# Flexural Strength of Partially Concrete-Filled Steel Tubes Subjected to Lateral Loads by Experimental Testing and Finite Element Modelling

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Test Programme

^{2}at a strain of 0.2%, a tensile strength of 542 N/mm

^{2}at a strain of 20%, and a final elongation of 32%. The stress-strain behaviour obtained from steel coupon tests are shown in Figure 2. The cube compressive strength and elastic modulus of concrete were obtained by testing cylinder specimens. Three 150 mm (diameter) × 300 mm (height) concrete cylinders of the same concrete mix were prepared during the fabrication of the specimens and tested at the same time of testing the PCFST. The concrete mix had the ratio of cement:sand:coarse aggregates as 1:2.5:3, and the detailed properties were shown in Table 1. The average values of concrete cube compressive strength measured at 7, 14, 21 and 28 days were 25.20, 33.30, 39.80 and 41.00 N/mm

^{2}, respectively, as shown in Figure 3, and the elastic modulus of concrete was 25.50 GPa. The concrete strength measured at 28 days of 41.00 N/mm

^{2}and the corresponding strain was 0.002 were used in this study.

#### Testing Procedure

## 3. Finite Element Modelling

^{2}, the Poisson’s ratio ν = 0.3, the density ρ = 7890 kg/m

^{3}, the yield stress σ

_{y}= 382 N/mm

^{2}and the tensile strength σ

_{t}= 542 N/mm

^{2}. The von Mises yield criterion with “isotropic hardening” rule was employed to define the elastic limit of the tube steel when it was subjected to multi-axial stresses. The response of the steel tube was modelled by an elastic-plasticity theory with associated flow rule.

_{f}. Upon cracking, the stresses went to zero, the material lost all load-carrying capacity in tension.

#### 3.1. Concrete in Compression

_{1}and J

_{2}were the principal stress invariants.

^{2}and the corresponding strain was 0.002. Other material properties of concrete were the elastic modulus of concrete E = 25500 N/mm

^{2}, the Poisson’s ratio ν = 0.2, the density ρ = 2500 kg/m

^{3}. The material properties of confined concrete were usually used for the concrete filling by assuming that the filled concrete was subjected to laterally confining pressure from the steel tube. Therefore, the equivalent model of the confined concrete materials, proposed by Mander and other researchers [38], was employed. In this study, the uniaxial compressive strength was 41 N/mm

^{2}and the corresponding strain was 0.002, as forementioned in Section “Experimental test programme”. The value of the compressive strength and the corresponding strain of this confined concrete model, were calculated as 46 N/mm

^{2}and 0.003, respectively, using Equations (A1) and (A2), as shown in Appendix A. Therefore, these values were used as the input compressive strength for concrete elements in the FE models in this study. However, to investigate the effect of the compressive strength on the concrete filling’s and the PCFST’s flexural capacities, different values of the compressive strength were also used as the input strengths for concrete. They included the compressive strength of 41 N/mm

^{2}and 36 N/mm

^{2}. Because the interaction (contact) between the concrete filling and stell tube was modelled, the filled concrete was subjected to laterally confining pressure from the steel tube. Hence, the confined compressive stresses and strains in concrete could be predicted by the FE models and compared with those determined by theoretical models.

#### 3.2. Concrete in Tension

_{f}required to completely open a crack was defined as

_{c}was the crack band width and was equal to the width of the element in the direction perpendicular to the crack; σ

_{n}was the stress normal to the crack plane; ε

_{n}was the strain normal to the crack plane; ε

_{0}was the strain at end of softening curve; G

_{f}was the fracture energy per unit area determined from experimental data or from codes of practices. In this study, the linear softening model was used, as shown in Figure 8. The tension-softening modulus was assumed to be 5 to 10% of the modulus of elasticity as usually adopted in literature. When the element size l

_{c}, the fracture energy G

_{f}and tensile strength of concrete were known, the stress-strain relationship of the linear softening model could be determined by Equation (2). The crack initiation and propagation in the concrete was modelled by specifying cracking stress, tension-softening modulus, and shear retention values for the concrete. After the crack was formed following the train-softening path, the load-carrying capacity of concrete in tension across the crack diminished, as illustrated in Figure 8.

- (i)
- Mesh density: two different meshes, MESH I and MESH II, with the same small size in the region around the loading plate, were used for the concrete filling. MESH I had 49,938 solid elements of 50 mm × 50 mm × 50 mm, and MESH II was a very fine mesh, of 82,656 elements of 50 mm × 50 mm × 25 mm. The simulation results revealed that the maximum loads in MESH I differed from those of MESH II by less than 2%, but the analysis time was twice expensive for MESH II. Therefore, MESH I was considered accurate enough to be used in this study.
- (ii)
- Compressive strength: values of 41 N/mm
^{2}and 46 N/mm^{2}were used for the compressive strength of concrete material in the FE models. The first value was for the concrete obtained from testing, and the second value was for confined concrete, proposed in [38], as presented in Appendix A. The two FE models, however, obtained similar results in terms of load-displacement relationship and maximum loads despite a small difference in their compressive strength values. It was because under the applied loads, the concrete filling was confined by the steel tube based on their frictional contact and structural configuration. This resulted in a significant increase in the compressive strength in the concrete filling after the yield point; it meant there was not crushing in the compressive region of concrete. - (iii)
- Shear retention factor: in the literature on nonlinear Finite Element modelling of reinforced concrete structures, a range of values between 0.1 to 0.5 was suggested for the shear retention factor [42,43]; therefore, these values were investigated in this study. It was found that when values of shear retention factor were relatively small, they led to numerical instabilities in the Finite Element analysis; when values were from 0.2 to 0.5 the maximum load values differed to the experimental result from 1% to 3%, respectively. For these reasons, the typical value of 0.2 was used for the shear retention factor in this study.
- (iv)
- Cracking strength: the values of cracking stress ${\mathrm{f}}_{\mathrm{t}}$ in the range of 5 to 10% of the compressive strength ${\mathrm{f}}_{\mathrm{c}}$ [38,41] which were of 2.5 to 4.6 N/mm
^{2}were investigated. It was found that the maximum load values were greater than the experimental values from 1% to 3% for the cracking strength of 2.5 and 4.5 N/mm^{2}, respectively. In this study, the value of cracking stress was taken as 2.5 N/mm^{2}. - (v)
- Tension softening modulus: The tension-softening modulus ${\mathrm{E}}_{\mathrm{t}}$ was assumed to be 5 to 10% of the modulus of elasticity. The investigation also included the case that the tension softening modulus was specified to zero for modelling concrete as a complete brittle material (the sudden cracking with a complete loss of the stiffness upon cracking). It was found that the maximum load values differed to the experimental result from −12% (when ${\mathrm{E}}_{\mathrm{t}}$ was zero) to 2% (when ${\mathrm{E}}_{\mathrm{t}}$ was 10% of the modulus of elasticity). In this study, the tension softening modulus ${\mathrm{E}}_{\mathrm{t}}$ was assumed to be 1930 N/mm
^{2}which had a maximum difference of less than 1% to the experimental value. - (vi)
- Loading step: an initial fraction of loading time of 0.001 s and a maximum fraction of loading time of 0.005 s were used for the solution to be stable and reliable. The time step was selected through a trial-and-error procedure by continuously reducing the time step until similar results were obtained. In particular, the first analysis started with a maximum time step of 0.02 s and then it reduced time step to a smaller one until a time step of 0.001 s. The amount of time step in one reduction varied from 0.0005 s up to 0.05 s was selected from comparing the result of the second analysis to the first analysis’s. The compared results were the slopes of load-displacement curves and the ultimate forces in the bending simulations of the PCFST specimen. The maximum difference in the slopes and in the ultimate forces was set up within 5%. It was found that converged results were observed when time step was small enough, and similar results were obtained for the analysis with a time step of 0.001 s and 0.002 s. In this study, a maximum time step of 0.001 s was selected for accuracy reasons. It should be noted that this time step was selected in association with a reasonable finite element mesh (refer to 3(i)), an implicit analysis type with a full iteration method (refer to 3(viii)), and a strict convergence tolerance (force tolerance of 0.01 and displacement tolerance of 0.01, refer to 3(vii)). The convergence and consistency of the numerical results seem to justify such time step. Excellent agreements when comparing with experimental results from the bending test further validated this time step selection.
- (vii)
- Analysis tolerance: a relative force tolerance of 0.01 and a relative displacement tolerance of 0.01 were used for the solution to be reliable.
- (viii)
- Analysis type: an implicit, static analysis was employed. A full Newton-Raphson method was used for the iterative procedure.

## 4. Parametric Study

_{f}/L, the compressive strength of concrete, and the yield stress and tensile strength of steel tube. For each parametric study, the equivalent envelope (the load-displacement curve of the PCFST when it was subjected to monotonic loading) was obtained, and the ultimate load was extracted from this load-displacement curve.

#### 4.1. The Tube Diameter-to-Thickness Ratio D/t

#### 4.2. The Concrete Filling Length-to-Total Length Ratio L_{f}/L

_{f}varied that the ratio L

_{f}/L was set at six different values: 0, 0.34, 0.50, 0.60, 0.66 and 1. The value of zero means there was no concrete filling whilst the unity value means the concrete filling length equals the tube length. The material properties of concrete and steel were assumed the same with those used in the FE modelling of the experimental test. The ultimate loads of the FE models with different values of parameters D/t and L

_{f}/L were obtained for evaluating the effects of these parameters on the flexural strength of PCFST structures. They were also used to predict the optimal length of partial concrete filling after which there was no gain to the maximum load of the PCFST.

#### 4.3. The Compressive Strength of Concrete f’c

^{2}, 41 N/mm

^{2}and 46 N/mm

^{2}. The value of 36 N/mm

^{2}was an assumed value. The value of 41 N/mm

^{2}was obtained from testing for the unconfined concrete while the value of 46 N/mm

^{2}was obtained from the unconfined concrete of 41 N/mm

^{2}with confinement consideration, using the theoretical model, proposed by [38]. By selecting these values, the effects of the compressive strength on the PCFST’s flexural strength could be studied. Furthermore, the confined strength of concrete filling predicted by the FE modelling and by the theoretical model could be compared. For each compressive strength, there were three different sets of steel grades: σ

_{y}= 334 N/mm

^{2}and σ

_{t}= 448 N/mm

^{2}, σ

_{y}= 382 N/mm

^{2}and σ

_{t}= 542 N/mm

^{2}, and σ

_{y}= 458 N/mm

^{2}and σ

_{t}= 650 N/mm

^{2}.

#### 4.4. The Steel’s Yield Stress σ_{y} and Tensile Strength σ_{t}

_{y}= 334 N/mm

^{2}and σ

_{t}= 448 N/mm

^{2}, σ

_{y}= 382 N/mm

^{2}and σ

_{t}= 542 N/mm

^{2}, and σ

_{y}= 458 N/mm

^{2}and σ

_{t}= 650 N/mm

^{2}. The second set was material properties of steel tube used in the FE modelling of the experimental test and was considered as the reference set. The values of the steel strengths in the first set were 10% smaller than those of the reference set whilst the values of the steel strengths in the third set were 10% greater than those of the reference set. For set of steel grades, there were three different values of compressive strength, ${\mathrm{f}}_{\mathrm{c}}^{\prime}$ = 36 N/mm

^{2}, 41 N/mm

^{2}and 46 N/mm

^{2}.

## 5. Result and Discussion

#### 5.1. Experimental Results

#### 5.2. Finite Element Modelling Results and Validation

#### 5.3. Parametric Study Results

#### 5.3.1. Effects of the Tube Diameter-to-Thickness Ratio D/t

_{f}/L, using the FE model predictions. The ultimate loads were obtained from the envelope of maximum load values of the FE load-displacement curves. Figure 18 illustrates all enveloped load-displacement curves of different concrete filling length ratios at the tube diameter-to-thickness ratio D/t = 78. Figure 19 presents the relationship between the concrete filling length ratio and the ultimate load with different values of D/t ratio. It could be seen from Table 2 that for each concrete filling length, the ultimate load increased as the D/t ratio decreased, but at different rates. Decreasing the values of D/t significantly increased the ultimate flexural loads. The results show that the ultimate load increased as the D/t ratio decreased, but at different rates: when D/t ratio decreased twice, from 140 to 78, the maximum load increased 58%; however, when D/t ratio decreased twice, from 78 to 39, the maximum load increased 95%, that was nearly two times the previous change in D/t ratio. When the D/t ratio increased the tube was more vulnerable to fail by local buckling (buckled shapes formed around the loading point). It indicated the significant effects of increasing the tube’s thickness on increasing the PCFST’s ultimate flexural strength.

#### 5.3.2. Effects of the Concrete Filling Length-to-Total Length Ratio Lf/L and the Optimal Length

_{f}/L increased from 0.00 (no concrete filling) to 0.34, there was unnoticeable increase in the PCFST’s ultimate flexural loads. However, noticeable increase was shown when the values of L

_{f}/L increased from 0.34 to 0.66. After the value of 0.66, it reached a plateau equivalent to the totally concrete filled tube capacity; it was where the optimal concrete filling ratio was determined. From Figure 19, it is evident that the optimal concrete filling ratio was dependable on the diameter-to-thickness ratio of PCFSTs.

_{f}/L ratio decreased and D/t ratio increased, the tubes were more prone to local buckling failure; therefore, it was necessary to increase the filling ratio up to the optimal concrete fill length as discussed earlier.

#### 5.3.3. Effects of the Concrete Compressive Strength ${\mathrm{f}}_{\mathrm{c}}^{\prime}$

^{2}to 41 N/mm

^{2}, the maximum increase of the PCFST’s flexural load was 2%; when the compressive strength increased from 41 N/mm

^{2}to 46 N/mm

^{2}, the maximum increase of the PCFST’s flexural load was 1%. It concluded that for the same steel grade, the compressive strength of concrete had insignificant effects on the PCFST’s flexural strength. Figure 20 shows the enveloped load-displacement curves of different compressive strength values while Figure 21 illustrates the stress-strain curves of concrete at the material points, for the case of the steel strength σ

_{y}= 382 N/mm

^{2}and σ

_{t}= 542 N/mm

^{2}. It could be seen that when the compressive strength changed from 41 N/mm

^{2}to 46 N/mm

^{2}, their load-displacement curves and the PCFST’s ultimate loads were almost identical.

#### 5.3.4. Effects of the Steel’s Yield Stress σ_{y} and Tensile Strength σ_{t}

_{y}= 334 N/mm

^{2}and σ

_{t}= 448 N/mm

^{2}, σ

_{y}= 382 N/mm

^{2}and σ

_{t}= 542 N/mm

^{2}, and σ

_{y}= 458 N/mm

^{2}and σ

_{t}= 650 N/mm

^{2}. Table 3 shows the effects of the steel strengths on the PCFST’s ultimate load, and the confined strength of concrete predicted. All studies were carried out at the reference PCFST model with the concrete filling length ratio of 0.66 and the tube diameter-to-thickness ratio D/t = 78. The results show that for the same concrete grade, when the steel strengths increased from σ

_{y}= 334 N/mm

^{2}and σ

_{t}= 448 N/mm

^{2}to σ

_{y}= 382 N/mm

^{2}and σ

_{t}= 542 N/mm

^{2}, the maximum increase of the PCFST’s flexural load was 12–14%; when the steel strengths increased from σ

_{y}= 382 N/mm

^{2}and σ

_{t}= 542 N/mm

^{2}to σ

_{y}= 450 N/mm

^{2}and σ

_{t}= 650 N/mm

^{2}, the maximum increase of the PCFST’s flexural load was 13–14%. It showed the significant effects of the material properties of steel tube on the PCFST’s flexural strength capacity.

#### 5.3.5. The Confined Strength of Concrete ${\mathrm{f}}_{\mathrm{cc}}^{\prime}$

_{f}/L on the confined strength of concrete ${\mathrm{f}}_{\mathrm{cc}}^{\prime}$ and the corresponding strain ${\mathsf{\epsilon}}_{\mathrm{cc}}^{\prime}$ predicted by Finite Element Analysis (only the concrete filling length ratios of 0.66 and 1.00 were presented on Table 4 as the confinement effects was not significant for shorter concrete filling lengths). Figure 21 shows the confined strength of concrete at different values of the compressive strength of concrete, for the case of the steel strength σ

_{y}= 382 N/mm

^{2}and σ

_{t}= 542 N/mm

^{2}. It could be seen from Table 3 that for the same steel grade, the confined stresses developed in the concrete filling $\left({\mathrm{f}}_{\mathrm{cc}\_\mathrm{FEA}}^{\prime}\right)$ were increased when the compressive strength increased, with a maximum difference of 9%; however, this did not significantly affect the PCFST’s ultimate strength, as shown in Table 3. For the case of the steel strength σ

_{y}= 382 N/mm

^{2}and σ

_{t}= 542 N/mm

^{2}, the confined strengths of concrete at different compressive strengths were shown in the stress-strain curves of concrete in Figure 21. For the same concrete compressive grade, the confined stresses developed in the concrete filling $\left({\mathrm{f}}_{\mathrm{cc}\_\mathrm{FEA}}^{\prime}\right)$ were increased when the steel strengths increased, with a maximum difference of 5%; however, this significantly affected the PCFST’s ultimate strength, as shown in Table 3. It could be observed from Table 4 that when the concrete filling length was equal or greater than 66% of the tube length, the confined strengths of concrete were the same. When the tube’s thicknesses were increased from 9 mm to 18 mm (reducing D/t from 78 to 39), the confined stresses developed in the concrete filling $\left({\mathrm{f}}_{\mathrm{cc}\_\mathrm{FEA}}^{\prime}\right)$ increased significantly, by 2.4 times; however, when the tube’s thicknesses were increased from 5 mm to 9 mm (reducing D/t from 39 to 140), the confined stresses were not significantly changed.

^{2}in comparison with 64 N/mm

^{2}). This deemed to be reasonable for the assumption that there was a perfect bond between steel tube and the concrete filling. However, the strains predicted by FE analysis were much lower than those calculated by the theoretical model. The reasons could be due to the greater stiffness in the FE models as explained earlier (the concrete model used in the FE modelling did not have a constitutive damage model that was able to capture the micro-damage in the concrete) or different confinement conditions considered in the theoretical model (concrete was confined by spiral/ hoop steel reinforcements in axial compression tests rather than by steel tube under flexural testing). It was revealed that, for the case of the steel strength σ

_{y}= 382 N/mm

^{2}and σ

_{t}= 542 N/mm

^{2}, when the compressive strength of concrete was 41 N/mm

^{2}, the confined strength was 46 N/mm

^{2}, and when the compressive strength of concrete was 46 N/mm

^{2}, the confined strength was 50 N/mm

^{2}. However, the load-displacement curves and the PCFST’s ultimate loads for these two cases were almost identical, as shown in Figure 20. This confirmed the discussions in the Section “Finite Element Modelling” (ii) Compressive strength, and in the Section “Finite Element modelling results and validation”.

## 6. Conclusions

- The comparisons show excellent agreements between the experimental test and Finite Element results, including load-vertical displacement curves, longitudinal strain measurements at some cross-sections, buckling failure modes, and neutral axis positions in the transition area between steel tube and steel tube with concrete fill. Based on this validation, a comprehensive parametric study was then conducted to investigate the effects of the tube diameter-to-thickness ratio, the concrete filling length ratio, the compressive strength of concrete, and the tube steel’s yield and tensile strengths on the PCFST’s ultimate flexural strength and the confined strength of the concrete filling. Based on these studies, the confined strength of concrete predicted by FE modelling was evaluated considering the effects of different parameters and validating against the theoretical values.
- The results show that the ultimate load significantly increased when the tube diameter-to-thickness ratio D/t decreased, i.e., when D/t ratio decreased from 78 to 39, the ultimate load increased more than 95. When the D/t ratio increased, the tube became more vulnerable to failure by local buckling (buckled shapes formed around the loading point). It indicated the significant effects of increasing the tube’s thickness on increasing the PCFST’s ultimate flexural strength.
- An optimal length of the partial concrete filling could be determined considering the PCFST dimensions through D/t ratios. It was found that when the optimal length was about 66% of the tube length, the ultimate flexural strength of the PCFST was the same with that of the tube fully filed with concrete for its entire length. The PCFST with this optimal length of concrete filling defined the best flexural strength-to-weight ratio for the PCFST.
- The compressive strength of concrete did not have significant effects on the PCFST’s flexural strength, i.e., changing the compressive strength by 30% only resulted in a maximum difference of 2% for the PCFST’s ultimate loads.
- The material properties of the steel tube (yield stress and tensile strength) had significant effects on the PCFST’s flexural strength, i.e., when the steel strengths increased by 10%, the PCFST’s ultimate load could increase up to 14%.
- The confined stresses developed in the concrete filling predicted by FE modelling increased when the compressive strength increased, with a maximum difference of 9% for the same steel grade but this did not significantly affect the PCFST’s ultimate strength.
- The confined stresses developed in the concrete filling predicted by FE modelling increased when the steel’s yield stress and tensile strength increased, with a maximum difference of 5% and this significantly affected the PCFST’s ultimate strength, i.e., a maximum increase of 14%.
- The confined stresses developed in the concrete filling predicted by FE modelling remained unchanged when the concrete filling length was equal or greater than 66% of the tube length. However, it increased significantly, by 2.4 times when the tube’s thicknesses increased.
- The confined strengths predicted by FE analysis were in excellent agreement with those calculated by the theoretical model while the corresponding strains were lower than those calculated by the theoretical model.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Confined Concrete Model and Properties

_{1}and k

_{2}were constants and could be obtained from experimental data. In this study, values of k

_{1}and k

_{2}were set as 4.1 and 20.5 based on the previous studies [44]. The value of ${\mathsf{\epsilon}}_{\mathrm{c}}^{\prime}$ was usually in the range of 0.002 to 0.003. A representative value used in the analysis was ${\mathsf{\epsilon}}_{\mathrm{c}}^{\prime}$ = 0.002. The equivalent uniaxial stress-strain curve for concrete was assumed to be linear before yielding when the concrete strain ${\mathsf{\epsilon}}_{\mathrm{c}}$ was less than ${\mathsf{\epsilon}}_{\mathrm{cc}}^{\prime}$. When ${\mathsf{\epsilon}}_{\mathrm{c}}$ was greater than ${\mathsf{\epsilon}}_{\mathrm{cc}}^{\prime}$ a linear descending line was used to model the softening behaviour of concrete, as shown in Figure 8. Assuming that the descending line was to be terminated at the point where ${\mathrm{f}}_{\mathrm{c}}={\mathrm{k}}_{3}{\mathrm{f}}_{\mathrm{cc}}^{\prime}$ and ${\mathsf{\epsilon}}_{\mathrm{c}}=11{\mathsf{\epsilon}}_{\mathrm{cc}}^{\prime}$ in which k

_{3}was considered as the material degradation parameter. The two parameters ${\mathrm{f}}_{\mathrm{l}}$ and ${\mathrm{k}}_{3}$ were depended on the diameter-to-thickness ratio (D/t), cross-sectional shape, and stiffening mean. Their appropriate values were determined through the following equations which were obtained by matching the numerical results with experimental results (Hu et al., 2005).

^{2}and 0.7698, respectively. The concrete compressive strengths of 46 N/mm

^{2}at a strain of 0.003 was also used in the FE models.

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**Figure 1.**Maximum bending moment distributions for a free-head pile-column with the above ground height H = 10 m and the tube diameter b = d = 1.5 m (Gerolymos et al. [29]).

**Figure 4.**Three-point bending test setup (

**a**) schematic diagram of test setup, (

**b**) a three-dimensional view of the test setup. The small box shows a closer view of spiral ribs inside the steel tube and concrete filling.

**Figure 5.**Locations of strain gauges and displacement sensors (

**a**) in longitudinal direction, strain gauges are on cross sections A-H and displacement sensors (defined by arrow symbols with numbers 1–8), (

**b**) strain gauges’ locations on cross sections.

**Figure 7.**FE model of the bending test setup including boundary conditions and a closer view of the mesh around the mid-span of the PCFST specimen.

**Figure 9.**Material points of steel tube and concrete filling at the outer fibres at mid-span cross section where the FE stress-strain data were extracted for investigating the PCFST behaviour and concrete confinement (

**a**) mid-span cross section and material points, (

**b**) steel tube, and (

**c**) concrete filling. The contour blue and red colours represent the maximum compressive and tensile stresses in the tube’s longitudinal direction.

**Figure 10.**Strain data at cross section H (

**a**) strain measurement locations, and (

**b**) strain-load curves for longitudinal strains (even numbers) and hoop strains (odd numbers). The red square marked the position of the ultimate load.

**Figure 11.**The comparison of load-vertical displacement curves between experimental results and FE analysis (presented for the first 14 cycles of the loading-unloading process) “Envelope” is the envelope of load values of the experimental curve not including unloading-loading values.

**Figure 12.**The stress-strain curves at material points of (

**a**) steel tube where the box shows a closer view of stress-strain curves in compression, and (

**b**) concrete filling. The positive values are for “tensile” stresses and the negative values are for “compressive” stresses.

**Figure 13.**The local buckling of the PCFST specimen: (

**a**) testing, and (

**b**) FE deformed shape, and (

**c**) stress distribution (red colours indicate maximum stress in tension, and blue colours indicate maximum stress in compression).

**Figure 14.**Stress contour and local buckling shapes of the PCFST (

**a**) at the end of the 1st cycle, (

**b**) at the end of the 2nd cycle, and (

**c**) at the end of the 12th cycle, in which blue areas indicate large compression stresses. In this figure, Z is the longitudinal axis of the PCFST specimen.

**Figure 15.**The behaviour of concrete filling in the PCFST specimen at the end of the 12th cycle (

**a**) normal stress distribution, and (

**b**) cracking strain distribution with a crack band under loading point, in which yellow and red areas indicate large cracking strain. Negative values are for compressive stresses.

**Figure 16.**Experimental and FE analysis’s load-longitudinal strain responses at the cross-section H position (strains were measured at loads up to 900 kN). Compression strain values are negative, and tension strain values are positive.

**Figure 17.**Neutral axis positions in the transition region measured by experiment and predicted by Finite Element analysis.

**Figure 18.**Enveloped load-displacement curves of different concrete filling length ratios at the tube diameter-to-thickness ratio D/t = 78.

**Figure 19.**Variation of maximum loads with concrete fill length ratio for different tube dimensions D/t.

**Figure 20.**Enveloped load-displacement curves at different values of the compressive strength of concrete, for the case of the steel strength σ

_{y}= 382 N/mm

^{2}and σ

_{t}= 542 N/mm

^{2}.

**Figure 21.**Stress-strain curves at material points of concrete at different values of the compressive strength of concrete, for the case of the steel strength σ

_{y}= 382 N/mm

^{2}and σ

_{t}= 542 N/mm

^{2}.

Material | Weight (kg/m^{3}) | Volume (m^{3}) |
---|---|---|

Water | 179 | 0.179 |

Cement | 350 | 0.112 |

Coarse aggregates (size: 5–20 mm) | 1010 | 0.015 |

Sand | 835 | 0.319 |

Admixtures | 1.4 | 0.001 |

Air (1.5%) | ||

Water/Cement ratio = 51% | ||

Cement/Sand ratio = 42% |

**Table 2.**Effects of the tube diameter-to-thickness ratio and the concrete filling length on the PCFST’s ultimate loads.

Concrete Filling Length Ratio | $\mathbf{Ultimate}\mathbf{Load}{\mathbf{P}}_{\mathbf{u}}\left(\mathbf{k}\mathbf{N}\right)$ | ||
---|---|---|---|

$\frac{{\mathbf{L}}_{\mathbf{f}}}{\mathbf{L}}$ | $\frac{\mathbf{D}}{\mathbf{t}}=39$ | $\frac{\mathbf{D}}{\mathbf{t}}=78$ | $\frac{\mathbf{D}}{\mathbf{t}}=140$ |

(1) | (2) | (3) | (4) |

0.00 | 1780 | 807 | 505 |

0.34 | 1824 | 817 | 515 |

0.50 | 2029 | 888 | 560 |

0.60 | 2300 | 1105 | 705 |

0.66 | 2333 | 1200 | 760 |

1.00 | 2335 | 1210 | 795 |

**Table 3.**Effects of the compressive strength of concrete and yield stress and tensile strength of steel tube on the PCFST’s ultimate load and the confined strength of concrete predicted by Finite Element Analysis and Theoretical Analysis. ${\mathrm{f}}_{\mathrm{c}}^{\prime}$ is the compressive strength of concrete, ${\mathrm{f}}_{\mathrm{cc}\_\mathrm{FEA}}^{\prime}$ is the confined strength of concrete predicted by FEA, ${\mathsf{\epsilon}}_{\mathrm{cc}\_\mathrm{FEA}}^{\prime}$ is the strain at the confined strength ${\mathrm{f}}_{\mathrm{cc}\_\mathrm{FEA}}^{\prime}$, ${\mathrm{f}}_{\mathrm{cc}\_\mathrm{FEA}}^{\u2033}$ is the compressive strength at the PCFST’s ultimate load predicted by FEA, ${\mathrm{f}}_{\mathrm{cc}\_\mathrm{THEO}}^{\prime}$ is the confined strength of concrete predicted by theoretical model [38], ${\mathsf{\epsilon}}_{\mathrm{cc}\_\mathrm{THEO}}^{\prime}$ is the strain at the confined strength ${\mathrm{f}}_{\mathrm{cc}\_\mathrm{THEO}}^{\prime}$.

Compressive Strength | Steel Yield and Ultimate Strength | Ultimate Load | Finite Element Analysis (FEA) | Theoretical Analysis (THEO) | Comparison | ||||
---|---|---|---|---|---|---|---|---|---|

${\mathbf{f}}_{\mathbf{c}}^{\prime}\left(\mathbf{N}/\mathbf{m}{\mathbf{m}}^{2}\right)$ | ${\mathsf{\sigma}}_{\mathbf{y}}\mathbf{a}\mathbf{n}\mathbf{d}{\mathsf{\sigma}}_{\mathbf{t}}\left(\mathbf{k}\mathbf{N}\right)$ | ${\mathbf{P}}_{\mathbf{u}}\left(\mathbf{k}\mathbf{N}\right)$ | ${\mathbf{f}}_{\mathbf{c}\mathbf{c}\_\mathbf{F}\mathbf{E}\mathbf{A}}^{\prime}$ | ${\mathbf{\epsilon}}_{\mathbf{c}\mathbf{c}\_\mathbf{F}\mathbf{E}\mathbf{A}}^{\prime}$ | ${\mathbf{f}}_{\mathbf{c}\mathbf{c}\_\mathbf{F}\mathbf{E}\mathbf{A}}^{\u2033}$ | ${\mathbf{f}}_{\mathbf{c}\mathbf{c}\_\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{O}}^{\prime}$ | ${\mathbf{\epsilon}}_{\mathbf{c}\mathbf{c}\_\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{O}}^{\prime}$ | $\frac{{\mathbf{f}}_{\mathbf{c}\mathbf{c}\_\mathbf{F}\mathbf{E}\mathbf{A}}^{\prime}}{{\mathbf{f}}_{\mathbf{c}\mathbf{c}\_\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{O}}^{\prime}}$ | $\frac{{\mathbf{\epsilon}}_{\mathbf{c}\mathbf{c}\_\mathbf{F}\mathbf{E}\mathbf{A}}^{\prime}}{{\mathbf{\epsilon}}_{\mathbf{c}\mathbf{c}\_\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{O}}^{\prime}}$ |

(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) |

σ_{y} = 334, σ_{t} = 488 | 1049 | 40 | 0.0015 | 52 | 40 | 0.0033 | 1.00 | 0.45 | |

36 | σ_{y} = 382, σ_{t} = 542 | 1173 | 42 | 0.0015 | 53 | 41 | 0.0035 | 1.02 | 0.43 |

σ_{y} = 458, σ_{t} = 650 | 1343 | 42 | 0.0015 | 53 | 43 | 0.0038 | 0.98 | 0.39 | |

σ_{y} = 334, σ_{t} = 488 | 1053 | 44 | 0.0017 | 56 | 45 | 0.0031 | 0.98 | 0.55 | |

41 | σ_{y} = 382, σ_{t} = 542 | 1200 | 46 | 0.0017 | 58 | 46 | 0.0033 | 1.00 | 0.52 |

σ_{y} = 458, σ_{t} = 650 | 1359 | 47 | 0.0017 | 60 | 48 | 0.0035 | 0.98 | 0.49 | |

σ_{y} = 334, σ_{t} = 488 | 1064 | 49 | 0.0019 | 59 | 50 | 0.0030 | 0.98 | 0.63 | |

46 | σ_{y} = 382, σ_{t} = 542 | 1200 | 50 | 0.0019 | 60 | 51 | 0..0031 | 0.98 | 0.61 |

σ_{y} = 334, σ_{t} = 488 | 1364 | 52 | 0.0019 | 64 | 53 | 0.0034 | 0.98 | 0.56 |

**Table 4.**Effects of the tube diameter-to-thickness ratio and concrete filling length ratio on the PCFST’s ultimate load and the confined strength of concrete predicted by Finite Element Analysis and Theoretical Analysis. ${\mathrm{f}}_{\mathrm{cc}\_\mathrm{FEA}}^{\prime}$ is the confined strength of concrete predicted by FEA, ${\mathsf{\epsilon}}_{\mathrm{cc}\_\mathrm{FEA}}^{\prime}$ is the strain at the confined strength ${\mathrm{f}}_{\mathrm{cc}\_\mathrm{FEA}}^{\prime}$, ${\mathrm{f}}_{\mathrm{cc}\_\mathrm{FEA}}^{\u2033}$ is the compressive strength at the PCFST’s ultimate load predicted by FEA, ${\mathrm{f}}_{\mathrm{cc}\_\mathrm{THEO}}^{\prime}$ is the confined strength of concrete predicted by theoretical model [38], ${\mathsf{\epsilon}}_{\mathrm{cc}\_\mathrm{THEO}}^{\prime}$ is the strain at the confined strength ${\mathrm{f}}_{\mathrm{cc}\_\mathrm{THEO}}^{\prime}$.

Diameter-to-Thickness Ratio | Concrete Filling Length Ratio | Ultimate Load | Finite Element Analysis (FEA) | Theoretical Analysis (THEO) | Comparison | ||||
---|---|---|---|---|---|---|---|---|---|

$\frac{\mathbf{D}}{\mathbf{t}}$ | $\frac{{\mathbf{L}}_{\mathbf{f}}}{\mathbf{L}}$ | ${\mathbf{P}}_{\mathbf{u}}\left(\mathbf{k}\mathbf{N}\right)$ | ${\mathbf{f}}_{\mathbf{c}\mathbf{c}\_\mathbf{F}\mathbf{E}\mathbf{A}}^{\prime}$ | ${\mathbf{\epsilon}}_{\mathbf{c}\mathbf{c}\_\mathbf{F}\mathbf{E}\mathbf{A}}^{\prime}$ | ${\mathbf{f}}_{\mathbf{c}\mathbf{c}\_\mathbf{F}\mathbf{E}\mathbf{A}}^{\prime \prime}$ | ${\mathbf{f}}_{\mathbf{c}\mathbf{c}\_\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{O}}^{\prime}$ | ${\mathbf{\epsilon}}_{\mathbf{c}\mathbf{c}\_\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{O}}^{\prime}$ | $\frac{{\mathbf{f}}_{\mathbf{c}\mathbf{c}\_\mathbf{F}\mathbf{E}\mathbf{A}}^{\prime}}{{\mathbf{f}}_{\mathbf{c}\mathbf{c}\_\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{O}}^{\prime}}$ | $\frac{{\mathbf{\epsilon}}_{\mathbf{c}\mathbf{c}\_\mathbf{F}\mathbf{E}\mathbf{A}}^{\prime}}{{\mathbf{\epsilon}}_{\mathbf{c}\mathbf{c}\_\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{O}}^{\prime}}$ |

(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) |

39 | 0.66 | 2333 | 120 | 0.0011 | 120 | 64 | 0.0058 | 1.88 | 0.19 |

1.00 | 2335 | 123 | 0.0011 | 123 | 64 | 0.0058 | 1.92 | 0.19 | |

78 | 0.66 | 1200 | 50 | 0.0019 | 60 | 51 | 0.0031 | 0.98 | 0.61 |

1.00 | 1210 | 50 | 0.0019 | 60 | 51 | 0.0031 | 0.98 | 0.61 | |

140 | 0.66 | 760 | 50 | 0.0019 | 50 | 48 | 0.0024 | 1.04 | 0.79 |

1.00 | 795 | 50 | 0.0019 | 56 | 48 | 0.0024 | 1.04 | 0.79 |

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## Share and Cite

**MDPI and ACS Style**

Nguyen, T.T.T.; Nguyen, V.B.; Thai, M.Q.
Flexural Strength of Partially Concrete-Filled Steel Tubes Subjected to Lateral Loads by Experimental Testing and Finite Element Modelling. *Buildings* **2023**, *13*, 216.
https://doi.org/10.3390/buildings13010216

**AMA Style**

Nguyen TTT, Nguyen VB, Thai MQ.
Flexural Strength of Partially Concrete-Filled Steel Tubes Subjected to Lateral Loads by Experimental Testing and Finite Element Modelling. *Buildings*. 2023; 13(1):216.
https://doi.org/10.3390/buildings13010216

**Chicago/Turabian Style**

Nguyen, Thi Tuyet Trinh, Van Bac Nguyen, and Minh Quan Thai.
2023. "Flexural Strength of Partially Concrete-Filled Steel Tubes Subjected to Lateral Loads by Experimental Testing and Finite Element Modelling" *Buildings* 13, no. 1: 216.
https://doi.org/10.3390/buildings13010216