lateral displacements in both the x-axis and y-axis directions. Hence, the bending moments in both directions were considered to analyze the mechanical characteristics of the member. Simultaneously, the crushing damage and local buckling occurred in the compression area of aluminium foam-filled steel tube column, which resulted in the sharply decreasing axial bearing capacity of the member. Therefore, the ultimate bending moment value of the filled steel tube column with an eccentricity of 150 mm was less than that of the member with an eccentricity of 50 mm or 100 mm.

Conclusion

The results derived from the eccentric compression experiments and numerical simulation of the steel tube column filled with aluminium foam can be summarized as follows:

Firstly, the lateral bending amplitude of the steel tube column filled with aluminium foam is smaller than that of the pure steel tube column under the eccentric compression experiments, and the lateral bending of the specimen is dramatically constrained by aluminum foam as the porosity decreases. The initial stiffness of the filled specimen is rarely affected by aluminium foam filling, and the yield and ultimate load of these components remarkably increase with the decrease of aluminium foam porosity. The strain value of the filled steel tube column in the tension and compression zone gradually decreases with the diminution of porosity.

Secondly, the ultimate loads of the aluminium foam-filled steel tube columns with porosities of 90%, 80% and 70% respectively increased by 87.13%, 93.41% and 104.11% compared with the pure steel tube column. The failure deformation mode of the specimen is dominantly influenced by filling aluminium foam, and the tensile or compressive strains of the pure steel tube column and that filled with aluminium foam respectively appeared symmetric and asymmetric phenomena. The corresponding failure phenomena of these tubes were observed to transition from local buckling occurring at the loading or fixed end to overall bending deformation along the column length with the gradual increase of eccentricity.

Thirdly, the failure mode of specimens was remarkably affected by the loading eccentricity in the finite element numerical simulation, and the failure pattern of the filled steel tube column at axial compression and eccentric compression respectively appeared progressive crushing symmetric and bending deformation mode. The stress concentration phenomenon of the outer steel tube column and aluminium foam in the compression area is increasingly severe with the gradual increase of load eccentricity.

Lastly, the ultimate vertical loads of the filling specimens with eccentricities of 30 mm, 50 mm, 100 mm and 150 mm decreased by 48.78%, 60.35%, 80.19% and 88.61%, the initial stiffness values decreased by 75.61%, 87.80%, 95.45% and 97.57% and the yield loads decreased by 54.68%, 63.64%, 84.52% and 88.90%, respectively compared with the steel tube column at the axial compression. The bending moment of the aluminium foam-filled steel tube column increases with the gradual decrease of load eccentricity at the elastic stage, and the final bending moment of the filling specimens with different eccentricities are slightly different.

Footnotes

Acknowledgements

This work was financially supported by National Natural Science Foundation of China (12462010), Qiandongnan Science and Technology Plan Project (No. 0019, Qiandongnan Science and Technology Joint Support No. 2024), Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX24_4133), Qiandongnan Science and Technology Plan Project (Qiandongnan Science and Technology Foundation (2023) No. 03), the “14th Five-Year Plan” Discipline Professional Platform Team Integration Construction Project of Kaili College (YTH-PT202403). However, any opinions, findings, conclusions, and recommendations presented in this paper are those of the writers and do not necessarily reflect the views of the sponsors.

ORCID iDs

Dongmei Li

Zhanguang Wang

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Funding was provided by “National Natural Science Foundation of China (12462010), Qiandongnan Science and Technology Plan Project (No. 0019, Qiandongnan Science and Technology Joint Support No. 2024), Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX24_4133), Qiandongnan Science and Technology Plan Project (Qiandongnan Science and Technology Foundation (2023) No. 03), the “14th Five-Year Plan” Discipline Professional Platform Team Integration Construction Project of Kaili College (YTH-PT202403) and Scientific Research Project of Higher Education Institutions of Guizhou Provincial Department of Education (No. 372 of Guizhou Education Technology [2022])”.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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