Abstract
JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 2021, VOL. 20, NO. 2, 193–209 https://doi.org/10.1080/13467581.2020.1782213 BUILDING STRUCTURES AND MATERIALS Strength gains of the axially and laterally loaded composite columns based on the concrete confinements provided by steel cores encased in structural concrete Dinh Han Nguyen and Won-Kee Hong Department of Architectural Engineering, Kyung Hee University, Yongin, Republic of Korea ABSTRACT ARTICLE HISTORY Received 14 December 2019 Most of the design codes for the composite columns including American Concrete Institute Accepted 29 May 2020 do not account for the concrete confinements offered by the steel cores encased in the concrete section. However, the flexural capacities of the axially and laterally loaded com- KEYWORDS posite columns were substantially underestimated as the axial load increases if the con- Double confining effects; crete confinement provided by a steel core is not considered. The aim of this study was to steel columns encased in identify the structural gains of the composite columns with the cross-, H- and T-shaped structural concrete; steel cores encased in the concrete when the concrete confinement provided by the steel confinement by steel sections; equivalent cores is taken into consideration. The formulation of the flexural strength considering the confinement factors; confining effects provided by the transverse reinforcements and cross-shaped steel cores cross-shaped steel section for the maximum load limit state was presented with a 50% axial load of the nominal column capacity. The formulation for an ultimate load limit state with a 40% axial load of the nominal column capacity is listed in the Appendix. The design charts were presented for the strain ranges between 0.001 and 0.01, leading to gaining the strengths of the composite columns that have been lost when the confining effects provided by the steel cores were ignored. 1. Introduction strength of the composite columns with the steel cores. According to Chen and Lin 2006; El-Tawil and 1.1. Previous research and objectives Deierlein 1999, the presence of the large steel cores A design strategy for the composite columns that encased in the concrete columns provides an reduces the likelihood of the collapse under the enhanced concrete confinement, leading to an severe seismic conditions must allow the concrete to increased strength and ductility. However, the study absorb as much energy as possible. A composite of the large steel cores and their contributions to the steel-concrete structure will help achieve the cost- composite columns was limited, and few research effective construction due to its high stiffness and results are available. The present study explored the ductility, strong earthquake resistance, and savings underestimated flexural strengths when the steel in terms of the construction time (Ricles and cores contributing to confining concrete were Paboojian 1994; Griffis and Lawrence 1986; Chen, neglected. For the composite columns, the nominal Astaneh-Asl, and Moehle 1992). Experimental investi- axial capacities were computed by summing all forces gations of the composite elements were performed in the structural components, including reinforcing by (Mirza and Skrabek 1992; Mirza, Hyttinen, and steel and steel cores encased in concrete. The nom- Hyttinen 1996; El-Tawil and Deierlein 1999; Dundar inal moment capacities were then calculated by sum- et al. 2008;; SSRC 1979). The analytical models of the ming the axial forces about the geometric centroid of concrete-encased steel composite elements have the composite column sections. The interactions been developed by (Munoz and Hsu 1997; Furlong between the axial forces and moments for the con- 1968; Kato 1996; Virdi and Dowling 1973; Roik and crete are based on a maximum compressive strain of Bergmann 1990; Chen and Lin 2006). A performance- 0.003, which corresponds to the compressive con- based design is receiving wide attention; however, crete failure. This study found the flexural strengths only a few studies of the axial force and moment of the composite columns are not significantly influ - interactions that consider the concrete confinements enced by the confining effect provided by the cross- provided by both the transverse reinforcements and shaped steel sections at a concrete strain of 0.003 or steel cores has been performed. The consequently lower. The present study explored the influence of the inaccurate results underestimated the flexural concrete confined by cross-shaped steel cores on the CONTACT Won-Kee Hong hongwk@khu.ac.kr Department of Architectural Engineering, Kyung Hee University, Yongin 17104, Republic of Korea This article has been republished with minor changes. These changes do not impact the academic content of the article. © 2020 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the Architectural Institute of Japan, Architectural Institute of Korea and Architectural Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 194 D. H. NGUYEN AND W.-K. HONG flexural strength of the composite columns at the area of the highly confined concrete, influencing strains between 0.001 and 0.01, which led to identify- the strength and ductility of the columns. The ing the strength gains of the composite columns measured material properties used to obtain the when the steel cores were considered. (Chen and Lin fracture surfaces of the concrete-encased steel 2006) presented an analytical model for predicting composite columns with the cross-, H- and the axial capacity and the behavior of the concrete T-shaped flange sections are presented in Table 1. encasing H/cross-shaped steel cores. They performed The confinement effects of the steel cores loaded the structural tests to validate their study but did not axially and laterally on the strength and ductility of test the performance of the test specimens by the the composite columns were evaluated up to flexural loadings. The present study included the lat- a concrete strain of 0.01. The selected results eral loads to extend a study proposed by (Chen and were verified through the comparison with the Lin 2006). Valuable test data were also obtained by test data of the prior researchers. The readers are (El-Tawil and Deierlein 1999), who applied both the referred to the authors’ previous studies (Nguyen axial and lateral loads to the composite columns with and Hong 2020a) of the structural concrete con- the H-shaped steel cores. The present study included finements offered by the encased cross-shaped the cross-, H- and T-shaped steel cores with those steel cores, in which the area inside the confined proposed by (El-Tawil and Deierlein 1999). In this cores was idealized as equivalent to the confined study, the strain distributions between 0.001 and region representing highly confined and partially 0.01, and the corresponding neutral axes, were used confined regions proposed by (Chen and Lin 2006). to analytically calculate the compression block and The performance-based flexural strength ignoring the stress of the structural components. The strength the steel cores was underestimated. The spacing of gains contributed by the confining effects of both the hoops was also correlated with the confining transverse reinforcements and steel cores at their effects. ultimate limit state were discovered based on the axial force and moment interaction. 2. Modeling of the concrete confinement offered by the steel cores 1.2. Significance for the prediction of the 2.1. Material properties and boundary condition nonlinear structural behavior of the steel-concrete composite columns The material properties of concrete, steel and rebar including hoops are shown in Figure 1(a). The At a maximum load limit state, the strength gains boundary conditions and loads of the FEM model of the axially and laterally loaded composite col- are indicated in Figure 1(b) which shows the combi- umns based on the concrete confinements pro- nation between axial load with two cases (30% and vided by the steel cores encased in the structural 60% axial load column capacity) and the lateral load concrete were explored. The cores formed by the that applied on the column. However, the lateral cross-shaped steel sections resulted in the largest Table 1. Geometrical properties of the composite columns and material properties (SRC5 and CL-TO). SRC5 (Cross-shaped steel core, 29.8 MPa with a hoop of 75 mm) Composite columns section (mm x mm) D × B 280 × 280 Height of concrete column (mm) D 280 Width of concrete column (mm) B 280 Vertical part Horizontal part Depth of H-steel section (mm) h 175 175 Width of H-steel section (mm) b 90 90 Web thickness of H-steel section (mm) t 5 5 Flange thickness of H-steel section (mm) t 8 8 Rebar diameter (mm) Ø 15.96 Concrete compressive strength (MPa) f’ 29.8 Yield strength of rebar (MPa) f 350 yR Yield strength of steel (MPa) f 345 yS CL-TO (T-shaped steel core) Composite columns section (mm x mm) D × B 300 × 300 Height of concrete column (mm) D 300 Width of concrete column (mm) B 300 Vertical part Horizontal part Depth of H-steel section (mm) h 100 125 Width of H-steel section (mm) b 50 60 Web thickness of H-steel section (mm) t 5 6 Flange thickness of H-steel section (mm) t 7 8 Rebar diameter (mm) Ø 19.05 Concrete compressive strength (MPa) f’ 22.9 Yield strength of rebar (MPa) f 388 yR Yield strength of steel (MPa) f 333 yS JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 195 load was controlled by the lateral displacement cores with different shapes. However, the authors’ pre- using loading protocol (Nzabonimpa and Hong vious study divided the confined regions into the two 2019). regions including an equivalent confining region inside the transverse stirrups and an unconfined region out- side the transverse stirrups. The columns with the hoop 2.2. Idealization of the concrete confinement spacings of 75 mm were used with the application of the 50% of the nominal axial strength to determine an In (Chen and Lin 2006), the confined regions were equivalent confinement effect representing the con- divided into (1) a highly confined region inside the fined and unconfined regions provided by the cross- arches formed by the steel section, (2) a partially con- shaped steel cores. The confined equation proposed fined region outside the highly confined concrete by Mander, Priestley, and Park 1988 for the concrete region and inside parabolic arches formed by the long- sections was obtained from the equivalent confining itudinal bars, and (3) an unconfined region outside the factors. Equation 1 is used to calculate the concrete parabolic arches formed by the longitudinal bars. (El- strengths, for which the equivalent confinement factor Tawil and Deierlein 1999) proposed parabolic arches (K based on the highly confined and partially confined formed by the longitudinal bars and structural steel e, Figure 1. Strain compatibility based on equivalent confining factors. 196 D. H. NGUYEN AND W.-K. HONG Figure 1. Continued. regions) accounts for the confining effects offered by 2.3. Local buckling of the longitudinal bars and the steel cores. Figure 1(c) shows the confined stress– structural steel strain relationships modified based on the equivalent A constitutive model for the longitudinal reinforcing confining factors when the concrete was confined by rebars in the compression zone was proposed by the cross-shaped (refer to Legends 2 to 5) and T-shaped (Bayrak and Sheikh 2001), who observed the load- (Legend 5) steel cores. The unconfined stress–strain carrying capacity and inelastic buckling of the longitu- relationship proposed by the Mander is represented by dinal rebars under compression. The longitudinal rein- Legend 1. The stress–strain relationship of the T-shaped forcing steel and steel sections with an elasto-plastic steel core (Legend 5) with K = 1.26 is slightly greater constitutive relationship were used to model the com- than that of the cross-shaped steel core (refer to posite members subjected to a flexural bending Legend 2) with K = 1.2. The equivalent confining factor moment in the tension zone. The local buckling of the is defined in Equation 1. longitudinal rebars and steel cores was assumed to occur after the crushing of the partially confined con- f ¼ K f (1) cc e co crete. Figure 1(d) illustrates the axial-strain relationships JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 197 for the concrete (refer to columns tested by Chen et al. C ¼ C C þ C ¼ α � c c c1 c1 c2 1 1 0 0 (Chen et al. 1999; Chen and Lin 2006)) with a hoop � B � f α � c � B � f þ α 1 1 2 2 2 c c spacing of 100 mm (22.9 MPa) confined by the core ð cm1 steel with T-shaped steel sections (CL-TO) when the f dε c1 c equivalent confining factor (K ) is 1.26. � c � B � f ¼ � c � B � e 2 2 1 1 f ε cm1 cm2 (6) f dε c1 c 0 0 2.4. Formulation of the flexural strength (M ) nd f � c � B � f 2 2 c c f ε cm2 considering the confining effects provided by both cm2 transverse reinforcements and steel cores f dε c2 c þ � c � B � f The equilibrium and neutral axes of the composite sec- 2 2 0 c f ε cm2 cc tions based on the two simplified zones with the concrete stress-strain profiles at the maximum load and ultimate The calculations of mean stress factors (α) for the two load limit state were calculated based on Equations 7 and zones can be found in the previous paper of the 8. The entire composite section was explored to find the authors (Nguyen and Hong 2020a). neutral axes and strains of all structural components con- The neutral axis (c ) was calculated from the equili- sisting of the composite sections. The nominal flexural brium equations given in Equations 7 and 8. strength was defined as the maximum load limit state. The substantial concrete was lost at the ultimate load limit F þ C þ F þ F ¼ F þ F axial c R steel R steel compression compression tension tenstion state. (7) or 2.5. Maximum load limit F ¼ F þ F C F F axial R steel c R steel tension tenstion compression compression The formulation of the flexural strength (M ) con- nd (8) sidering the confining effects provided by the trans- verse reinforcements and cross-shaped steel cores Equations 9 to 12 were used to determine the inter- for the maximum load limit state with a 50% axial nal force capacity, including the c10ontribution from load of the nominal column capacity was presented. the steel cores at a maximum load limit state with The formulation for an ultimate load limit state with a 50% axial load of the nominal column capacity; a 40% axial load of the nominal column capacity is listed in the Appendix. The unconfined and equiva- F ¼ F þ F ¼ A � E � ε þ A � E � ε R r3 r4 r3 r r3 r4 r r4 tension lent confined concrete zones based on the constitu- d c r3 1 ¼ A � E � ε � þ A � E � ε tive equation developed by (Mander, Priestley, and r3 r cm1 r4 r cm1 Park 1988) were calculated using Equations 2, 3, and d c r4 1 4, where c is the neutral axis of the section at the maximum load limit state. The depth of the com- (9) pressive concrete block (c ) represents the equiva- lent zone confined by the steel section. The compressive forces C & C’ were determined for F ¼ F þ F ¼ 0:5� A � f þ A c1 c1 steel s8 s9 s8 s8 s9 tenstion the unconfined concrete, and the compressive force � f ¼ 0:5� A � E � ε þ A � E � ε ¼ 0:5 s9 s8 s s3 s9 s s4 C was calculated for the equivalent confined c2 � ½ðh þ d c t Þ� t � 1 s 1 w1 f 12 concrete. � � h þ d c t 1 s 1 f 12 � E � ε � s cm1 C ¼ α � c � B � f (2) c c1 1 1 1 � � h þ d c 0:5t 1 s 1 f 12 þ b � t � E � ε � f 1 f 12 s cm1 0 0 0 C ¼ α � c � B � f (3) 2 2 c1 1 c ðh þ d c t Þ 1 s 1 f 12 ¼ 0:5� � t � w1 0 c C ¼ α � c � B � f (4) c2 2 2 2 cc E � ε þ b � t � E s cm1 f 1 f 12 s � � Relationships between the compressive concrete h þ d c 0:5t 1 s 1 f 12 � ε � blocks and the neutral axes c and c are also estab- cm1 1 2 lished in Equation 5. (10) c ¼ c x (5) 2 1 1 The compressive concrete block is then calculated by F ¼ F þ F ¼ ðA þ A Þ� f (11) R r1 r2 r1 r2 yR compression Equation 6. 198 D. H. NGUYEN AND W.-K. HONG � � F ¼ F þ F þ F þ F þ F þ F steel s1 s2 s3 s4 s5 s6 compression � E � ε � � ðd þ h t þ 0:5c Þ d s cm1 s 1 1 c f 12 þ F ¼ A � f þ A � f þ A � f � � s7 s1 yS s2 yS s3 yS h þ d c 0:5t 1 s 1 f 12 þ A � f þ A � f þ 0:5A � f þ 0:5A þ b � t � E � ε � f 1 f 12 s cm1 s4 s1 s5 s5 s6 yS s5 � f ¼ ðb � t Þ� f s7 f 1 f 11 yS � ðd þ h 0:5t d Þ s 1 f 12 c � � yS (17) þ c c � d t � t 1 1 s f 11 w1 cm1 � � yS � f þ 0:5b þ c 0:5D c � yS f 2 1 1 M ¼ A � f � ðd d Þþ A � f R r1 yR 1 c r2 yR ε compression cm1 � ðd d Þ (18) 2 c � ðt þ t Þ� f þðh t t t Þ f 21 f 22 yS 2 f 21 f 22 w1 c 0:5D � t � E � ε � w2 s cm1 M ¼ A � f � ðd d Þþ A � f � � steel s1 yS s1 c s2 yS compression yS þ 0:5Dþ 0:5b c � � ðt þ t Þ � ðd d Þþ A � f � ðd d Þþ A � f f 2 1 f 21 f 22 s2 c s3 yS s3 c s4 s1 cm1 � � � ðd d Þþ A � f � ðd d Þþ 0:5A � f s4 c s5 s5 s5 c s6 yS c 0:5D 0:5b 1 f 2 � E � ε � s cm1 � ðd d Þþ 0:5A � f � ðd d Þ ¼ ðb � t Þ s6 c s5 s7 s7 c f 1 f 11 � � � f � ðd þ 0:5t d Þ yS s f 11 c yS � � þ 0:5 c c � � t � f 1 1 w1 yS ε yS cm1 þ c c � d t � t � � 1 1 s f 11 w1 ε cm1 yS � � þ 0:5 0:5Dþ 0:5b c � � ðt þ t Þ f 2 1 f 21 f 22 ε yS cm1 � f � 0:5c 0:5c � þ 0:5d þ 0:5t d � � yS 1 1 s f 11 c c 0:5D 0:5b cm1 1 f 2 � � � E � ε ε � s yS cm1 c yS þ 0:5b þ c 0:5D c � � ðt þ t Þ� f 1 1 yS f 2 f 21 f 22 cm1 (12) � � yS Equation 13 is used to find the nominal moment � 0:5c þ 0:25D 0:5c � 0:25b d 1 1 f 2 c cm1 strength at the maximum load limit state; þðh t t t Þ� t � E � ε 2 f 21 f 22 w1 w2 s cm1 M M þ M M (13) nominal¼ R=centroid steel=centroid Conc=centroid c 0:5D � � ð0:5D d Þ The flexural moment capacities provided by the struc- � � yS tural components with respect to the centroid are þ 0:5Dþ 0:5b c � f 2 1 shown in Equations 14 to 20; cm1 � � c 0:5D 0:5b 1 f 2 M ¼ M M (14) � ðt þ t Þ� E � ε � R=centroid R R tension compression f 21 f 22 s cm1 � � yS M ¼ M M (15) steel=centroid steel steel � 0:25Dþ 0:25b þ 0:5c 0:5c � d tension compression f 2 1 1 c cm1 � � yS M ¼ A � E � ε � ðd d Þþ A � E � ε R r3 r r3 3 c r4 r r4 þ 0:5 c c � tension 1 1 cm1 d c � � 3 1 � ðd d Þ ¼ A � E � ε � � ðd d Þ 1 2 ε 4 c r3 r cm1 3 c ys 1 � t � f � c þ c � d w1 yS 1 1 c 3 3 ε cm1 d c 4 1 � � þ A � E � ε � � ðd d Þ r4 r cm1 4 c yS þ 0:5 0:5Dþ 0:5b c � f 2 1 cm1 (16) � ðt þ t Þ� E f 21 f 22 s � � M ¼ 0:5� A � f � ðd d Þ c 0:5D 0:5b steel s8 s8 s8 c 1 f 2 tenstion � ε ε � yS cm1 þ A � f � ðd d Þ ¼ 0:5� A � E � ε � s9 s9 s9 c s8 s s3 � � ðd d Þþ A � E � ε � ðd d Þ yS s8 c s9 s s4 s9 c � 0:5Dþ 0:5b þ 2c þ 2c � f 2 1 1 3 ε cm1 ðh þ d c t Þ 1 s 1 f 12 ¼ 0:5� � t w1 (19) 1 JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 199 3.2. A Three-dimensional finite element model for M ¼ α � c � B � f � ðγ � c d Þ 1 1 1 1 c Conc=centroid 1 steel-concrete composite column 0 0 α � c � B � f � ðγ � c þ x d Þþ α � c 1 2 2 2 1 c 2 2 cm1 In most of the three-dimensional FE model, the three f dε c1 c modelling techniques are available, and they are cohe- � B � f � ðγ � c þ x d Þ ¼ 2 2 1 c c 2 0 sive (tie), embedded model and bond-slip model f ε cm1 (Nzabonimpa and Hong 2017; Delso et al. 2011; Hong � c � B � f 1 1 2019). A cohesive zone model-based cohesive contact is modeled with cohesive constitutive relations indicated ð ð 00 1 1 ε ε cm1 cm2 by traction-separation law. The contact behavior was ε f dε f dε c c1 c c1 c BB C C identified based on the interface stiffnesses, K , K , and nn ss 0 0 BB C C � 1 � c d 1 c ε 0 @@ cm1 A A f ε K , for the uncoupled traction-separation behavior cm2 tt ε f dε cm1 c1 c (Nzabonimpa and Hong 2017) in which interfacial stiff - � c � B � f ness parameters are shown for the constant contact 2 2 00 ð 1 1 cm2 resisting strength. Both cohesive and tie model allowed ε f dε c c1 c BB C C the rotational degrees of freedom contributing to the BB C C � 1 ð � c þ x d ε 2 1 c cm2 @@ A A release of the loads, whereas embedded model yielded ε f dε cm2 c1 c greater loads because the rotational degrees of freedom cm2 between concrete and rebar, steel sections were not f dε c2 c accounted. However, the SRC precast frames tested by þ � c � B � f 0 2 2 f ε cm2 the authors in their previous study showed that the cc 00 1 1 cm2 bond-slip behavior was not obvious. Hong (2019) ε f dε c c2 c BB C C 0 showed that the load–displacement relationships are BB C C � 1 ð � c þ x d 2 1 c @@ cm2 A A better predicted by the embedded model than by cohe- ε f dε cm2 c2 c 0 sive (tie) and bond-slip models for frames, when neither (20) bond slippage nor rotation takes place between steel section and concrete. An algorithm was developed to calculate the neutral In the finite element model, a tie-modeling approach axis and the corresponding nominal moment capacity was introduced to predict the structural behavior of the of the steel-concrete composite section with axial loads. proposed mechanical joints. Reinforcing bars and cross- The entire section from the top to the bottom of the shaped steels were tied to the concrete surface using section was explored to locate the neutral axis and was the tie-contact approach (Structural Stability Research used to calculate the internal loads exerted from the Council (SSRC) Task Group 20 1979). The surfaces of the rebars and steel sections to design the composite cross-shaped steels and reinforcing bars were defined as sections. a master surface, and the concrete surface was defined as a slave surface. The assigned tie-constraint method fuses these two surfaces together to eliminate the rela- 3. Verification analysis tive motion between them. The tie approach can be 3.1. Verification with cross-shaped steel sections used to model the steel beams encased in the concrete and axial loads members under the seismic loads. The geometrical and material properties of the composite columns and mate- (Chen and Lin 2006) tested concrete-encased steel col- rial properties used in the analytical prediction are sum- umns with cross-shaped steel sections. In Figure 1(d), marized in Table 1. Chen and Lin 2006 stated that the the authors compared the axial load–strain relationships confining effect was enhanced by the structural steel, analytically proposed by (Chen and Lin 2006) with those indicating that the confinement factor K for the highly presented by the study of the authors for the columns confined concrete and the confining stress on the con- with the hoop spacings of 75 mm (SRC5; 29.8 MPa with crete core were influenced by the shape of the structural a hoop of 75 mm) and 35 mm (SRC6; 29.5 MPa with steel section. a hoop of 35 mm). The confining effects provided by the cross-shaped steel cores were accounted for by equiva- lent confining factors of 1.55 (35 mm), 1.35 (75 mm) and 4. Fracture criteria based on 1.2 (140 mm), demonstrating a strong correlation with three-dimensional P-M interaction diagram the results of (Chen and Lin 2006). The equivalent con- 4.1. Idealized equivalent confining factors fining factors of 1.11, 1.23, and 1.50 for the confinement considering double confinements provided by the H-shaped steel sections (Nguyen and Hong 2020b) were smaller than those obtained for the Fracture surfaces of the moments interacting with the confinement provided by the cross-shaped sections axial forces obtained for the cross-shaped columns (1.2, 1.35, and 1.55). based on the analytical model developed in the 200 D. H. NGUYEN AND W.-K. HONG authors’ study (Nguyen and Hong 2020a) were reinforcements (35 mm, 75 mm and 150 mm) are obtained for the hoop spacings of 35 mm, 75 mm, shown for the varied axial loads. The equivalent con- and 140 mm proposed by (Chen and Lin 2006). In fining factors decreased as the hoop spacing increased, Figure 2(a), the equivalent confining factors (K ) of indicating that the confining factor was approximately 1.55 (29.5 MPa), 1.35 (29.8 MPa), and 1.20 (29.8MPa) 1.2 at a hoop spacing of 140 mm (Legend 1 of Figure 2 accounting for the confinement provided by the cross- (a)) whereas the confining factor of 1.55 was calculated shaped steel sections were presented for all concrete at a hoop spacing of 35 mm (Legend 3 of Figure 2(a)) strains between 0.001 and 0.01. The equivalent confin - when the cross-shaped steel cores were used. The ing factors regardless of the types of the steel cores did influences of the confinements by the cross-shaped, not vary over the compressive concrete strains H-shaped, and T-shaped core steels (refer to the Chen between 0.001 and 0.01 for all axial loads, as illustrated et al. test (Chen et al. 1999; Chen and Lin 2006) on the in Figure 2(a) where the equivalent confining factors equivalent confining factors are idealized in Figure 2(b) were identified for the cross-shaped, H-shaped and as a function of the hoop spacing. The calculated T-shaped core steels tested by (Chen et al. 1999; Chen equivalent confining factors of the columns confined and Lin 2006). The concrete confinements were influ - by H-shaped steel section (El-Tawil and Deierlein 1999) enced mostly significantly by the hoop spacing. In relative to the concrete strength are also shown in Figure 2(a), the relationships between the equivalent Legend 4 of Figure 2(b), which indicates that confining confining factors and the spacing of the hoop effects decrease when the concrete strengths increase. Figure 2. Idealized equivalent confining factors considering confinements offered by the steel cores with and without lateral loads. JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 201 steel cores were more significant than those without 4.2. Fracture criteria considering them when the rebar buckling was con- In Figure 3(a), the moment–curvature relationships of sidered as the axial loads increased. The concrete cur- the columns proposed by Chen and Lin 2006, both vatures are compared with the compressive concrete with and without rebar buckling, are shown for 0%, strains at the extreme fiber in Figure 3(c). The rebar 30%, and 60% axial loads of nominal axial load capa- buckling was considered for the axial loads of 0%, 30% city. The concrete strains were evaluated at the and 60% of nominal axial strength, demonstrating the extreme fiber in the compressive concrete regions. underestimation of the curvatures when the confine - The moment–curvature relationships relative to the ment provided by the steel section is ignored. The curvatures at the extreme fiber considering the con- curvature–strain relationships indicated by Legends 7 crete confinements offered by the core steel section and 8 of Figure 3(c) were obtained numerically based were larger than those ignoring the effect of the core on the finite element analysis considering the concrete steel section. The flexural strengths were larger than plasticity for the axial loads of 30% and 60%, and those obtained when rebar buckling was considered. agreed well with those of the analytical model. In Figure 3(b), the moment-compressive concrete strain relationships at the extreme fiber considering the concrete confinements offered by the core steel 4.3. Performance-based capacity with respect to section are greater than those obtained without con- the strength and ductility sidering the concrete confinements provided by the 4.3.1. Definition of the capacity reduction ratio (R) core steel section. As shown in the failure surfaces The capacity reduction ratio (R) was defined as the depicted in Figure 3(b), the moment strengths indi- ratio between the flexural strength ignoring the con- cated by Legends 4 (K = 1.0) and 6 (K = 1.0) were e e fining effects provided by the steel cores and the substantially underestimated compared with the strength including the confining effects. In Figure 4 moment strengths represented by Legends 3 (K (a), the capacity reduction ratio is computed as flexural = 1.35) and 5 (K = 1.35) for the strains beyond those strength (M ) without accounting for the confining corresponding to the maximum flexural strength as ns effects provided by steel sections divided by M con- the axial loads increased (30% to 60%) when the con- nd sidering the confining effects provided by both trans- finement effect was ignored. Figure 3(a,b) suggests verse reinforcements and steel cores. that the rebar buckling be considered when the R indicates the fraction of the interaction dia- moment–curvature relationships are evaluated relative gram of the columns for the specified concrete to the curvature (or compressive concrete strains), strains and axial loads. It was obtained at the axial especially when axial loads increase. The underesti- loads between 10% and 100% of the nominal axial mates of the moment-compressive concrete strains strength of the full composite sections which were relationships considering the confinements by the Figure 3. Moment-curvature relationships accounting for vs. ignoring the confining effects by steel sections encased in columns of (cross-shaped column of Chen and Lin 2006). 202 D. H. NGUYEN AND W.-K. HONG Figure 4. Performance-based capacity of composite columns. based on the confining effects provided by the property for the rebars in the compression region. transverse reinforcements and steel sections The M indicated a flexural moment capacity for ns encased in structural concrete. The elasto-plastic the axial loads of between 10% and 100% of the rebars and steel sections encased in the structural nominal axial strength obtained from the concrete were used, permitting the elasto-buckling P-M diagram (Figure 4(a)) when the confining JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 203 Figure 4. Continued. effects provided by the steel sections in the con- concrete confined by T-shaped steel cores, as crete were not considered. The flexural nominal shown by the R (0.17) for the same concrete strain moment capacity (M ) was calculated similarly to (0.01) and axial load (50%). But, for the lower con- nd M , but accounting for the confining effects pro- crete strains, the R value was close to 1 as indicated ns vided by the hoops and steel sections in the in Legends 21 and 22 of Figure 4(c)-(1). No signifi - concrete. cant underestimates of the flexural capacity were observed at the concrete strains below 0.002 or 0.003; however, the underestimates increased as the 4.3.2. Capacity reduction ratio (R) vs. strain level of concrete strains increased. The composite columns the concrete confined by the H-shaped steel core, confined by the cross and H steel cores under the cross-shaped steel core and T-shaped steel core low axial loads (Legends 1 to 8 of Figure 4(b)-(1)) Figure 4(b)-(1) illustrates R values obtained for showed no significant underestimate of the flexural a hoop spacing of 75 mm of the column including capacity at the concrete strains below 0.003 whereas SRC5 (a hoop spacing of 75 mm with 29.8 MPa) the underestimate of the flexural capacity under the tested by (Chen and Lin 2006). In Figure 4(b)-(1) low axial loads was significant for the composite and (c)-(1), the flexural capacity was underestimated columns with the large concrete strains (refer to for the composite columns (refer to the columns Legends 1 to 8 of Figure 4(b)-(1)). tested by Chen and Lin 2006 with the cross-shaped (28.1 MPa), H-shaped (29.8 MPa) and T-shaped steel cores). Equations 13 to 20 were used to locate an 4.3.3. Capacity reduction ratio (R) vs. axial loads of R of 0.76 corresponding to an axial load of 50% at the composite columns confined by the H-shaped the concrete strain of 0.006, resulting in the under- steel core, cross-shaped steel core) and T-shaped estimate of 24% with Legend 11 (SRC5; cross-shaped steel core steel, a hoop spacing of 75 mm with 29.8 MPa) and The columns with the cross and H steel cores with the Legend 12 (SRC2; H-shaped steel, a hoop of 75 mm high axial loads were also associated with the under- with 28.1 MPa) as shown in Figure 4(b)-(1) and (c)-(1). estimates of the flexural strength, and the underesti- An underestimate of 65% of the flexural strength mates increased rapidly as the axial loads increased as (R of 0.35) at the higher concrete strain of 0.01 shown in Figure 4(b)-(1). An R of 0.9 (a flexural capacity was seen with Legends 11 (SRC5), 12 (SRC2) of underestimate of 10%) is seen for the column with Figure 4(b)-(1) and 4(c)-(1) when an axial load of a cross-shaped steel core (Legend 1) at a concrete 50% was considered. As shown in Figure 4(c)-(1), strain of 0.01, whereas an R of 0.95 (a flexural capacity a greater underestimate (83%) was found for the underestimate of 5%) for the column with an H-shaped 204 D. H. NGUYEN AND W.-K. HONG steel core (Legend 2) was observed at a concrete strain 4.3.5. Recommendation for the current design of 0.01, when no axial load was applied. The under- practices based on the performance-based capacity estimates of the flexural capacity became even greater of composite columns when the axial load increased, indicating that the strain The flexural load-bearing capacities of the composite for the concrete columns with the axial loads greater columns can be determined, not only at a concrete than 50% (Legends 13 to 22) did not reach 0.01. strain of 0.003 but at larger strains in the vicinity of the In Figure 4(b)-(1), the columns loaded with a 50% of ultimate limit states based on the performance-based the nominal axial strength exhibited, 66% (R = 0.34, strength of the composite columns considering the Legend 11 of the H-shaped wide flange steel) and 62% confinement by the steel cores. The flexural capacity (R = 0.38, Legend 12 of cross-shaped steel) capacity of the steel-concrete composite columns can be underestimates, at the strains of 0.01. The underesti- greater when the fracture surfaces are based on the mate identified for the T steel cores was less than that confining factors provided by hoop reinforcements for the H-shaped wide flange steel and cross-shaped and steel sections. The current design practices under- steel cores when the strains were small, as shown in estimated the performance-based flexural capacities of the R values represented by Legends 28 to 30. But for the composite columns when the steel sections con- the axial loads exceeding 80% (R values in Legend 31 fining concrete section were not accounted for. This to 33), the underestimates similar to those of the col- underestimate was more significant when the higher umns with the H-shaped wide flange steel and cross- axial loads were applied to the composite columns at shaped steel cores were found. Figure 4(c)-(1) shows the large strains. that the underestimates of the flexural capacity of the columns exerted with a 60% the nominal axial strength 4.3.6. Design charts for determining the flexural are 90% (R = 0.10) and 80% (R = 0.2) for the cross- capacities of the composite columns with the shaped (Legend 17 of Figure 4(c)-(1)) and the H-shaped concrete confinement offered by the steel cores sections (Legend 18 of Figure 4(c)-(1)), respectively, In Figure 4(b)-(3) and 4(c)-(3), design charts are pre- when the strains of 0.009 were reached. R value sented to help determine the flexural capacities of the becomes 0 when the composite columns were loaded columns proposed by (El-Tawil and Deierlein 1999) for with a 64% of the nominal axial strength of the full the various concrete strengths and hoop spacing. The composite sections. However, the underestimates of underestimates of the flexural strengths were also a 22% flexural capacity increase to R of 0.78 for both identified when the concrete confinement offered by the cross-shaped and H-shaped sections (Legends 9 the steel cores encased in the structural concrete was and 10 of Figure 4(c)-(1)) when the strains of the com- ignored. The strains of the columns of 69 MPa and 110 posite columns loaded with 60% of the nominal axial MPa with the 40% of axial load capacities (Legends 14 strength reached 0.005. Figure 4(b)-(1) and 4(c)-(1) and 15 of Figure 4(b)-(3)) only reached 0.0084 and illustrate the underestimates of the flexural capacity 0.0064, respectively, whereas the 28 MPa column increase as the strains of columns and axial loads of the reached a strain of 0.01 (Legend 13 of Figure 4(b)-(3)) columns increase. The more significant underestimates with an R of 0.53 (a strength underestimate of 47%). were found for the composite columns with the cross- The R values (referring to the underestimates of flex - shapes steel cores than for those with the columns ural strength) decreased significantly as the concrete with the H-shaped steel cores when the large axial strengths increased (Legends 16, 17, and 18 of Figure 4 loads were applied. The less ductility was demon- (b)-(3) and (c)-(3), representing 28 MPa, 69 MPa, and strated for the highly loaded columns, as shown in 110 MPa, respectively). The strength and ductility of Figure 4(b)-(1), whereas the R values decrease rapidly the high-strength concrete section were underesti- for the large strains as shown in Figure 4(c)-(1). mated more significantly than in the normal concrete when R was calculated without the consideration of 4.3.4. Level of the axial load vs. the ductility the steel cores encased in the structural concrete. In demands Figure 4(b)-(1), (2), and 4(c)-(1), (2), R values of the Figure 4(c)-(1) illustrates that the level of the axial load specimens shown with the legends are identified at can be determined once the ductility demands of the the specified strains and axial loads. R values of the composite columns are established. For example, the Specimen SRC5 are indicated at the maximum limit composite columns should be designed with the axial state when the columns are loaded with a 50% of forces less than the 60% of the nominal axial strength axial load capacity. of the full composite sections to reach the concrete strain of 0.01, as shown in Figure 4(c)-(1). The charts 4.4. Use of high-strength hoops for high-strength shown in Figure 4(b)-(2) and 4(c)-(2) were for the hoop concrete spacings of 35 mm (SRC6, 29.5 MPa), 75 mm (SRC5, 29.8 MPa) and 140 mm (SRC4, 29.8 MPa) for the cross- The hoop spacings of 35 mm, 75 mm, and 140 mm of shaped steel sections. the column proposed by Chen and Lin 2006, which JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 205 yielded the equivalent confining factors of 1.55, 1.35, fabrication and placement of the longitudinal and trans- and 1.2, resulted in congested column sections related verse reinforcements around the structural steel cores. to the fabrication and placement of the longitudinal/ The use of the high-strength hoops can minimize the transverse reinforcements around the structural steel congestion of the hoop reinforcements in the compo- cores. The use of the high-strength hoops may mini- site sections, as (Muguruma and Watanabe 1990; mize the congestion of hoop reinforcements in the Nishiyama, Watanabe, and Muguruma 1991), and others composite sections. (Muguruma and Watanabe 1990; suggest. According to them, the large strains and ducti- Nishiyama, Watanabe, and Muguruma 1991), and lity of the composite columns with the high-strength others suggest that effective confinement by the high- concrete can be achieved through the use of the high- strength concrete be made available via the high- strength hoop reinforcements when the concrete con- strength hoop reinforcements, even if the confining finements provided by the steel cores encased in the effects are not fully activated owing to the lower com- structural concrete are included in the strength pressive-dilatational characteristics of the higher- calculation. strength concrete. However, the high-strength hoop reinforcements were effective in reaching the large Nomenclature strains of the sections when the concrete confine - ments were provided by the steel cores encased in 2 A area of the rebar layer i (i=1-4), mm ; ri the structural concrete. A area of the part i of H-steel section, mm ; si B width of the concrete section, mm; B width of the unconfined, equivalent confined 5. Conclusions concrete area (i =1 - 2), mm; c height of the concrete compression zone of The influence of the axial loads and confining effects unconfined, equivalent confined concrete provided by cross-, H- and T-shaped steel cores on the area (i =1 - 2), mm; flexural strength of composite columns resulted in C compressive force given by unconfined, ci equivalent confined concrete area (i=1,2), kN; underestimates of capacity when the flexural strength C’ compressive force given by the equivalent c1 utilizing the confinement provided by the steel cores confined concrete area inside, kN; was neglected. The performance of the steel-concrete D height of the concrete section, mm; composite columns was even more underestimated as d distance from the rebar layer i (i=1-4) to top of the axial loads of the columns increased. The ductility the concrete section, mm; d distance from the centroid to top of the con- in the high-strength concrete sections was underesti- c crete section, mm; mated much more significantly than it was in normal d distance from the top flange of H-steel to top concrete when the steel cores encased in structural of the concrete section, mm; concrete were not considered for the concrete con- d distance from the force given by the part i of si finement. The additional strengths contributed by the the H-steel to top of the concrete section, mm; concrete confined by the cross-shaped steel cores at E Young’s modulus of steel, MPa; E Young’s modulus of rebar, MPa; the strains larger than 0.003 are available. This study F force given by rebar layer i (i=1,4), kN; ri presented the design charts defining the axial load– F force given by part i of H-steel section (i=1,8), si moment relationships for the composite columns to kN; identify the underestimates of the flexural capacity of ε strain at fiber of the unconfined, equivalent cmi the composite columns with the cross-, H- and confined concrete area (i =1- 2); ε yield strain of the rebar; T-shaped steel cores for the strain ranges between yR ε yield strain of the steel; yS 0.001 and 0.01. The design charts lead to gaining the ε strain of the rebar layer i (i=1-4); ri strengths of the composite columns that have been ε strain respect to the part i of the H-steel si lost when the confining effects provided by the steel section; cores were ignored. No significant underestimate of f yield strength of the rebar, MPa; yR the flexural capacity was observed at concrete strains f yield strength of the steel, MPa; yS f’ compressive strength of the unconfined con- below 0.002 or 0.003, indicating that the underesti- crete, MPa; mates increased with the concrete strains. The compo- f’cc compressive strength of the equivalent con- site columns with the moderate axial loads fined concrete, MPa; demonstrated no significant underestimate of the flex - f concrete compressive stress in term of the c1 ural capacity at a concrete strain of around 0.003, concrete strain of the unconfined area, MPa; f concrete compressive stress in term of the whereas then columns with the high axial loads c2 concrete strain of the equivalent confined demonstrated substantial underestimates of the flex - area, MPa; ural strength, which increased rapidly with the axial h depth of the vertical H-steel section part of the loads. cross-shape steel section, mm; The design charts can also be used to size congested h depth of the horizontal H-steel section part of column sections with the hoops that make for difficult the cross-shape steel section, mm; 206 D. H. NGUYEN AND W.-K. HONG K confinement factors for the highly confined engineering with hybrid composite structures. He has pro- concrete; vided many useful solutions to issues in current structural K confinement factors for the partially confined design and construction technologies as a result of his concrete; research combining structural engineering with construction K confinement factors for the equivalent con- technologies. He is the author of numerous papers and fined concrete; patents, both in Korea and the USA. Currently, Dr. Hong is t top flange thickness of the vertical H-steel developing new connections that can be used with various f11 section part of cross-shape steel section, mm; types of frames, including hybrid steel–concrete precast t bottom flange thickness of the vertical H-steel composite frames, precast frames and steel frames. These f12 section part of cross-shape steel section, mm; connections would contribute to the modular construction t top flange thickness of the horizontal H-steel of heavy plant structures and buildings as well. He recently f21 section part of cross-shape steel section, mm; published a book titled as “Hybrid Composite Precast t bottom flange thickness of horizontal H-steel Systems: Numerical Investigation to Construction” (Elsevier). f22 section part of cross-shape steel section, mm; t web thickness of the vertical H-steel section w1 part of the cross-shape steel section, mm; References t web thickness of the horizontal H-steel section w2 part of the cross-shape steel section, mm; Bayrak, O., and S. A. Sheikh. 2001. “Plastic Hinge Analysis.” x distance from the edge of the concrete 1 Journal of Structural Engineering 127 (9): 1092–1100. equivalent confined areas to the top of the doi:10.1061/(ASCE)0733-9445(2001)127:9(1092). concrete section, mm; Chen, C., A. Astaneh-Asl, and J. P. Moehle. 1992. “Behavior w width of the vertical H-steel section part of the 1 and Design of High Strength Composite Columns.” In cross-shape steel section, mm; Structures Congress 92: 820–823. w width of the horizontal H-steel section part of 2 Chen, C. C., and N. J. Lin. 2006. “Analytical Model for the cross-shape steel section, mm; Predicting Axial Capacity and Behavior of Concrete α stress factors for the concrete areas i (i=1,2); i Encased Steel Composite Stub Columns.” Journal of α’ stress factors for the concrete areas inside; 1 Constructional Steel Research 62 (5): 424–433. γ centroid factor for the concrete areas i (i=1,2); i doi:10.1016/j.jcsr.2005.04.021. γ’ centroid factor for the concrete areas inside; 1 Chen, C. C., C. C. Weng, I. M. Lin, and J. M. Li. 1999. “Seismic Behavior and Strength of Concrete Encased Steel Stub Columns and Beam-columns.” Report nº. MOIS 881012-1. Author contributions Architecture and Building Research Institute. Delso, J. M., A. Stavridis, B. Shing, M. Papadrakakis, Won-Kee Hong conceived the idea; Won-Kee Hong and Dinh M. Fragiadakis, and V. Plevris. 2011. “Modeling the Han Nguyen derived the equations; Won-Kee Hong wrote Bond-slip Behavior of Confined Large Diameter and prepared the original draft; Won-Kee Hong and Dinh Reinforcing Bars.” In In III ECCOMAS Thematic Conf. On Han Nguyen reviewed & edited the manuscript. Computational Methods in Structural Dynamics and Earthquake Engineering COMPDYN (Vol. 14). Corfu, Greece. Disclosure statement Dundar, C., S. Tokgoz, A. K. Tanrikulu, and T. Baran. 2008. “Behavior of Reinforced and Concrete-encased Composite The authors declare that they have no conflict of interest. Columns Subjected to Biaxial Bending and Axial Load.” Building and Environment 43: 1109–1120. doi:10.1016/j. buildenv.2007.02.010. Funding El-Tawil, S., and G. G. Deierlein. 1999. “Strength and Ductility of Concrete Encased Composite Columns.” Journal of This work was supported by the National Research Structural Engineering, ASCE 125 (9): 1009–1019. Foundation of Korea (NRF) grant funded by the Korea gov- doi:10.1061/(ASCE)0733-9445(1999)125:9(1009). ernment (MSIT) (No. 2019R1A2C2004965). Furlong, R. W. 1968. “Design of Steel-encased Concrete Beam–columns.” Journal of Structural Division, ASCE 94 (1): 267–281. Notes on contributors Griffis,, and G. Lawrence. 1986. “Some Design Considerations for Composite-frame Structures.” Engineering Journal, AISC Dinh Han Nguyen is currently enrolled as a Ph.D. candidate in 23: 59–64. the Department of Architectural Engineering at Kyung Hee Hong, W. K. 2019. Hybrid Composite Precast Systems: University, Republic of Korea. His research interests include Numerical Investigation to Construction. Woodhead precast composite structures. Publishing, Elsevier. Kato, B. 1996. “Column Curves of Steel-concrete Composite Won-Kee Hong is a Professor of Architectural Engineering at Members.” Journal of Constructional Steel Research 39 (2): Kyung Hee University. Dr. Hong received his Masters and 121–135. doi:10.1016/S0143-974X(96)00030-2. Ph.D. degrees from UCLA, and he worked for Englelkirk and Mander, J. B., M. J. Priestley, and R. Park. 1988. “Theoretical Hart, Inc. (USA), Nihhon Sekkei (Japan) and Samsung Stress-strain Model for Confined Concrete.” Journal of Engineering and Construction Company (Korea) before join- Structural Engineering 114 (8): 1804–1826. doi:10.1061/ ing Kyung Hee University (Korea). He also has professional (ASCE)0733-9445(1988)114:8(1804). engineering licenses from both Korea and the USA. Dr. Hong Mirza, S. A., V. Hyttinen, and E. Hyttinen. 1996. “Physical Tests has more than 30 years of professional experience in struc- and Analyses of Composite Steel-concrete Beam–col- tural engineering. His research interests include new umns.” Journal of Structural Engineering, ASCE 122 (11): approaches to construction technologies based on value JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 207 1317–1326. doi:10.1061/(ASCE)0733-9445(1996)122:11 Assemblages.” Proceedings of the Pacific Conference on (1317). Earthquake Engineering New Zealand 20: 217–228. Mirza, S. A., and B. W. Skrabek. 1992. “Statistical Analysis of Nzabonimpa, J. D., and W. K. Hong. 2017. “Evaluation of Slender Composite Beam–column Strength.” Journal of Hybrid Joints Strengthened by Carbon Plates Connecting Structural Engineering, ASCE 118 (5): 1312–1331. New Steel Frames with Existing Concrete Slabs.” doi:10.1061/(ASCE)0733-9445(1992)118:5(1312). Composite Structures 174: 348–365. doi:10.1016/j. Muguruma, H., and F. Watanabe. 1990. “Ductility compstruct.2017.04.074. Improvement of High-strength Concrete Columns with Nzabonimpa, J. D., and W. K. Hong. 2019. “Experimental Lateral Confinement.” Special Publication 121: 47–60. Investigation of Hybrid Mechanical Joints for L-shaped Munoz, P. R., and C. T. Hsu. 1997. “Biaxially Loaded Columns Replacing Conventional Grouted Sleeve Concrete-encased Composite Columns: Design Connections.” Journal of Engineering Structure 185: Equation.” Journal of Structural Engineering ASCE 123 (12): 243–277. doi:10.1016/j.engstruct.2019.01.123. 1576–1585. doi:10.1061/(ASCE)0733-9445(1997)123:12 Ricles, J. M., and S. D. Paboojian. 1994. “Seismic Performance (1576). of Steel-encased Composite Columns.” Journal of Nguyen, D. H., and W. K. Hong. 2020a. “Part I: Failure Criteria Structural Engineering 120 (8): 2474–2494. doi:10.1061/ of the Steel-concrete Columns (SRC Columns) Confined by (ASCE)0733-9445(1994)120:8(2474). Cross-shaped Flange Sections”. Journal of Asian Roik, K., and R. Bergmann. 1990. “Design Method for Architecture and Building Engineering. doi: 10.1080/ Composite Columns with Unsymmetrical Cross-sections.” 13467581.2020.1782211 Journal of Constructional Steel Research 33: 153–172. Nguyen, D. H., and W. K. Hong. 2020b. “An Analytical Model doi:10.1016/0143-974X(90)90046-J. Computing Flexural Strength and Performance of Structural Stability Research Council (SSRC) Task Group 20. 1979. Concrete Columns Confined by Both Transverse “A Specification for the Design of Steel-concrete Composite Reinforcements and Steel Sections”. Journal of Asian Columns.” AISC Engineering Journal 16 (4): 101–115. Architecture and Building Engineering. doi: 10.1080/ Virdi, K. S., and P. J. Dowling. 1973. “The Ultimate Strength of 13467581.2020.1775603 Composite Columns in Biaxial Bending.” Proceedings Nishiyama, M., F. Watanabe, and H. Muguruma. 1991. “Seismic Institution of Civil Engineers Part 2 (55): 251–272. Performance of Prestressed Concrete Beam-Column Joint doi:10.1680/iicep.1973.4958. 208 D. H. NGUYEN AND W.-K. HONG Appendix At ultimate load limit state with 40% axial load column capacity The internal forces contributed by the structural components of the composite columns are shown as follows: d c 3 1 F ¼ F þ F ¼ A � E � ε þ A � f ¼ A � E � ε � þ A � f (A1) R r3 r4 r3 r r3 r4 yR r3 r cm1 r4 yR tension � � yS F ¼ F þ F ¼ 0:5� A � f þ A � f ¼ 0:5� c � � t � f þ b � t � f (A2) steel s6 s7 s6 yS s7 yS 1 w1 yS f 1 f 12 yS tenstion cm1 F ¼ F þ F ¼ ðA þ A Þ� f (A3) R r1 r2 r1 r2 yR compression F ¼ F þ F þ F þ F þ F ¼ A � f þ A � f þ A � f þ A � f þ 0:5A � f steel s1 s2 s3 s4 s5 s1 yS s2 yS s3 yS s4 yS s5 yS compression � � yS ¼ b � t � f þ c c � d t � t � f f 1 f 11 yS 1 1 s f 11 w1 yS cm1 � � (A4) yS þ 0:5b 0:5Dþ c c � � ðt þ t Þ� f þðh t t t Þ� t � f f 2 1 1 f 21 f 22 yS 2 f 21 f 22 w1 w2 yS cm1 yS þ 0:5c � � ðt þ t þ t Þ� f 1 f 21 f 22 w1 yS cm1 The compressive forces C & C’ are obtained by the green region for the unconfined concrete based on the Mander curve c1 c1 (Mander, Priestley, and Park 1988) whereas the compressive force C is obtained by the orange region for the equivalent c2 confined concrete based on the Mander curves (Mander, Priestley, and Park 1988). 0 0 0 0 0 C ¼ C C þ C ¼ α � c � B � f α � c � B � f þ α � c � B � f c c1 c1 c2 1 1 1 1 2 2 2 2 2 c c c ð ð 0:006 0:006 f dε f dε c1 c c1 c 0 0 0 0 ¼ � 0:6c � B � f � 0:6c � B � f 0 1 1 0 1 2 c c 0:006f 0:006f c c cm2 (A5) f dε c2 c þ � c � B � f 2 2 f ε cm2 cc ð ð 0:006 ε cm2 f dε f dε c1 c c2 c 0 0 0 0 ¼ � 0:6c � f � ðB B Þþ � c � B � f 1 1 2 2 2 0 0 c c 0:006f f ε cm2 c cc The depth of each compressive concrete block was obtained from the neutral axis c ; as follows: c ¼ c x ; (A6) 2 1 1 The nominal moment strength at the maximum load limit state is then obtained using Equation A7; M M þ M M (A7) nominal ¼ R=centroid steel=centroid Conc=centroid where the flexural moment capacities provided by the structural components with respect to the centroid are shown as follows: M ¼ M M (A8) R R R=centroid tension compression M ¼ M M (A9) steel=centroid steel steel tension compression d c 3 1 M ¼ A � E � ε � ðd d Þþ A � f � ðd d Þ ¼ A � E � ε � � ðd d Þþ A � f � ðd d Þ R r3 r r3 3 c r4 yR 4 c r3 r cm1 3 c r4 yR 4 c tension (A10) M ¼ 0:5� A � f � ðd d Þþ A � f � ðd d Þ steel s6 yS s6 c s7 yS s7 c tenstion � � � � (A11) ε 2 ε yS yS ¼ 0:5� c � � t � f � c þ c � d þ b � t � f � ðd þ h 0:5t d Þ 1 w1 yS 1 1 c f 1 f 12 yS s 1 f 12 c ε 3 ε cm1 cm1 JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 209 M ¼ A � f � ðd d Þþ A R r1 yR 1 c r2 compression (A12) � f � ðd d Þ yR 2 c M ¼ A � f � ðd d Þþ A � f � ðd d Þþ A � f � ðd d Þ steel s1 yS s1 c s2 yS s2 c s3 yS s3 c compression þ A � f � ðd d Þþ 0:5A � f � ðd d Þ s4 yS s4 c s5 yS s5 c � � yS ¼ b � t � f � ðd þ t d Þþ c c � d t f 1 f 11 yS s f 11 c 1 1 s f 11 cm1 � � � � ε ε yS yS � t � f � 0:5c 0:5c � þ 0:5d þ 0:5t d þ 0:5b 0:5Dþ c c � w1 yS 1 1 s f 11 c f 2 1 1 ε ε cm1 cm1 (A13) � � yS � ðt þ t Þ� f � 0:5c 0:5c � þ 0:25D 0:25b d þðh t t t Þ f 21 f 22 yS 1 1 f 2 c 2 f 21 f 22 w1 cm1 yS � t � f � ð0:5D d Þþ 0:5c � � ðt þ t þ t Þ� f w2 yS c 1 f 21 f 22 w1 yS cm1 � � 2 ε yS � c c � d 1 1 c 3 ε cm1 0 0 0 0 M ¼ α � c � B � f � ðγ � c d Þ α � c � B � f � ðγ � c þ x d Þ Conc=centroid 1 1 1 1 1 c 1 2 2 1 2 1 c c c þ α � c � B � f � ðγ � c þ x d Þ 2 2 2 2 1 c c 2 0 0 1 1 ð ð 0:006 0:006 f dε ε f dε B B C C c1 c c c1 c B B C C 0 0 ¼ � 0:6c � f � ðB B Þ� 0:4c þ 0:6 1 � c d B B ð C C 0 1 1 2 1 1 c 0:006 @ @ A A 0:006f c (A14) 0:006 f dε c1 c ð ð 00 1 1 ε ε cm2 cm2 f dε ε f dε c2 c c c2 c BB C C 0 0 BB C C þ � c � B � f � 1 ð � c þ x d 2 2 2 1 c 0 ε c @@ cm2 A A f ε cm2 cc ε f dε cm2 c2 c
Journal
Journal of Asian Architecture and Building Engineering
– Taylor & Francis
Published: Mar 4, 2021
Keywords: Double confining effects; steel columns encased in structural concrete; confinement by steel sections; equivalent confinement factors; cross-shaped steel section
