*3.1. Failure Modes and Crack Propagation*

The failure modes of the FITRC and CRC columns under large eccentric loads were similar. The failure of all specimens was manifested by longitudinal bar yielding in the tension zone, followed by concrete crushing in the compression zone (Figure 8). The eccentricity of the FITRC45 and CRC45 columns was 0.60 and that of the FITRC35 and CRC35 columns was 0.70. In a previous study [55], the eccentric compressive behaviour of iron tailings sand RC columns was investigated, and similar results were obtained.

Cracks occurred in the tension zone at the midspan point along the lateral depth of the FITRC45 and CRC45 columns when the load reached approximately 0.2 *Nu* and propagated with increasing load, resulting in new microcracks. When the load reached approximately 0.9 *Nu*, vertical cracks were observed in the lateral compression zone of the column specimens and began to propagate. When the ultimate load *Nu* was reached, the concrete cracks in the compression zone elongated and were accompanied by the appearance of new microcracks. Some of the concrete was crushed as the test specimen continued to be loaded, and the ultimate compressive strain of the concrete in the compression zone was reached; thus, the axial load on the column specimens decreased. Similar failure processes were observed for FITRC35 and CRC35 columns.

Regardless of whether the eccentricity was 0.60 or 0.70, the final failure of all the RC columns was demonstrated by the crushed concrete in the outer layer of the stirrups and the intact concrete in the inner layer of the stirrups. This can be attributed to the fact that the stirrups significantly restrained the concrete in the inner layer and inhibited its deformation. In addition, Figure 8 shows that the area of crushed concrete was related to the eccentricity of the specimen section; that is, a smaller eccentricity indicated a larger area of crushed concrete.

Figure 9 shows the load–maximum crack width curves of the RC columns. The crack width propagation trends of FITRC and CRC columns remained similar for eccentricities of 0.60 and 0.70, and the curves of the four RC columns overlapped. Considering that the splitting tensile strength of FITC was lower than that of CC, the crack width of FITRC columns was larger than that of CRC columns at the same load level.

**Figure 9.** Load–maximum crack width curves of RC columns: (**a**) *e*0/*h*<sup>0</sup> = 0.60; (**b**) *e*0/*h*<sup>0</sup> = 0.70.

#### *3.2. Load–Deflection Relationships*

Figure 10 shows the axial load–deflection relationship (axial and lateral displacements) of RC columns under large eccentric loads. Axial and lateral displacements were measured using a vertical displacement gauge and a displacement gauge at the midspan point along the column depth, respectively. The negative and positive values in the figure indicate the axial and lateral displacements, respectively, at the midspan point along the column depth. Prior to longitudinal bar yielding, there was an approximately linear relationship between the loads and displacements of all the RC columns. After tensile yielding of the longitudinal bars, the stiffness of the RC columns decreased and the load-deflection curve increased nonlinearly until the peak load was reached when the concrete cover was crushed and spalled. After the peak load, the curve shows a decreasing trend; the load of the RC columns showed a rapid reduction of 5–10% followed by a more gradual reduction, and ductile failures occurred in the specimens. The vertical and lateral displacements of the CRC columns were clearly smaller than those of the FITRC columns. In particular, there were smaller lateral displacements compared to vertical displacements, considering that the modulus of elasticity of CC was approximately 19% greater than that of FITC (Table 6). Similarly, the peak loads of the CRC columns were slightly higher than those of the FITRC columns (Table 8).

**Figure 10.** Load–deflection curves of RC columns: (**a**) *e*0/*h*<sup>0</sup> = 0.60; (**b**) *e*0/*h*<sup>0</sup> = 0.70.


**Table 8.** Main performance indices of RC columns.

Figure 11 shows the lateral displacements of four typical RC columns distributed along the column depth (measured using five displacement gauges). The RC columns exhibited similar lateral displacements at the different loading stages. Furthermore, the columns showed slow and rapid increases in lateral displacement in the early loading stages and as the peak load approached, respectively. The lateral deformations of the RC columns were caused by first- and second-order moments. In accordance with the literature [56], the lateral deformations and sinusoidal waveforms of columns hinged at both ends remained similar. Therefore, the lateral deformation of the RC columns can be expressed as follows:

$$D\_L = \Delta\_p \sin\left(\frac{\pi \cdot l}{L}\right) \tag{1}$$

where Δ*<sup>p</sup>* is the maximum displacement at the midspan point along the column depth, *l* is the location along the column depth, *L* is the column depth, and *DL* is the lateral displacement of the column at l. Figure 11 compares the lateral deformations of the RC columns and the sinusoidal model, which agree well for eccentricities of both 0.60 and 0.70, thereby indicating that the sinusoidal model can effectively predict the lateral deformations of FITRC columns at different loading stages.

#### *3.3. Deformation and Ductility*

Table 8 lists the primary test results of the RC columns, including the peak load *Nu*, displacement at the corresponding peak load Δ*p*, and yield displacement Δ*y*. The behaviour parameters of the RC columns were quantified to determine the deformation capacity of the RC columns. In seismic design, the inelastic deformation capacity of specimens is generally

quantified using the displacement ductility factor [57] and deformation coefficient [58]. The displacement ductility factor indicates the ductility behaviour of the specimens, which can be calculated as follows.

$$
\mu = \frac{\Delta\_{0.85}}{\Delta\_y} \tag{2}
$$

The deformation coefficient indicates the deformation capacity of the specimens after reaching the peak load and can be calculated as follows.

$$
\lambda = \frac{\Delta\_{0.85}}{\Delta\_p} \tag{3}
$$

**Figure 11.** Lateral deformation along the section depth of typical RC columns: (**a**) FITRC45-2 column specimen; (**b**) CRC45 column specimen; (**c**) FITRC35-2 column specimen; (**d**) CRC35 column specimen.

Here, Δ0.85 is the corresponding displacement at *Nu* of 0.85 in the load decreasing stage [57], Δ*<sup>p</sup>* is the peak lateral deflection at the peak load, and Δ*<sup>y</sup>* is the corresponding displacement at the yield load when the limit is reached in the elastic stage. The yield displacement Δ*<sup>y</sup>* can be obtained using a graphical method [57] (Figure 12).

**Figure 12.** Definitions of yield displacement Δ*<sup>y</sup>* and ultimate displacement Δ0.85.

Table 8 shows that the mean ductility factors of the FITRC45 and FITRC35 columns were 1.78 and 2.82, respectively, and the ductility factors relative to the CRC45 and CRC35 columns were reduced by 36.6% and 19.5%, respectively. Therefore, an increase in section size and eccentricity reduced the difference in ductility between FITRC35 and CRC35 columns. A previous study [52] examined the axial compressive properties of full iron tailings RC columns and also found that the ductility coefficients of FITRC columns were lower than those of CRC columns. The mean deformation coefficients of the FITRC45 and FITRC35 columns were 1.33 and 1.35, respectively, and the deformation coefficients relative to CRC45 and CRC35 columns were reduced and increased by 5.27% and 2.80%, respectively. In the load-decreasing phase, the deformation capacities of FITRC and CRC columns were identical.

As can be seen from Table 8, the lateral deflection Δ*<sup>p</sup>* corresponding to the peak load of the FITRC columns was greater than that corresponding to the peak load of the CRC columns, and the mean values of Δ*<sup>p</sup>* of the FITRC45 and FITRC35 columns increased by 25% and 21%, respectively, compared to the CRC45 and CRC35 columns.

#### *3.4. Load–Strain Relationships*

Figure 13 shows the load–rebar strain curves of typical RC columns, where *c* is *Asc* of the compressive rebars, which were close to the axial compressive force with a negative strain value, and *t* is *Ast* of the tensile rebars, which were away from the axial compressive force (Figure 7) with a positive strain value. ε*y*<sup>14</sup> and *εy*<sup>16</sup> are the tensile yield strains of rebars with diameters of 14 and 16 mm, respectively. Figure 13 shows that the strains of *Ast* in the RC columns were greater than ε*y*<sup>14</sup> and *εy*16, and the yield strain of rebars in the compression zone was reached before the peak load was reached. The stress state is a typical feature of compression failure at large eccentricities. In addition, the strain of rebars in the FITRC columns was significantly higher than that in the CRC columns for both *Asc* and *Ast*, because FITC had a lower modulus of elasticity than CC. In addition, the prismatic compressive and cracking strengths of FITC were slightly lower than those of CC. As a result, the ultimate load of the FITC columns was lower than that of the CRC columns, while the lateral deflection and axial displacement of the FITC columns are greater than those of the CRC columns.

Concrete strain gauges were placed along the section depth of the RC column, as shown in Figure 7a, to study the strain distribution of the concrete during loading. Figure 14 shows the strain distribution of typical concrete. Under large eccentric loads, the concrete in the tension zone of the RC columns cracked, resulting in failure of the concrete strain gauges near the tension zone. In particular, the strain distributions in the concrete before and after cracking were recorded. Figure 13 shows that the concrete strain was linearly distributed along the depth of the RC column section. Therefore, FITRC columns satisfied the planar section assumption, and the flexural capacity of FITRC columns can be theoretically calculated according to the planar section assumption.

**Figure 13.** Load–rebar strain curves of typical RC columns: (**a**) *e*0/*h*<sup>0</sup> = 0.60; (**b**) *e*0/*h*<sup>0</sup> = 0.70.

**Figure 14.** Typical concrete strain distribution along the section depth at the mid-span point: (**a**) FITRC45-2 column specimen; (**b**) CRC45 column specimen; (**c**) FITRC35-2 column specimen; (**d**) CRC35 column specimen.

#### **4. Analysis of Sectional Capacities**

#### *4.1. Moment Magnification Factor*

Considering the axial and flexural deformations that occurred in the RC columns under large eccentric loads during the experiment, the axial capacity, moment, and crack resistance should be calculated across the section at the mid-span of the column depth. Figure 15 shows a schematic of the second-order effects of the RC columns. Under the eccentric axial load N with initial eccentricity *e*0, the lateral displacement Δ*<sup>p</sup>* of the RC columns across the column depth at the midspan allowed the eccentricity of the axial load relative to the centre of mass of the section to reach Δ*<sup>p</sup>* + *e*0. Meanwhile, the moment of the RC columns increased from *M*<sup>1</sup> = *Ne*<sup>0</sup> to *Mmax* = *M*<sup>1</sup> + *M*<sup>2</sup> = *N* (*e*<sup>0</sup> + Δ*p*), which is known as the second-order effect of eccentrically loaded columns [59,60]. During design, the second-order effects are accounted for using the moment augmentation factor [61].

**Figure 15.** Schematic diagram of the second-order effect.

According to [62], the relationship between the ultimate sectional curvature *ϕ<sup>p</sup>* and Δ*<sup>p</sup>* of RC columns can be expressed as:

$$
\varphi\_p = \Delta\_p \left(\frac{\pi}{L}\right)^2 \tag{4}
$$

This equation is valid only if the first- and second-order deformations of a column can be expressed as sinusoidal shapes.

Therefore, the moment augmentation factor *η* can be expressed as:

$$\eta = \frac{e\_0 + \Delta\_p}{e\_0} = 1 + \frac{\varphi\_p L^2}{e\_0 \pi^2} \tag{5}$$

In the design code, an additional eccentricity *ea* is introduced because of uncertain load locations, uneven concrete quality, and construction variations, and a value of 20 mm is considered with an eccentricity of *ei* = *e*<sup>0</sup> + *ea*.

#### *4.2. Bearing Capacity*

In general, the bearing capacity under eccentric compression is calculated using the equivalent rectangular stress diagram [61], assuming that the FITRC columns satisfy the plane section assumption, and the theoretical calculation can be performed according to GB 50010-2010 [61]. Furthermore, to simplify the calculation process, the tensile strength of the concrete was ignored. Figure 16 shows a simplified diagram for calculating the section of an RC column subjected to compression failure under a large eccentric load.

**Figure 16.** Simplified diagram for calculation of the section subjected to compression failure under large eccentric loading: (**a**) strain diagram of the specimens; (**b**) equivalent calculation diagram.

The compressive stress curve of the concrete in the compression zone was replaced with an equivalent rectangular diagram. According to the equilibrium conditions of the forces, the following formula was obtained.

$$N\_u = a\_1 f\_c b x + f\_{yc} A\_{sc} - f\_{yt} A\_{st} \tag{6}$$

Here, *α*<sup>1</sup> is the equivalent rectangular stress block coefficient of concrete in the compression zone, defined as 1.0; *b* and *h* are the sectional dimensions of the RC columns; *fyt* and *fyc* are the yield strengths of the tensile and compressive longitudinal bars, respectively; *Ast* and *Asc* are the sectional areas of the tensile and compressive longitudinal bars, respectively. According to the resultant point of various forces on the tensile specimens in Figure 16b, the moment equilibrium conditions were determined and expressed as:

$$N\_{\rm u}e = a\_1 f\_c bx(h\_0 - 0.5x) + f\_{\rm yc} A\_{\rm sc} (h\_0 - a\_{\rm sc}) \tag{7}$$

$$e = \eta e\_i + 0.5h - a\_{st} \tag{8}$$

where *e* is the distance from the point of axial force to the resultant point of *Ast* of tensile longitudinal bars, *h*<sup>0</sup> is the distance from the resultant point of *Ast* of tensile longitudinal bars to the edge of the compressive concrete, *x* is the depth of concrete in the compression zone, *ast* is the distance from the resultant point of *Ast* of tensile longitudinal bars to the edge of the tensile concrete, and *asc* is the distance from the resultant point of *Asc* of compressive longitudinal bars to the edge of the compressive concrete.

#### *4.3. Crack-Resistant Load*

According to the SL191-2008 standard in China [63], the crack resistance of RC columns under eccentric compression should be calculated as follows:

$$N\_{cr} \leq \frac{\gamma\_m a\_{ct} f\_t A\_0 \mathcal{W}\_0}{e\_i A\_0 - \mathcal{W}\_0} \tag{9}$$

where *γ<sup>m</sup>* is the plastic section modulus, with the rectangular section set to 1.55; *αct* is the tensile stress limit coefficient of concrete, offset to 0.85; *A*<sup>0</sup> is the area of the transformed section; and *W*<sup>0</sup> is the elastic section modulus of the tensile edge of the transformed section, calculated as:

$$A\_0 = bh + \alpha\_E A\_{st} + \alpha\_E A\_{sc} \tag{10}$$

$$\mathcal{W}\_0 = I\_0 / (h - y\_0) \tag{11}$$

$$I\_0 = (0.0833 + 0.19\alpha\_E \rho) b h^3 \tag{12}$$

$$y\_0 = (0.5 + 0.425\alpha\_E \rho)h \tag{13}$$

where *I*<sup>0</sup> is the moment of inertia of the column section, *y*<sup>0</sup> is the distance from the axis of gravity of the transformed section to the compression edge, and *ρ* is the reinforcement ratio of the tensile longitudinal bars.

#### *4.4. Experimental Verification of Theoretical Predictions*

According to the aforementioned formulae, the moment augmentation factor and bearing capacity of the RC columns were calculated using the measured mechanical parameters of the rebar and concrete, respectively, and compared with the experimental results. As shown in Table 9, the theoretically calculated cracking load and ultimate bearing capacity were in good agreement with the measured results, with a maximum error of only 16%. In addition, there was a strength safety margin, which verifies that the calculation formulae are effective for FITRC columns under large eccentric loads.

**Table 9.** Bearing capacity of RC columns.


Table 10 lists the moments of the RC columns and the calculated moment augmentation factor. According to the structural design code [61], the moment augmentation factor *η* should be 1.0 when *L*/*h* ratios do not exceed 5.0. Furthermore, the moment augmentation factor of the FITRC columns is greater than that of the CRC columns in the control group. The measured moment augmentation factors of the six FITRC columns are close to each other, with a maximum value of 1.079, whereas the *η* value of the FITRC columns increases by a maximum of 1.5% compared to the CRC columns. This is because the *L*/*h* ratios of the RC columns in this study do not exceed 5.0 and the impact of the second-order effects is marginal.



In addition, Table 10 shows that the second-order moments of the FITRC columns are larger than those of the CRC columns. Furthermore, the average ratio of the second-order moments of the three FITRC45 columns to the CRC45 columns is 1.14, and that of the three FITRC35 columns to the CRC35 columns is 1.15. Therefore, if the second-order effect is to be considered in the structural design, the moment augmentation factor *η* of the FITRC columns should be 1.15 for safety reasons.

#### **5. Conclusions**

This study investigated the structural behaviour of FITRC columns under large eccentric loading. Six FITRC columns and two CRC column specimens were examined to investigate the effect of different raw materials, section dimensions, and eccentricities, on the mechanical behaviour of RC columns under large eccentric compression. The following conclusions were drawn from this study:


The ductility factor of FITRC columns is much lower than that of CRC columns, so the ductility of FITRC columns can theoretically be improved by stirrup reinforcement. However, this hypothesis requires further investigation.

**Author Contributions:** X.M.: Conceptualisation, methodology, validation, investigation, data curation, and writing of the original draft. J.S.: Conceptualisation, methodology, writing—review and editing, and supervision. F.Z.: Validation, data curation, resources, and supervision. J.Y.: Investigation and resources. M.Y.: Investigation and resources. Z.M.: Validation and visualisation. Y.B.: Investigation and resources. Y.L.: Investigation and resources. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Hebei Province Key Research and Development Project, China [grant number 19211502D], and the Hebei Province Graduate Innovation Funding Project, China [grant number CXZZBS2018109].

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
