# Numerical Investigation on Load-carrying Capacity of High-strength Concrete-encased Steel Angle Columns

- Chang-Soo Kim
^{1}and - Hyeon-Jong Hwang
^{2}Email author

**12**:11

https://doi.org/10.1186/s40069-018-0238-7

© The Author(s) 2018

**Received: **14 January 2017

**Accepted: **7 January 2018

**Published: **31 January 2018

## Abstract

To investigate the load-carrying capacity of high-strength concrete-encased steel angle (CES-A) columns, in which corner steel angles are encased in concrete and transmit column loads directly, a numerical study was performed by using a proposed analysis model. The proposed model considered the strain compatibility, confinement effect, local buckling, and premature cover-spalling, and was verified against previous experimental study results. To investigate the effect of design parameters, a parametric study was conducted, and based on the parametric study results, a simple approach was also discussed to predict the residual strength (2nd peak load) after spalling of concrete cover at corners (1st peak load). The numerical investigations showed that when steel contribution and confinement efficiency are high, CES-A columns exhibit relatively large load-carrying capacity even after cover-spalling, due to the maintained strength of confined concrete and yielding of steel angles, and the proposed simple approach gave a good prediction for the residual strength.

## Keywords

## 1 Introduction

Steel–concrete composite columns such as concrete-encased steel (CES) and concrete-filled steel tube (CFT) columns have large load-carrying capacity and high local stability due to composite action, and high-strength materials improve structural safety and space efficiency. Thus, the use of high-strength composite columns is growing in the construction of high-rise and long-span structures.

When high-strength steel is used for conventional CES columns (consisting of a wide-flange steel core and concrete encasement), early crushing of concrete encasement needs to be considered, because the steel core may not develop its full plastic strength at the concrete failure, particularly under flexural loading (Kim et al. 2012, 2014). On the other hand, CFT columns using high-strength steel show excellent structural performance, because the steel tube provides good lateral confinement to concrete core and the concrete core restrains local buckling of the steel tube (Kim et al. 2014). However, in terms of fire proofing, local instability, diaphragm connections, and concrete compaction, CES columns still have advantages over CFT columns. Thus, further studies are necessary for high-strength CES columns.

Structural steel angles for columns have been widely studied and used in various ways: (1) to externally strengthen existing RC columns with batten plates (i.e., steel jacketing for reconstruction or seismic retrofitting) (Critek 2001; Zheng and Ji 2008a, b; Montuori and Piluso 2009; Nagaprasad et al. 2009; Calderon et al. 2009; Badalamenti et al. 2010; Garzon-Roca et al. 2011a, b, 2012; Campione 2012a, b; Khalifa and Al-Tersawy 2014; Tarabia and Albakry 2014; Cavaleri et al. 2016); (2) to replace wide-flange members with the built-up members connected by battens (Hashemi and Jafari 2004); (3) to reduce laborious fieldwork of composite columns by prefabrication (Kim et al. 2011; Eom et al. 2014; Hwang et al. 2015, 2016); and/or (4) to improve structural capacity and cost efficiency of composite columns by maximizing the contribution of high-strength steel (Kim et al. 2014, 2017). According to the primary purpose, steel angles could be either encased in concrete or exposed, and could transmit column loads or not (providing confinement only). Especially for the steel jacketing, extensive studies are available. However, the discussions in the existing studies were limited to (1) low-strength materials (because the steel jacketing is to strengthen deficient columns) and (2) confinement of non-buckled steel angles (because the externally attached steel angles are generally not subjected to high compression), and (3) the effect of concrete cover was not involved (because the externally attached steel angles are exposed).

In the present study, to investigate the load-carrying capacity of high-strength CES-A columns, in which corner steel angles are encased in concrete and transmit column loads directly (i.e., composite columns), a numerical study was performed using a proposed analysis model. The proposed model considered the strain compatibility, confinement effect, local buckling, and premature cover-spalling. For verification, the predictions by the proposed model were compared with previous experimental study results (Kim et al. 2014, 2017; Eom et al. 2014; Hwang et al. 2016), and to supplement the test results, a numerical parametric study was conducted for various design parameters. To predict residual strength after spalling of concrete cover at corners, a simple approach was also discussed.

## 2 Previous Experimental Studies

*f*

_{ ys }= 444 MPa and compressive strength of concrete \( f^{\prime}_{c} \) = 23.5 MPa). The test parameters included the sectional ratio of steel angles and vertical spacing of links to investigate the confinement effect and the flexural, shear, and bond resistances (Table 1; Figs. 2, 3). Additionally, to evaluate seismic performance, Hwang et al. (2015, 2016) conducted cyclic load tests for CES-A columns and beam-column joints. The test results showed that, in the case of the concentrically loaded specimens, the uniaxial strength was greater than the design strength \( P_{o} = 0.85f^{\prime}_{c} A_{c} + f_{ys} A_{s} + f_{yl} A_{l} \) (

*A*

_{ c },

*A*

_{ s },

*A*

_{ l }= area of concrete, steel, or longitudinal bars, and

*f*

_{ yl }= yield strength of longitudinal bars) due to the confinement effect. In the case of the flexurally loaded specimens, crushing of concrete cover (corresponding to the ultimate compressive strain of

*ε*

_{ cu }= 0.003) did not control the maximum deformation due to the confinement effect, and ductile behavior was maintained until steel angles reached their fracture strain (ε

_{ cs }= 0.015).

Tests specimens (total 16) | E1–E4: Eccentric axial load tests (Eccentricity = 120 mm) C1–C5: Concentric axial load tests F1–F7: Flexural tests |

Dimensions (square section) (see Fig. 2) | \( b \) = 260 mm, \( b_{c} \) = 230 mm, \( L_{k} \) = 2620 mm for E’s \( b \) = 500 mm, \( b_{c} \) = 400 mm, \( L_{k} \) = 1500 mm for C1–C3 and F1–F4 \( b \) = 400 mm, \( b_{c} \) = 300 mm, \( L_{k} \) = 1500 mm for C4–C5 and F5–F7 |

Concrete | \( f^{\prime}_{c,u} \) = 103.6 MPa, \( \varepsilon_{co,u} \) = 0.0027, \( \varepsilon_{cu,u} \) = 0.003 for E1 \( f^{\prime}_{c,u} \) = 96.6 MPa, \( \varepsilon_{co,u} \) = 0.003, \( \varepsilon_{cu,u} \) = 0.003 for E2 and E3 \( f^{\prime}_{c,u} \) = 98.7 MPa, \( \varepsilon_{co,u} \) = 0.0032, \( \varepsilon_{cu,u} \) = 0.0032 for E4 \( f^{\prime}_{c,u} \) = 23.5 MPa, \( \varepsilon_{co,u} \) = 0.002, \( \varepsilon_{cu,u} \) = 0.003 for C’s and F’s |

Steel angles (equal-leg) | \( f_{ys} \) = 812 MPa, \( f_{us} \) = 868 MPa, \( b_{s} \) = 60 mm, \( t_{s} \) = 15 mm for E1 \( f_{ys} \) = 759 MPa, \( f_{us} \) = 884 MPa, \( b_{s} \) = 60 mm, \( t_{s} \) = 15 mm for E2 – E4 \( f_{ys} \) = 444 MPa, \( f_{us} \) = 689 MPa, \( b_{s} \) = 90 mm, \( t_{s} \) = 7 mm for C’s and F’s |

Transverse reinforcement (see Fig. 3) | Lattices of \( \phi 7 \) Bars (\( f_{yt} \) = 531 MPa, \( s_{t} \) = 50 mm) for E1 Links of D13 Bars (\( f_{yt} \) = 481 MPa, \( s_{t} \) = 100 mm) for E2 Battens of 6 \( \times \) 60 mm Plates (\( f_{yt} \) = 418 MPa, \( s_{t} \) = 210 mm) for E3 Spirals of D13 Bars (\( f_{yt} \) = 567 MPa, \( s_{t} \) = 100 mm) for E4 Links of D10 Bars (\( f_{yt} \) = 522 MPa) for C’s and F’s \( s_{t} \) = 100 mm for C2, C4, F2, and F5 \( s_{t} \) = 200 mm for C1, C3, C5, F1, F3, and F6 \( s_{t} \) = 300 mm for F4 and F7 |

Longitudinal reinforcement | 8-D13 (\( f_{yl} \) = 513 MPa, \( f_{ul} \) = 634 MPa) for E1 4-D19 (\( f_{yl} \) = 523 MPa, \( f_{ul} \) = 650 MPa) for C1 and F1 |

Also, the previous experimental study results (Kim et al. 2014, 2017; Eom et al. 2014; Hwang et al. 2015, 2016) showed that the strain compatibility method of ACI 318-14 (2014) with* ε*_{
cu
} = 0.003 underestimated the load-carrying capacity of the CES-A columns by neglecting the confinement effect of steel angles and transverse reinforcement on concrete core. On the other hand, the load-carrying capacity was less than the prediction by the plastic stress distribution method of Eurocode 4 (2005) due to failure (spalling or crushing) of concrete cover earlier than yielding of steel angles. As such, since the load-carrying capacity of CES-A columns is strongly affected by the behavior of concrete cover and steel angles, further studies are required for those effects.

## 3 Nonlinear Numerical Analysis

### 3.1 Concrete and Confinement

- (1)Concrete cover is a protection of steel and reinforcement against corrosion and fire (American Concrete Institute 2014), and premature spalling of concrete cover may occur due to shrinkage of concrete and weakness planes between concrete cover and concrete core, which are created by longitudinal and transverse reinforcements (Collins et al. 1993; Cusson and Paultre 1994). This phenomenon is more obvious when higher strength concrete and denser reinforcement are used (Collins et al. 1993; Cusson and Paultre 1994). Especially in CES-A columns, the premature cover-spalling is more pronounced at corners due to the smooth surface of steel angles (Kim et al. 2014, 2017). Thus, for the stress \( f_{c,u} \) of concrete cover at corners, the proposed model assumed Eq. (2) (case 1 in Fig. 5). In the absence of experimental data, the ultimate strain of \( \varepsilon_{cu,u} \) = 0.003 is recommended based on test results (Kim et al. 2014, 2017).$$ f_{c,u} = 0\;{\text{if}}\;\varepsilon_{c} > \varepsilon_{c,u} \;{\text{for}}\;{\text{concrete}}\;{\text{cover}}\;{\text{at}}\;{\text{corners}} $$(2)
- (2)Steel angles provide good confinement to concrete core (Calderon et al. 2009; Montuori and Piluso 2009; Nagaprasad et al. 2009; Badalamenti et al. 2010; Kim et al. 2014, 2017; Eom et al. 2014; Hwang et al. 2015, 2016), but the effect of local buckling on the confinement should be also considered. Thus, unlike the existing models (Calderon et al. 2009; Montuori and Piluso 2009; Nagaprasad et al. 2009; Badalamenti et al. 2010), in which the full leg \( b_{s} \) of steel angles was assumed to exert the confining pressure in whole analysis steps, the proposed model assumed that only the effective leg \( b_{{s,{\text{eff}}}} \) exerts the confining pressure after local buckling of steel angles (Fig. 6). The effective leg also varies in vertical, but concrete failure occurs at the weakest point (i.e., at the mid-height within a buckling length). Thus, in the calculation of the geometrical effectiveness coefficient \( k_{2} \) (see Appendix), Eq. (3) was used for the ineffective width \( w_{i} \) (\( b_{c} \) = dimension of confined concrete) to consider the reduction in confining pressure by local buckling.$$ w_{i} = b_{c} - 2b_{{s,{\text{eff}}}} $$(3)

### 3.2 Steel Angles and Local Buckling

Stress distribution | Local buckling coefficient | Elastic effective width | |
---|---|---|---|

SS \( \ge \) Free | \( \psi \ge 0 \) | \( k_{b} = \left({\frac{0.578}{\psi + 0.34}} \right) + C_{k} \left({\frac{{b_{s}}}{{s_{t}}}} \right)^{2} \) \( C_{k} = 2.5 - 2.5\,\psi + \,1.0\psi^{2} \) | \( \rho =\frac{{\left({1 - \alpha \frac{1}{\lambda}} \right)}}{\lambda} \) |

\( \psi < 0 \) | \( k_{b} = \left({1.7 - 5\,\psi + \,17.1\psi^{2}} \right) +C_{k} \left({\frac{{b_{s}}}{{s_{t}}}} \right)^{2} \) \( C_{k} = 2.5 - 1.2\,\psi - 0.6\psi^{2} \) | \( \rho =\left({1 + \psi} \right)\frac{{\left({1 - \alpha \frac{1}{\lambda}} \right)}}{\lambda} \) | |

Free \( \ge \) SS | \( \psi \ge 0 \) | \( k_{b} = \left({0.57 - 0.21\,\psi + \,0.07\psi^{2}} \right) + C_{k} \left({\frac{{b_{s}}}{{s_{t}}}} \right)^{2} \) \( C_{k} = 1.25 - 0.25\,\psi \) | \( \rho =\frac{{\left({1 - \alpha \frac{1}{\lambda}} \right)}}{\lambda} \) |

\( \psi < 0 \) | \( \rho =\left({1 - \psi} \right)\frac{{\left({1 - \alpha \frac{{\left({1 - \psi} \right)}}{\lambda}} \right)}}{\lambda} \) |

Bambach and Rasmussen (2004a, b) calculated the local buckling coefficient by finite strip analysis with large half-wavelengths (or for long plates). However, since the local buckling coefficient is affected by boundary conditions and plate geometry, the effect of transverse reinforcement needs to be considered (Timoshenko and Gere 1985). Thus, the local buckling coefficient was modified using CUFSM (Li and Schafer 2010), which is an open source finite strip elastic stability analysis program, and regression analysis. The modified equations for the local buckling coefficient considering the spacing of transverse reinforcement are given in Table 2 (the newly introduced second term \( C_{k} \left({b_{s}/s_{t}} \right)^{2} \) is the modification).

The critical buckling strain \( \varepsilon_{bs} \) of steel angles was assumed to be greater than the peak strain \( \varepsilon_{co,u} \) of concrete cover (i.e., \( \varepsilon_{bs} \ge \varepsilon_{co,u} \)) because concrete cover restrains local buckling of steel angles (Chen and Lin 2006), and local bucking of steel angles was assumed to incorporate spalling of concrete cover at corners (i.e., \( f_{c,u} = 0 \) if \( \varepsilon_{s} \ge \varepsilon_{bs} \)) (case 2 in Fig. 5).

### 3.3 Longitudinal Bars and Local Buckling

### 3.4 Bond Strength between Concrete and Steel

In the case of CES-A sections, frictional bond between concrete and steel is insignificant due to the smooth surface of steel angles, but transverse reinforcement provides two mechanical bond mechanisms (concrete bearing and dowel action). The authors experimentally investigated the bond resistance of CES-A beams (Eom et al. 2014), and it was found that the bond resistance is mainly provided by concrete bearing at a small flexural deformation and by dowel action at a large flexural deformation, and bond-slip of tension steel angles affects the structural behavior. Thus, to prevent premature bond-slip, a denser spacing of transverse reinforcement needs to be used. However, in the case of CES-A columns, the effect of bond-slip is less pronounced because large axial compression is expected to apply and external forces are transferred to concrete and steel angles directly. Thus, in the present study, the bond-slip was not considered for simplicity.

### 3.5 Second-order Effect

### 3.6 Verifications

For verification, the nonlinear numerical analysis results by the proposed model were compared with the previous test results (Kim et al. 2014; 2017; Eom et al. 2014; Hwang et al. 2016). Table 1 and Figs. 2, 3 present the material and geometric properties, section types, and transverse reinforcement types of the previous test specimens.

Specimens | Maximum load, \( P \) (kN) for E1–E4 and C1–C5 or \( M_{m} \) (kN-m) for F1–F7 | Secant stiffness at maximum load, \( P /\varepsilon_{c} \) (kN) for E1–E4 and C1–C5 or \( M_{m} /\kappa_{m} \) (kN-m | ||||
---|---|---|---|---|---|---|

T | A | A/T | T | A | A/T | |

E1 | 3343 | 3587 | 1.07 | 637700 | 491312 | 0.77 |

E2 | 3614 | 3328 | 0.92 | 449287 | 416056 | 0.93 |

E3 | 3472 | 3344 | 0.96 | 462683 | 405290 | 0.88 |

E4 | 3598 | 3366 | 0.94 | 371730 | 420757 | 1.13 |

C1 | 8081 | 8101 | 1.00 | 2711782 | 4050492 | 1.49 |

C2 | 7684 | 8267 | 1.08 | 2677178 | 2755614 | 1.03 |

C3 | 6722 | 7581 | 1.13 | 3280720 | 3790417 | 1.16 |

C4 | 5842 | 5941 | 1.02 | 1794820 | 1980334 | 1.10 |

C5 | 5680 | 5405 | 0.95 | 1897557 | 2702699 | 1.42 |

F1 | 572 | 572 | 1.00 | 11838 | 20637 | 1.74 |

F2 | 497 | 498 | 1.00 | 12697 | 7866 | 0.62 |

F3 | 412 | 487 | 1.18 | 31588 | 12820 | 0.41 |

F4 | 394 | 486 | 1.23 | 8649 | 13098 | 1.51 |

F5 | 377 | 338 | 0.90 | 6816 | 6014 | 0.88 |

F6 | 345 | 326 | 0.95 | 8133 | 9904 | 1.22 |

F7 | 351 | 325 | 0.92 | 8718 | 9911 | 1.14 |

Accuracy | Mean and standard deviation of A/T = 1.02 and 0.10 | Mean and standard deviation of A/T = 1.09 and 0.35 |

For more detailed investigation, the strength contributions of unconfined (cover) concrete, confined (core) concrete, steel angles, and longitudinal bars obtained from numerical analysis are separately presented in Fig. 9 (thin dashed lines with markers). In the case of the eccentrically loaded specimens (E1–E4 in Fig. 9a–d), the axial load reached its maximum even after cover-spalling (see the marker of *u*) due to the maintained strength of confined concrete (see the marker of *c*) and yielding of steel angles (see the marker of *s*). On the other hand, in the case of the concentrically loaded specimens (C1–C5 in Fig. 9e–i), cover-spalling determined the maximum load, since it occurred around the entire perimeter. In the case of the flexurally loaded specimens (F1–F7 in Fig. 9j–p), the flexural strength was maintained after cover-spalling due to the ductile behavior of confined concrete and steel angles.

The large load-carrying capacity of CES columns after failure of concrete cover (and local buckling of longitudinal bars) was also reported by Naito et al. (2011), and this beneficial effect could be more pronounced in the case of using high-strength and compact steel angles in CES-A columns.

## 4 Parametric Study and Discussion

### 4.1 Effects of Design Parameters

Since the 2nd peak load is developed by the maintained strength of confined concrete and yielding of steel angles, generally \( \alpha \) was increased as the steel contribution and confinement efficiency increased. In more detail, as the concrete strength \( f^{\prime}_{c,u} \) increased, \( \alpha \) was decreased (\( \alpha \) = 1.09 for \( f^{\prime}_{c,u} \) = 40 MPa, 1.02 for 60 MPa, and 0.98 for 80 MPa in Fig. 10a), since the use of higher strength concrete resulted in higher strength-loss by cover-spalling and less ductile behavior. As the yield strength \( f_{ys} \) or sectional ratio \( \rho_{s} \) of steel angles increased, \( \alpha \) was increased (\( \alpha \) = 0.94 for \( f_{ys} \) = 315 MPa, 0.96 for 450 MPa, and 1.02 for 650 MPa, or \( \alpha \) = 0.98 for \( \rho_{s} \) = 3.2%, 1.02 for 3.8%, and 1.08 for 4.8% in Fig. 10b). However, as the width-to-thickness ratio increased, \( \alpha \) was decreased (\( \alpha \) = 1.06 for \( b^{\prime}_{s}/t_{s} \) = 7.0, 1.02 for 11.5, and 1.01 for 16.5 in Fig. 10b), because the slender section was more vulnerable to local instability. As the yield strength \( f_{yt} \) and thickness \( t_{t} \) of battens increased, \( \alpha \) was slightly increased, but their effects were not significant (Fig. 10c). On the other hand, the spacing \( s_{t} \) of battens had a great effect (\( \alpha \) = 1.15 for \( s_{t}/b_{c} \) = 0.3 (clear spacing \( s_{c} = s_{t} - h_{t} \) = 50 mm, and volumetric ratio of battens to the confined concrete core \( \rho_{t} = \varSigma \left({b_{t} t_{t} L_{t}} \right)/\left({b_{c,x} b_{c,y} s_{t}} \right) \) = 6.25%: \( L_{t} \) = length of battens), 1.02 for 0.5 (\( s_{c} \) = 150 mm, \( \rho_{t} \) = 3.75%), and 0.97 for 1.0 (\( s_{c} \) = 400 mm, \( \rho_{t} \) = 1.88%) in Fig. 10c). It is because that the spacing \( s_{t} \) of battens is related not only to confinement but also to local buckling of steel angles. In comparison with the effect of battens (Fig. 10c), the effect of steel angles (Fig. 10b) was also highly influential for the load-carrying capacity of CES-A columns. This parametric study result partly differs from the result of an existing study: in the case of RC columns strengthened by steel jacketing, thick and/or dense battens are much more effective than large steel angles for load-carrying capacity (Khalifa and Al-Tersawy 2014). This partly different result comes from the different purpose of steel angles: unlikely the existing study for steel jacketing, in which the primary purpose of steel angles is to provide lateral confinement, the steel angles in CES-A columns are used to transmit column loads directly as well as to provide confinement. Thus, the steel angles in CES-A columns are subjected to high compression, and their properties associated with local buckling and confinement are also important for load-carrying capacity. The eccentricity \( e_{0} \) of axial load also had a significant effect, but \( \alpha \) was not directly proportional to \( e_{0} \) (\( \alpha \) = 0.96 for \( e_{0}/b \) = 0.1, 1.02 for 0.3, and 1.06 for 0.5, whereas \( \alpha \) = 1.06 for a high eccentricity of \( e_{0}/b \) = 1.0 in Fig. 10d): under the higher eccentricities (or lower compression), strength-loss by cover-spalling was less pronounced, but the contribution of steel angles to axial load was decreased due to the increased bending moment. As the column length \( L_{k} \) increased, \( \alpha \) was decreased (\( \alpha \) = 1.02 for \( L_{k} \) = 4 m, 1.00 for 5 m, and 0.98 for 6 m in Fig. 10e) due to the increased slenderness and 2nd-order effect. On the other hand, the effect of the increased slenderness by using a smaller section was compensated by the increased steel contribution and confinement efficiency in the smaller section (\( \alpha \) = 1.08 for \( b \) = 500 mm, 1.02 for 600 mm, and 0.99 for 700 mm in Fig. 10e).

### 4.2 Spalling Load and Residual Strength

The passive confining pressure is generated by the laterally expanding concrete under compression due to the Poisson effect and the restraining forces in steel angles and transverse reinforcement (Saatcioglu and Razvi 1992; Razvi and Saatcioglu 1999; Nagaprasad et al. 2009; Badalamenti et al. 2010). Thus, the confinement effect is not fully developed until a column is subjected to sufficient compression and deformation, and the large compression and deformation lead to cover-spalling (American Concrete Institute 2014). As stated in the subsection of Concrete and Confinement, cover-spalling is more pronounced at the corners of CES-A columns. However, in well-confined sections, strength-gain in confined concrete may compensate or even exceed strength-loss in concrete cover (Cusson and Paultre 1994). Furthermore, as stated in the subsection of Verifications, in the case of using high-strength and compact steel angles, strength-gain after cover-spalling is more pronounced. That is, the 2nd peak load can be even greater than the 1st peak load depending on the steel contribution and confinement efficiency.

Figure 11 shows the numerical \( P-M \) interaction curves of the typical CES-A section (Fig. 10), which correspond to the 1st peak load (thick dashed lines) and 2nd peak load (thick solid lines). As expected, the 2nd peak load (or residual strength) was affected by the design parameters, and in some cases, the 2nd peak load was greater than the 1st peak load: (1) as the steel contribution increased (in the cases of using higher strength (Fig. 11c) and/or larger (Fig. 11d) steel angles), the residual strength in the tension-controlled zone (below the balanced failure point) was increased (by comparing with Fig. 11a or the shaded area in each figure); whereas (2) as the confinement efficiency increased (in the cases of using more compact steel angles (Fig. 11e), higher strength, thicker, and/or denser battens (Fig. 11f–h), the residual strength in the compression-controlled zone (above the balanced failure point) was increased. In the case of using higher strength concrete (Fig. 11b), the interaction curve expanded toward the compression-controlled zone, but the residual strength was decreased due to the decreased steel contribution. Especially in the practical range of axial load (generally in actual design, \( P \ge 0.1A_{g} f^{\prime}_{c,u} \) according to the definition of compression members and \( e_{0}/b \ge 0.1 \) to account for accidental eccentricity (American Concrete Institute 2014)), the residual strength was obviously greater than the 1st peak load.

The 2nd peak load or residual strength is a meaningful factor in seismic design and progressive collapse analysis. Thus, a rational approach is required to predict the residual strength after cover-spalling. It is noted that the 1st peak load (or spalling load) can be obtained by the strain compatibility method of ACI 318-14 (Kim et al. 2014; 2017), in which the linear strain distribution and ultimate compressive strain of * ε*_{
cu
} = 0.003 for concrete are used neglecting the confinement effect (American Concrete Institute 2014).

### 4.3 Simple Approach for Residual Strength

- (1)
The critical buckling strain \( \varepsilon_{bs} \) in the elastic range (\( \eta \) = 1 from Eq. (6) because \( E_{s,\sec} = E_{s,tan} = E_{s} \), \( \nu \) = 0.3 from Eq. (7), and \( f_{bs} = E_{s} \varepsilon_{bs} \) in Eq. (5)) and inelastic range (\( \eta = 2/3 \times \left({f_{ys}/\varepsilon_{bs}} \right)/E_{s} \) from Eq. (6) because \( E_{s,\sec} = f_{ys}/\varepsilon_{bs} \) and \( E_{s,tan} \) = 0, \( \nu \) = 0.5 from Eq. (7), and \( f_{bs} = f_{ys} \) in Eq. (5)) can be rewritten as Eq. (13): \( \varepsilon_{bs} = \varepsilon_{bs1} \) if \( \varepsilon_{bs1} \le \varepsilon_{ys} \), or \( \varepsilon_{bs} = \varepsilon_{bs2} \) if \( \varepsilon_{bs1} > \varepsilon_{ys} \).

$$ \varepsilon_{bs1} = \frac{{k_{b} \pi^{2}}}{{12\left({1 - 0.3^{2}} \right)}}\left({\frac{{t_{s}}}{{b_{s}}}} \right)^{2} = 0.904k_{b} \left({\frac{{t_{s}}}{{b_{s}}}} \right)^{2}\,\text{in the elastic range} $$(13a)where the local buckling coefficient can be conservatively taken as \( k_{b} = 0.43 + \left[{b_{s}/\left({s_{t} - h_{t}} \right)} \right]^{2} \) (for battens) from Table 2.$$ \varepsilon_{bs2} = \frac{2}{3} \times \frac{{k_{b} \pi^{2}}}{{12\left({1 - 0.5^{2}} \right)}}\left({\frac{{t_{s}}}{{b_{s}}}} \right)^{2} = 0.731k_{b} \left({\frac{{t_{s}}}{{b_{s}}}} \right)^{2}\,\text{in the inelastic range} $$(13b) - (2)
The effective confining pressure \( \sigma_{le} \) on confined concrete can be rewritten as Eq. (14) from the subsection of Concrete and Confinement and Appendix, and then \( \varepsilon_{cu,c} \) can be obtained by using \( \sigma_{le} \) and Eq. (10c).

where \( k_{2} \le 1.0 \), \( f_{t} \le f_{yt} \), \( A_{t1} = h_{t} \times t_{t} \) = area of a batten, \( w_{i} = b_{c} - 2b_{s} \) (before local buckling), and \( E_{t} \) = elastic modulus of battens.$$ \sigma_{le} = k_{2} \rho_{t} f_{t} = \frac{{0.3A_{t1}}}{{s_{t} \sqrt {w_{i} \left({s_{t} - h_{t}} \right)}}} \times E_{t} \left({0.0025 + 0.04\sqrt[3]{{\frac{{0.3A_{t1}}}{{f^{\prime}_{c,u} s_{t} \sqrt {w_{i} \left({s_{t} - h_{t}} \right)}}}}}} \right) $$(14)

The interaction curve for the residual strength can be obtained by increasing \( \varepsilon_{t} \) and summing up internal forces (axial force \( P \) and bending moment \( M \)) over the cross-section. As shown in Fig. 11, the interaction curve for the residual strength by the simple approach (thin solid lines) agreed well with that by the numerical analysis (thick solid lines) for all cases.

It is noted that, in the case of using the residual strength for design purpose, the partial factors for materials are recommended to be used: \( f^{\prime}_{cu,d} = f^{\prime}_{cu}/1.5 \), \( f^{\prime}_{cc,d} = f^{\prime}_{cc}/1.5 \), and \( f_{ys,d} = f_{ys}/1.1 \) (European Committee for Standardization 2008). Even though CES-A columns showed good performance under cyclic loading (Hwang et al. 2015, 2016; Zheng and Ji 2008a, b), further studies on ductility and post-yield stiffness as well as residual strength are required for seismic design and progressive collapse analysis.

## 5 Conclusions

- (1)
Considering the strain compatibility, confinement effect of steel angles and transverse reinforcement, and local buckling of steel angles and longitudinal bars, nonlinear numerical analysis was performed. In the analysis, the premature spalling of concrete cover at corners and the effect of local buckling of steel angles on confinement, which are the distinctive local failure mechanisms of CES-A columns, were also taken into account.

- (2)
For verification, the numerical analysis results were compared with the previous experimental study results. The proposed model gave fairly good predictions for the peak load, secant stiffness at the peak load, and post-beak behavior. To investigate the effect of design parameters (strength of concrete; strength, area, and compactness of steel angles; strength, thickness, and spacing of battens; eccentricity of axial load; and slenderness by varying column length and sectional size), a parametric study was also conducted.

- (3)
The numerical investigation showed that when the steel contribution is high (by using higher strength, and/or larger steel angles; or by using lower strength concrete) and the confinement efficiency is high (by using more compact steel angles; or by using higher strength, thicker, and/or denser battens), CES-A columns exhibit relatively large load-carrying capacity even after spalling of concrete cover at corners due to the maintained strength of confined concrete and yielding of steel angles. The eccentricity and slenderness were also highly influential for load-carrying capacity.

- (4)
To predict the residual strength (2nd peak load) after cover-spalling (1st peak load), a simple approach was proposed on the basis of the strain compatibility method considering the confinement effect. The residual strength was determined by local buckling of steel angles or crushing of confined concrete, whichever is earlier, and the proposed simple approach gave a good prediction.

## Declarations

### Acknowledgements

This research was supported by grants from the National Natural Science Foundation of China (Research Fund for International Young Scientists, Grant No. 51650110498), and the authors are grateful to the authority for the support.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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