materials Article

Uniaxial Compressive Constitutive Relationship of Concrete Confined by Special-Shaped Steel Tube Coupled with Multiple Cavities Haipeng Wu *, Wanlin Cao, Qiyun Qiao and Hongying Dong College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100124, China; [email protected] (W.C.); [email protected] (Q.Q.); [email protected] (H.D.) * Correspondence: [email protected]; Tel.: +86-152-0122-7267 Academic Editor: Jorge de Brito Received: 21 November 2015; Accepted: 19 January 2016; Published: 29 January 2016

Abstract: A method is presented to predict the complete stress-strain curves of concrete subjected to triaxial stresses, which were caused by axial load and lateral force. The stress can be induced due to the confinement action inside a special-shaped steel tube having multiple cavities. The existing reinforced confined concrete formulas have been improved to determine the confinement action. The influence of cross-sectional shape, of cavity construction, of stiffening ribs and of reinforcement in cavities has been considered in the model. The parameters of the model are determined on the basis of experimental results of an axial compression test for two different kinds of special-shaped concrete filled steel tube (CFT) columns with multiple cavities. The complete load-strain curves of the special-shaped CFT columns are estimated. The predicted concrete strength and the post-peak behavior are found to show good agreement within the accepted limits, compared with the experimental results. In addition, the parameters of proposed model are taken from two kinds of totally different CFT columns, so that it can be concluded that this model is also applicable to concrete confined by other special-shaped steel tubes. Keywords: constitutive relationship; confined concrete; special-shaped cross-section; concrete filled steel tube (CFT); multiple cavities

1. Introduction Concrete filled steel tubes (CFTs) combine steel and concrete, which results in tubes that have the beneficial qualities of high tensile strength and the ductility of steel as well as the high compressive strength and stiffness of concrete. Hence, they possess perfect seismic resistance property. In recent years, mega-frame structures have widely been applied to super high-rise buildings for their clear force transferring paths between primary and secondary structures, and flexible arrangement of structural members [1,2]. To fulfill the requirements of structural safety, architectural layout and economic efficiency, the mega CFT (concrete filled steel tube) columns are often designed as special shapes with multiple cavities, which are often very different from normal circular and rectangular CFTs [3–5]. The constitutive relationships play a pivotal role for both design and research in concrete materials and structures. Due to the diversity of concrete materials, inconformity of test methods and confinement of steel tubes, various stress-strain equations have been proposed in the past; most of them originated from classical theories. These models can be divided into two types, i.e., the uniaxial and the triaxial. The constitutive relationships for the case of uniaxial model are simple and are often used in fiber based models; whereas, for the case of triaxial models, these relationships are complex and are often used in finite element methods (FEMs) [6,7]. A typical stress-strain curve of steel tube confined concrete exhibits an ascending trend followed by a post-peak descending behavior [8–10]. In the case of circular Materials 2016, 9, 86; doi:10.3390/ma9020086

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tube confined concrete, Tang et al. [11] have established a stress-stain relationship for circular steel tube confined concrete by assuming the steel ratio, the tube width-to-thickness ratio and the material’s property on the strength and post-peak behavior. Xiao [12] conducted a series of tests on concrete filled steel tube stub columns, and then proposed a triaxial constitutive relationship, in which failure criterion and flow rule are expressed by octahedron element. Zhong and Han et al. [13,14] have conducted a number of tests on circular and rectangular CFT columns and established the constitutive relationships, which were derived by using regression analysis. Chen [15] has developed and modified the increment constitutive relationships based on plastic-fracturing mechanics, which were initially proposed by Banzant [16]. Susantha et al. [17] have proposed the calculation method of stress-strain relationships of CFT-subjected axial load and horizontal load. In this model, the existing formulas and the FEM method have been used to determine the pressure of confinement for steel tube to core concrete. In the cases of square and rectangular steel tube confined concrete, Hajjar et al. [18] have developed a triaxial constitutive relationship that was expressed in a polynomial order; this model can be used to estimate the behavior of CFT under the coupled effect of axial force and bending moment. Watanabe et al. [19] have proposed a stress-strain relationship, which is applicable to a rectangular CFT; in this model, the local buckling of component plate and initial imperfection are considered. Tomii et al. [20] have proposed a stress-strain relationship of concrete confined by square steel tubes; the ascent stage of the model adopted second-degree parabola, whereas the cylindrical strength is assumed as peak strength. On the basis of Mander [21], Long and Cai et al. [22] have established new models for confined concrete especially for the concrete confined by rectangular steel tubes along with binding bars. The above mentioned literature is only limited to either circular or rectangular steel tube confined concrete. Only few cases have been reported regarding the constitutive relationships of concrete that was confined by special-shaped steel tube with multiple cavities. The confinement action of special-shaped steel tubes with multiple cavities is different than that of normal steel tubes (either circular or rectangular including the quadratic shape). Hence, the constitutive relationships of concrete confined by normal steel tubes, available in the literature, are not suitable for the concrete confined by special-shaped steel tube with multiple cavities. The confinement pressure, to the core concrete in special-shaped steel tube with multiple cavities, can be determined by evaluating the cross-sectional shape, cavity construction, steel ratio of outer steel tube and inner cavity partition steel plates, steel ribs, steel bars in cavities, etc. It is very complex to estimate what extent confinement action can be considered. On the basis of an axial compression test of two groups of special-shaped CFT columns with multiple cavities, this article evaluates how each factor contributes to confinement action of core concrete, and proposes uniaxial stress-strain relationship based on Mander’s model. The theoretical results match well with the test results. 2. Model of Constitutive Relationship 2.1. Confinement Mechanism The normal CFT and special-shaped CFT coupled with multiple cavities may lead to generate longitudinal deformation as well as transversal deformation under axial loading. Owing to the fact that the Poisson’s ratio of concrete is smaller than that of steel during the initial loading stage, the steel tube and in-filled concrete make a trend of departure and there’s no squeezing between them. When the stress of steel tube is loaded to reach its proportional limit, the Poisson’s ratio of concrete is approximately equal to that of steel. When the stress of steel tube exceeds to its proportional limit, the Poisson’s ratio of concrete is greater than that of steel; a lateral interactional force generates along with squeezing trend between them, owing to the fact that the steel tube constrains concrete transversal deformation. The confinement action in case of a circular CFT is, generally, better than that of a rectangular CFT; whereas, it is of medium order for the case of regular polygonal CFT. It is strongly increased as

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the side number of a CFT increases. It also differs when the cross-section changes from a regular shape to an irregular shape. The research [14] shows that the confinement action at the corner of steel tube is steel tube is strong, whereas it is weak at the central part of the sides of steel tube. It is pertinent to  strong, whereas it is weak at the central part of the sides of steel tube. It is pertinent to note that the note that the confinement action increases in the case of a small interior angle that may be formed  confinement action increases in the case of a small interior angle that may be formed by the adjacent by the adjacent steel plates. In the case of a special‐shaped CFT coupled with multiple cavities, the  steel plates. In the case of a special-shaped CFT coupled with multiple cavities, the inner partition steel inner partition steel plate can effectively be used to reduce the interior angle, as well as to provide  plate can effectively befor  used toexternal  reduce the interior well as to provide transverse constraint transverse  constraint  the  steel  tube. angle, As  a as consequence,  the  confinement  action  of for a  the external steel tube. As a consequence, the confinement action of a special-shaped CFT coupled special‐shaped  CFT  coupled  with  multiple  cavities  is  strengthened,  and  the  bearing  capacity  and  with multiple cavities is strengthened, and the bearing capacity and ductility are improved. ductility are improved.  Analogous with normal reinforced confined concrete, the external steel tube of a special-shaped CFT Analogous with normal reinforced confined concrete, the external steel tube of a special‐shaped  coupled with multiple cavities performs as longitudinal reinforcement and transverse reinforcement, CFT  coupled  with  multiple  cavities  performs  as  longitudinal  reinforcement  and  transverse  while the inner partition steel plate performs as longitudinal reinforcement and tie bar. As it has been reinforcement, while the inner partition steel plate performs as longitudinal reinforcement and tie bar.  shown in Figure 1, the steel plate in longitudinal active confined region is equivalent to longitudinal As it has been shown in Figure 1, the steel plate in longitudinal active confined region is equivalent  reinforcement normal confined concrete, whereas the steel plate in transverse active confined to  longitudinal in reinforcement  in  normal  confined  concrete,  whereas  the  steel  plate  in  transverse  region is equivalent to tightened transversal reinforcement. Of course, the inactive confined region active confined region is equivalent to tightened transversal reinforcement. Of course, the inactive  has a similar confinement effect which is relatively weak. In this article, the equivalent method of confined region has a similar confinement effect which is relatively weak. In this article, the equivalent  lateral confining stress proposed by Mander has been applied to study the confinement action for method of lateral confining stress proposed by Mander has been applied to study the confinement  a special-shaped CFT coupledCFT  withcoupled  multiplewith  cavities. The difference concrete action  for  a  special‐shaped  multiple  cavities. from The  Mander’s difference confined from  Mander’s  model is that the special-shaped steel tube with multiple cavities does not bear only longitudinal force confined  concrete  model  is  that  the  special‐shaped  steel  tube  with  multiple  cavities  does  not  bear  but also transversal force. is atransversal  three dimensional stress status, and it is compressed in only  longitudinal  force  but Italso  force.  It complex is  a  three  dimensional  complex  stress  status,  longitudinal and radial directions, whereas it is pulled in a hooping direction. Keeping this in view, the and it is compressed in longitudinal and radial directions, whereas it is pulled in a hooping direction.  influence of longitudinal stress to hooping stress needs to be considered in the theoretical estimation Keeping this in view, the influence of longitudinal stress to hooping stress needs to be considered in  of an equivalent lateral confining stress. Zhong and Shamugam [13,23] have regarding the the  theoretical  estimation  of  an  equivalent  lateral  confining  stress.  Zhong  and found Shamugam  [13,23]  circular CFT that the radial stress of steel tube is relatively small as compared to longitudinal and have  found  regarding  the  circular  CFT  that  the  radial  stress  of  steel  tube  is  relatively  small  as  hooping stresses, and it can be neglected. Next, the confining stress of a special-shaped CFT coupled compared to longitudinal and hooping stresses, and it can be neglected. Next, the confining stress  with multiple cavitiesCFT  is complex and variable; its values areis  different at and  different locations, e.g., atare  the of  a  special‐shaped  coupled  with  multiple  cavities  complex  variable;  its  values  corner, and at the center at the stiffening ribs of different sides. In this article, an equivalent average different at different locations, e.g., at the corner, and at the center at the stiffening ribs of different  lateralIn  confining stressan  is applied to simplify situation and to reflect confinement actionthe  of sides.  this  article,  equivalent  average the lateral  confining  stress  is the applied  to  simplify  a special-shaped CFT coupled with multiple cavities. situation and to reflect the confinement action of a special‐shaped CFT coupled with multiple cavities. 

  Figure 1. Division of confined regions of special‐shaped concrete filled steel tube (CFT) column with  Figure 1. Division of confined regions of special-shaped concrete filled steel tube (CFT) column with multiple cavities.  multiple cavities.

2.2. The Proposed Model  2.2. The Proposed Model An expression (unified) of an equivalent uniaxial stress‐strain relationship of a concrete, which  An expression (unified) of an equivalent uniaxial stress-strain relationship of a concrete, which is is  confined by special‐shaped CFT coupled with multiple cavities  has  been  proposed.  It  is  mainly  confined by special-shaped CFT coupled with multiple cavities has been proposed. It is mainly based based  on  Mander’s  confined  concrete  model.  The  five  parameter  based  strength  criterion,  as  on Mander’s confined concrete model. The five parameter based strength criterion, as determined determined by William‐Warnke, is applied to evaluate the ultimate strength of a confined concrete.  by William-Warnke, is applied to evaluate the ultimate strength of a confined concrete. The Popovics The Popovics concrete stress‐strain curve has  been applied  to express the constitutive relationship.  concrete stress-strain curve has been applied to express the constitutive relationship. The model can The model can well reflect the characteristic of confined concrete that the strength and the strain at  maximum concrete stress increase, while the descending branch tends to slow. The expressions can  be described as follows: 

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well reflect the characteristic of confined concrete that the strength and the strain at maximum concrete stress increase, while the descending branch tends to slow. The expressions can be described as follows: fc “

f cc xr r ´ 1 ` xr

(1)

εc εcc

(2)

x“

εcc “ εc0 r1 ` ηp r“

f cc ´ 1qs f c0

Ec Ec ´ f cc {εcc d

f cc “ f c0 p´1.254 ` 2.254 1 `

(3) (4)

7.94 f l f ´2 l q f c0 f c0

(5)

where, f c and εc are the longitudinal compressive stress and strain of a core concrete respectively; a ˘ ` 105 f c0 , εco “ 700 ` 172 f co ˆ 10´6 and Ec “ [24] are the longitudinal compressive 2.2 ` 34.7{ f cu,k strength, the corresponding strain and the elasticity modulus of an unconfined concrete respectively; f cc and εcc are the longitudinal compressive strength and corresponding strain of a concrete that is confined by special-shaped steel tube with multiple cavities, respectively; η is the correction coefficient of the strain at maximum concrete stress; γ is the shape parameter of the curve; f l is an equivalent lateral confining stress. Mander et al. have suggested “η = 5” for a concrete confined by reinforcement [22]. However, the correction coefficient is not constant for the concrete that is confined by a special-shaped CFT with multi-cavities in accordance with experimental research. 3. Results and Discussion 3.1. Experimental Test Data The special-shaped CFTs coupled with multiple cavities are often applied in specific super high-rise buildings; the related experimental test data are seldom available. Only data of tests conducted by the authors of this paper are documented and available for review. Therefore, this article only focuses to discuss the data of axial compressive test for the six special-shaped CFT columns with multiple cavities, as these were conducted by the authors, to study an equivalent uniaxial stress-strain relationship for a confined concrete. By considering the cross-sectional shape, the cavity construction, the concrete strength, the steel strength, and the reinforcement arrangement in cavities differ in each column, the equivalent uniaxial stress-strain relationship that worked out from the test results has good applicability. 3.1.1. Construction Details Six special-shaped CFT columns coupled with multiple cavities were designed in accordance with actual CFT columns in super high-rise buildings. The six columns were divided into two groups as follows: (1) the group P that includes three irregular pentagonal CFT columns coupled with multiple cavities, and (2) the group H that includes three irregular hexagonal CFT columns coupled with multiple cavities. The scales of group P columns and group H columns, respectively, are 1/5 and 1/12. The real prototype mega column of group P has a cross sectional area of 45 m2 , whereas group H is approximately equal to 9 m2 . All the specimens were designed by using the geometric similarity principle. Group P columns were named CFT1-P, CFT2-P and CFT3-P, respectively. The cross-sectional geometric dimensions of external steel tubes that were welded by 12 mm steel plates are same. The vertical continuous stiffening ribs, whose cross sectional dimensions were 90 mm ˆ 6 mm, were welded to

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were  welded  to  an  inside  face  of  an  external  steel  tube,  whereas  five  story  horizontal  continuous  stiffening ribs were welded at a vertical spacing of 500 mm. For the case of the columns CFT2‐P and  an inside face of an external steel tube, whereas five story horizontal continuous stiffening ribs were CFT3‐P, the cross‐section was divided into two cavities by using a 10 mm thick solid partition steel  welded at a vertical spacing of 500 mm. For the case of the columns CFT2-P and CFT3-P, the cross-section plate, and, thereafter, it was divided into four cavities by using two 6 mm thick symmetrical lattice  was divided into two cavities by using a 10 mm thick solid partition steel plate, and, thereafter, it was partition steel plate on which rectangular holes were punched to make it easy for flow of concrete  divided cavitiescavities.  by usingGroup  two 6 mm thick symmetrical lattice partition plate on which between into the four adjacent  H  specimens  were  named  CFT1‐H, steel CFT2‐H,  CFT3‐H,  rectangular holes were punched to make it easy for flow of concrete between the adjacent cavities. respectively.  The  cross‐sectional  geometric  dimensions  of  steel  tubes,  which  were  welded  into  a  Group H specimens were named CFT1-H, CFT2-H, CFT3-H, respectively. The cross-sectional geometric hexagon along with six cavities by using a 5 mm steel plate, were the same. The vertical continuous  dimensions of steel tubes, were welded into a hexagon six cavities by using a 5of  mm stiffening  ribs  along  with which a  cross‐section  of  25  mm  ×  3  mm along were with welded  to  the  inside  face  an  steel plate, were the same. The vertical continuous stiffening ribs along with a cross-section of 25 mm ˆ 3 mm external steel tube and to the both faces of the partition steel plates. The studs along with a diameter  were welded to theof  inside face of an external60  steel tubemm  and were  to thewelded  both faces of the partition steel of  4  mm,  a  length  30  mm  and  a  spacing  mm×60  to  the  same  place  as  the  plates. The studs along with a diameter of 4 mm, a length of 30 mm and a spacing 60 mmˆ60 mm vertical continuous stiffening ribs. For the column specimens CFT1‐P, CFT3‐P, CFT1‐H and CFT3‐H,  were welded to the same place as vertical continuous stiffening For the column specimens the  longitudinal  reinforcement  is the arranged  into  cavities  by  using  a ribs. welded  spacer  bar  to  improve  CFT1-P, CFT3-P, CFT1-H and CFT3-H, the longitudinal reinforcement is arranged into cavities by the  shrinkage  of  mass  concrete  and  the  heat  of  hydration  issues  as  well  as  to  constrain  the  inner  using a welded spacer bar to improve the shrinkage of mass concrete and the heat of hydration issues concrete. The main parameters and the running parameters of the six specimens have been figured  as well as to constrain the inner concrete. The main parameters and the running parameters of the out, as these have been shown in Table 1. The construction details have also been shown in Figure 2,  six specimens have been figured out, as these have been shown in Table 1. The construction details whereas the construction photos have been shown in Figure 3.  have also been shown in Figure 2, whereas the construction photos have been shown in Figure 3. Table 1. Main parameters of the specimens.  Table 1. Main parameters of the specimens. Equivalent    Cross‐Sectional  Concrete  Steel Plate  Steel‐Bars  Steel    Equivalent Quantity    Area  Strength  Ratio  Ratio  Cross-Sectional Concrete Steel Plate Steel-Bars Strength  Quantity Steel Group  Name  Shape  of  Group Shape Area Strength Ratio Ratio Name of Strength Cavities  fcu,m  fc,m  ρ2   ρ3     ρ1   fy     A    Cavities (Mpa)  (Mpa)  (%)  (%)  (%)  (mm2)  f cu,m f c,m ρ1 ρ2 ρ3 fy (Mpa)  A (m2 ) (Mpa) (Mpa) (%)7.81  (%) (%) (Mpa) CFT1‐P  1  378.5  1.68  0.29  Irregular    P  CFT2‐P  4 1 0.354  51.1  38.8  385.8  7.81  1.68 3.60  0.290  CFT1-Ppentagon  378.5 7.81 Irregular 4 4 385.7  7.81  3.60 3.60  CFT2-P 385.8 7.81 00.29  0.354 51.1 38.8 P CFT3‐P  pentagon CFT1‐H  6 4 30.7  23.3  300.5  3.42  3.60 2.60  0.29 0.82  CFT3-P 385.7 7.81 Irregular    H  0.313  CFT2‐H  6  42.0  31.9  295.0  3.42  2.60  CFT1-Hhexagon  6 30.7 23.3 300.5 3.42 2.60 0.820  Irregular CFT3‐H  6 6 42.0  31.9  300.5  3.42  2.60 2.60  CFT2-H 42.0 31.9 295.0 3.42 00.82  0.313 H hexagon Note: fCFT3-H cu,m is the tested average concrete cubic strength (150 mm × 150 mm × 150 mm); f  = 0.76fcu,m0.82  is  6 42.0 31.9 300.5 3.42 c,m2.60

 

Note: f cu,m is the tested average concrete cubic strengthř (150 mm ˆ 150f yi mm Asi ˆ 150 mm); f c,m = 0.76f cu,m is the

the  average  concrete  axial  compressive  strength  [25];  f yi Afsiy  average concrete axial compressive strength [25]; f y “

ř

is  the  equivalent  steel  strength;   

is the equivalent steel strength; ρ1 is external Asi

Asi tube steel plate ratio; ρ2 is inner partition steel plate ratio.  is external steel tube steel plate ratio; ρ 2 is inner partition steel plate ratio.  ρ1steel

  Figure 2. Column construction details.  Figure 2. Column construction details.

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(a) welding plates 

(b) group P 

(c) latticed plates 

(d) group H 

Figure 3. Column construction pictures.  Figure 3. Column construction pictures.

3.1.2. Material Properties  3.1.2. Material Properties The  concrete  strength  has  been  shown  in  Table  1.  The  tested  yield  strength,  the  ultimate  The concrete strength has been shown in Table 1. The tested yield strength, the ultimate strength, strength,  the  tensile  elongation,  the  elasticity  modulus  of  reinforcement  and  the  steel  plates  have  the tensile elongation, the elasticity modulus of reinforcement and the steel plates have been shown in been shown in Table 2.  Table 2. Table 2. Mechanical properties of reinforcement and steel plates.  Table 2. Mechanical properties of reinforcement and steel plates.

Group 

Type 

Location fy (MPa) fu (MPa)  ρ (%)  Es (MPa) f y (MPa) f u (MPa) ρ (%) Es (MPa) Vertical and horizontal    and horizontal 5  6mm steel plate  Vertical stiffening ribs, lattice    416  528  27.5  2.10 × 10 6 mm steel plate stiffening ribs, lattice 416 528 27.5 2.10 ˆ 105 partition steel plate  partition steel plate 5  10mm steel plate  Solid partition steel plate  P  10 mm steel plate Solid partition steel plate 409 409  498 498  27.6 27.6  2.12 2.12 × 10 ˆ 105 P 5 5  1212mm steel plate  mm steel plate External steel tube 373 373  525 525  27.4 27.4  2.06 2.06 × 10 ˆ 10 External steel tube  5 Longitudinal reinforcement 382 582 31.3 ø6 reinforcement 2.07 ˆ 10 5  ø6 reinforcement  Longitudinal reinforcement  382  582  31.3  2.07 × 10 ø10 reinforcement Longitudinal reinforcement 310 473 36.7 2.05 ˆ 105 ø10 reinforcement  Longitudinal reinforcement  310  473  36.7  2.05 × 105  5 5mm steel plate  mm steel plate SteelSteel tube  tube 296 296  428 428  28.9 28.9  2.06 2.06 × 10 ˆ 105 5  Longitudinal reinforcement 334 445 24.5 ø8 reinforcement 2.05 ˆ 105 5  H ø8 reinforcement  Longitudinal reinforcement 363 334  446 445  26.3 24.5  2.05 × 10 ø10 reinforcement Longitudinal reinforcement 2.07 ˆ 105 H  5  ø10 reinforcement  Longitudinal reinforcement  ø12 reinforcement Longitudinal reinforcement 326 363  423 446  27.1 26.3  2.04 2.07 × 10 ˆ 105 Longitudinal reinforcement  326  423  27.1  2.04 × 105  Note: ø12 reinforcement  f is the tested yield strength; f is tested ultimate strength; ρ is the tested tensile elongation; E is the

Group

Type

Location

y

u

s

tested elastic modulus of steel. Note: f y is the tested yield strength; f u is tested ultimate strength; ρ is the tested tensile elongation; Es is  the tested elastic modulus of steel. 

3.1.3. Experimental Set-Up 3.1.3. Experimental Set‐Up  A 40,000 kN universal testing machine was used to conduct the axial compressive tests. The axial 40,000  kN  universal  testing  was  used  conduct  the  axial  compressive  tests.  The  load isA applied at the centroid of themachine  cross-section. The to  coordinate of the centroid is calculated by ř ř ř ř axial load is applied at the centroid of the cross‐section. The coordinate of the centroid is calculated  formulas x “ σi Ai xi { σi Ai and y “ σi Ai yi { σi Ai , where σi is the strength of concrete part or steel part, and xAi is the of concrete part. the upper lower end of the σi Ai xarea σi A  or σi Asteel σi Ai , On σ i   is and where  the the strength  of  concrete  by  formulas  i i   and  ypart i yi loading device, the spherical hinges have been arranged. During the testing, a cyclically uniaxial load part or steel part, and  Ai is the area of concrete part or steel part. On the upper and the lower end  was applied to the specimens to study the residual deformation each time the unloading was finished. of prevent the  loading  device, ofthe  hinges  have was been  arranged.  During  testing,  a  cyclically  To the overturn thespherical  specimens, the device unloaded to 2000 kN. the  During the initial stage uniaxial  load  was  applied  to  the  specimens  to  study  the  residual  deformation  each  time  the  (i.e., the elastic stage), the specimens were loaded at the intervals of one-sixth an estimated ultimate unloading was finished. To prevent the overturn of the specimens, the device was unloaded to 2000 kN.  load. After evident yield appeared on the load-displacement curves, the loading process principle During  the  initial  stage  (i.e.,  the  elastic  stage),  the  specimens  were  loaded  at  the  intervals  of  turned out to be controlled by displacement. one‐sixth an estimated ultimate load. After evident yield appeared on the load‐displacement curves,  The two displacement meters, which were used to measure the vertical displacement, were the loading process principle turned out to be controlled by displacement.  arranged in the central part of the specimens, where the deformation was uniform. The gauge length The  two  displacement  meters,  which  were gauges used  to  vertical  displacement,  were  of the displacement meters is 1600 mm. The strain tomeasure  measurethe  longitudinal deformation were arranged  of  the  specimens,  the  deformation  uniform.  gauge  also placedin  onthe  the central  exteriorpart  of steel plates of centralwhere  steel tubes in the verticalwas  direction. TheThe  real-time length  of  the  displacement  meters  is  1600  mm.  The  strain  gauges  to  measure  longitudinal  values of the load, the displacement and the strain were gathered by using a data gathering system; deformation  were  also  placed  on and the  exterior  steel  plates  central  steel  tubes  in  the  vertical  the buckling of external steel tube crack of of  welding seamsof  were recorded manually. The test direction. The real‐time values of the load, the displacement and the strain were gathered by using  scene photo has been shown in Figure 4. The arrangement of displacement meters has been shown in a  data  system;  the strain buckling  of  external  steel shown tube  and  crack 6.of  welding  seams  were  Figure 5;gathering  the distribution of the gauges has also been in Figure recorded  manually.  The  test  scene  photo  has  been  shown  in  Figure  4.  The  arrangement  of  displacement meters has been shown in Figure 5; the distribution of the strain gauges has also been  shown in Figure 6. 

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  Figure 4. Test scene. 

Figure 4. Test scene. Figure 4. Test scene. 

 

 

Figure 4. Test scene. 

  Figure 5. Displacement meters arrangement. 

 

  Figure 5. Displacement meters arrangement. 

Figure 5. Displacement meters arrangement. Figure 5. Displacement meters arrangement. 

(a) group P

(b) group H

Figure 6. The arrangement of strain gauges. 

3.1.4. Test Phenomenon 

(a)  group P (a)  group P

(b)  group H (b)  group H

Figure 6. The arrangement of strain gauges.  All  the  specimens  went  through  a  similar  failure ofprocess,  i.e.,  wrinkling  of  oil  painted  skin,  Figure 6. The arrangement of strain gauges.  Figure 6. The arrangement strain gauges. buckling of steel plates, cracking of welding seams, breaking of concrete, etc. The final failure patterns  3.1.4. Test Phenomenon  have been shown in Figure 7.  3.1.4. Test Phenomenon  3.1.4. Test Phenomenon All  the  specimens  went  through  a  similar  failure  process,  i.e.,  wrinkling  of  oil  painted  skin,  There are few differences between the two groups of specimens. The wrinkling of oil painted  All  the  specimens  went  through through  similar  failure  process,  i.e.,  wrinkling  of  painted  buckling of steel plates, cracking of welding seams, breaking of concrete, etc. The final failure patterns  skin  in  group  P  specimens  is  horizontal  cracks,  while process, that  in  group  H  specimens  is  45‐degree  All the specimens went aa  similar failure i.e., wrinkling of oil  oil painted skin,  skin, buckling of steel plates, cracking of welding seams, breaking of concrete, etc. The final failure patterns  staggered  cracks.  It  shows  that  the  vertical  strain  develops  faster than  the  hoop  strain  in group  P  have been shown in Figure 7.  buckling of steel plates, cracking of welding seams, breaking of concrete, etc. The final failure patterns specimens,  while  the  vertical  have been shown in Figure 7.  have beenThere are few differences between the two groups of specimens. The wrinkling of oil painted  shown in Figure 7. strain  develops  close  to  the  hoop  strain  in  group  H  specimens.  The  buckling  regions  of  group  P is  specimens  are  few  and  concentrate  in  only H  two  to  three is  regions,  but  skin  in  group  P  specimens  horizontal  cracks,  while  that  in  group  specimens  There are few differences between the two groups of specimens. The wrinkling of oil painted  There are few differences between the two groups of specimens. The wrinkling of oil45‐degree  painted skin staggered  cracks.  It  shows  that  the  vertical  strain  develops  faster than  the  hoop  strain  in group  P  each buckling region is large; by contrast, the buckling regions of group H specimens are numerous  skin  in  group  P  specimens  is  horizontal  cracks,  while  that  in  H  specimens  is  staggered 45‐degree  in group P specimens is horizontal cracks, while that in group H group  specimens is 45-degree specimens,  while  the  vertical  strain  develops  close  to  and  the  hoop  strain  in  group  H great  specimens.  The  and  scattered,  but  each  local  region  is  small  protruding.  There  differences  staggered  cracks.  It  shows  that buckling  the  vertical  strain  develops  faster than  the are  hoop  strain  P  cracks. It shows that the vertical strain develops faster than hoop strain group P in group  specimens, buckling  regions  of  group  specimens  are  few  and  in  only  two  to inthree  regions,  but  between  the  two  groups  of P specimens.  The  reason  is concentrate  that  the the steel  ratio  of  group  P  specimens  is  specimens,  while  the  vertical  strain  develops  close  to  the  hoop  strain  in  group  H  specimens.  The  whilehigher than that of group H specimens by 58.63% to 85.05%. The cross sectional moment of inertia  the vertical strain develops close to the hoop strain in group H specimens. The buckling regions each buckling region is large; by contrast, the buckling regions of group H specimens are numerous  buckling  regions  of  group  P  specimens  are  few  and  concentrate  in  only  two  to  three  regions,  but  of group P specimens few and concentrate in two three regions, but region of  group  P  specimens  in local  the  two  main region  directions  is  close  to  each  other,  whereas  it  is  buckling not  close  to  and  scattered,  but are each  buckling  is only small  and to protruding.  There  are each great  differences  each buckling region is large; by contrast, the buckling regions of group H specimens are numerous  is large; by contrast, the regions of reason  group H that  specimens numerous and scattered, each  other  in two  the groups  case buckling of of  group  H  specimens.  Owing  to  an  strong  vertical  and  between  the  specimens.  The  is  the arrangement  steel are ratio  of of  group  P  specimens  is  but and  scattered,  but  each  buckling  region  is  small  protruding.  There  are  great  differences  higher than that of group H specimens by 58.63% to 85.05%. The cross sectional moment of inertia  horizontal stiffening ribs, the stability of external steel tubes of group P specimens is better and the  each local buckling regionlocal  is small and protruding. Thereand  are great differences between the two groups between  the  two  groups  of  specimens.  reason  that  steel  ratio  of  group  P  specimens  is  of  group  P  specimens  the  two  directions  is is  close  to the  each  other,  whereas  it  is that not  close  to  H confinement effect to infill concrete is stronger.  of specimens. The reason in  is that the main  steelThe  ratio of group P specimens is higher than of group each  other  in  the  case  of  group  H  specimens.  Owing  to  an  arrangement  of  strong  vertical  and  higher than that of group H specimens by 58.63% to 85.05%. The cross sectional moment of inertia  specimens by 58.63% to 85.05%. The cross sectional moment of inertia of group P specimens in the horizontal stiffening ribs, the stability of external steel tubes of group P specimens is better and the  of  group  specimens  in  the  to  each  other,  whereas  is  not  close  H to  two main P  directions is close totwo  eachmain  other,directions  whereas itis isclose  not close to each other in theit  case of group confinement effect to infill concrete is stronger.  each  other  in  the  case  of  group  H  specimens.  Owing  to  an  arrangement  of  strong  vertical  and 

specimens. Owing to an arrangement of strong vertical and horizontal stiffening ribs, the stability of horizontal stiffening ribs, the stability of external steel tubes of group P specimens is better and the  external steel tubes of group P specimens is better and the confinement effect to infill concrete is stronger. confinement effect to infill concrete is stronger. 

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  (a) CFT1‐P 

(a) CFT1‐P 

(b) CFT2‐P  

 

(c) CFT3‐P 

(d) CFT1‐H

(e) CFT2‐H 

  (f) CFT3‐H 

(b) CFT2‐P  (c) CFT3‐P  (d) CFT1‐H (e) CFT2‐H  Figure 7. Failure patterns of specimens. 

(f) CFT3‐H 

Figure 7. Failure patterns of specimens. Figure 7. Failure patterns of specimens.  3.1.5. Load F‐Average Strain ε Curves 

3.1.5. Load F-Average Strain ε Curves 3.1.5. Load F‐Average Strain ε Curves  Under  a  cyclically  uniaxial  load,  the  tested  load  F‐average  strain  ε  curves  and  the  related  Under a cyclically uniaxial load, the tested load F-average strain ε curves and the related backbone backbone curves of the six columns have been shown in Figure 8. In Figure 8, F is the applied axial  Under  a  cyclically  uniaxial  load,  the  tested  load  F‐average  strain  ε  curves  and  the  related  load  during  the  testing  process,  the  average  strain  that  from  the load related  curves of the six columns have beenwhereas  shown ε  inis Figure 8. In Figure 8, F is  is switched  the applied axial during backbone curves of the six columns have been shown in Figure 8. In Figure 8, F is the applied axial  displacement  in  the  1600  mm  gauge  length  of  the  middle  of  columns.  The  tested  characteristic  the testing process, whereas ε is the average strain that is switched from the related displacement in load  points have been shown in Table 3. In Table 3, F during  the  testing  process,  whereas  ε  is  the ut is the tested peak load, whereas ε average  strain  that  is  switched utfrom  the  related   is the tested  the 1600 mm gauge length of the middle of columns. The tested characteristic points have been shown displacement  in  the ut1600  mm  gauge  length  of  the  middle  of  columns.  The  tested  characteristic  strain related to F ; Fu0 is the aggregate value bearing capacity s of concrete and the steel part.  in Table 3. In Table 3, Fut is the tested peak load, whereas εut is the tested strain related to Fut ; Fu0 is points have been shown in Table 3. In Table 3, Fut is the tested peak load, whereas εut is the tested  the aggregate value bearing capacity s of concrete and the steel part. strain related to Fut; Fu0 is the aggregate value bearing capacity s of concrete and the steel part. 

(a) CFT1‐P 

(b) CFT2‐P

(a) CFT1‐P 

(b) CFT2‐P

(e) CFT1‐H 

(f) CFT2‐H 

(c) CFT3‐P

(d) Backbone curves

(c) CFT3‐P

(g) CFT3‐H 

(d) Backbone curves

(h) Backbone curves 

Figure 8. Tested F‐ε curves and backbone curves of specimens.  Table 3. Peak results of specimens.  (f) CFT2‐H   (g) CFT3‐H 

(e) CFT1‐H 

(h) Backbone curves 

Columns  CFT1‐P  CFT2‐P  CFT3‐P  CFT1‐H  CFT2‐H  CFT3‐H  Figure 8. Tested F‐ε curves and backbone curves of specimens.  Tested F-ε32119  curves and backbone curves 17400  of specimens. FutFigure /kN  8. 26233  33496  14800  17557  εut/με  Fu0 

3316  4049  5650  3144  2495  Table 3. Peak results of specimens.  Table 3. Peak results of specimens. 24370  26268  26612  12636  14343 

2800  15138 

Columns  CFT1‐P  CFT2‐P  CFT3‐P  CFT1‐H  CFT2‐H  CFT3‐H  Columns 3.1.6. Backbone Curves of Load F‐Measured Strain ε i  Fut/kN  CFT1-P 26233  CFT2-P 32119  CFT3-P 33496 

CFT1-H 14800 

CFT2-H 17400 

CFT3-H 17557 

Fut 26233 32119 33496 14800 17400 17557 εut/kN /με  3316  4049  5650  3144  2495  2800  Parts of the tested backbone curves of load F‐measured strain ε i curves are shown in Figure 9.  ε /µε 3316 4049 5650 3144 2495 2800 ut In the figure, εaF  is the tested average strain switched from the related displacement.  24370  26268  26612  12636  14343  15138  Fu0u0  24370 26268 26612 12636 14343 15138 From the figure, it can be known that the change law of the measured stain by strain gauges is  similar  to  the  average  stain  switched  from  the  related  displacement.  Moreover,  when  the  3.1.6. Backbone Curves of Load F‐Measured Strain εi  3.1.6.longitudinal deformation of the columns reaches a larger value, local buckling of the steel plate may  Backbone Curves of Load F-Measured Strain εi Parts of the tested backbone curves of load F‐measured strain ε i curves are shown in Figure 9.  occur,  so  that  the  strain  measured  by  strain  gauges  may  not  be  accurate  any  more.  Thus,  in  this 

Parts of the tested backbone curves of load F-measured strain εi curves are shown in Figure 9. In the figure, εa is the tested average strain switched from the related displacement.  In the figure, εa is the tested average strain switched from the related displacement. From the figure, it can be known that the change law of the measured stain by strain gauges is  similar  to  the  average  stain  switched  from  the  related  displacement.  Moreover,  when  the  longitudinal deformation of the columns reaches a larger value, local buckling of the steel plate may  occur,  so  that  the  strain  measured  by  strain  gauges  may  not  be  accurate  any  more.  Thus,  in  this 

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From the figure, it can be known that the change law of the measured stain by strain gauges is similar to the average stain switched from the related displacement. Moreover, when the longitudinal deformation of the columns reaches a larger value, local buckling of the steel plate may occur, so that Materials 2016, 9, 86  9 of 19  the strain measured by strain gauges may not be accurate any more. Thus, in this paper, in order to simply calculations and reduce errors, the average strain switched from the related displacement is paper,  in  order  to  simply  calculations  and  reduce  errors,  the  average  strain  switched  from  the  applied to study the relationship of the concrete confined by special-shaped steel tube coupled with related displacement is applied to study the relationship of the concrete confined by special‐shaped  multiple cavities. steel tube coupled with multiple cavities. 

  (a) CFT1‐P

(b) CFT2‐P

(c) CFT3‐P 

(e) CFT2‐P 

(f) CFT3‐P 

  (d) CFT1‐P 

Figure 9. Backbone curves of F‐ε Figure 9. Backbone curves of F-εii. .

3.2. Determination of the Effective Confinement Coefficient ke  3.2. Determination of the Effective Confinement Coefficient ke The confinement action of core concrete that is confined by a special‐shaped CFT coupled with  The confinement action of core concrete that is confined by a special-shaped CFT coupled with multiple  cavities is  different  from one  that is confined by normal reinforcement. The external steel  multiple cavities is different from one that is confined by normal reinforcement. The external steel tube tube  has  strong  confinement  action  at  the  corners  and  the  locations  that  were  provided  with  has strong confinement action at the corners and the locations that were provided with partition steel partition  steel  plates,  longitudinal  stiffening  ribs  or  transversal  stiffening  ribs.  The  confinement  plates, longitudinal stiffening ribs or transversal stiffening ribs. The confinement action is relatively action  is  relatively  weak  at  the  central  part  of  the  external  steel  tube  plate  that  may  exist  either  weak at the central part of the external steel tube plate that may exist either between the two adjacent between the two adjacent longitudinal stiffening ribs or between the adjacent longitudinal stiffening  longitudinal stiffening ribs or between the adjacent longitudinal stiffening rib and the partition steel rib  and  the  partition  steel  plate.  Therefore,  the  cross‐sectional  boundary  in  transversal  direction  plate. Therefore, the cross-sectional boundary in transversal direction between the active confined between the active confined region and the inactive confined region is assumed to be a parabola, the  region and the inactive confined region is assumed to be a parabola, the same as occurs in lateral same as occurs in lateral cross‐section due to similar confinement features between the two adjacent  cross-section due to similar confinement features between the two adjacent transverse stiffening ribs. transverse stiffening ribs.  The cavity construction and the arrangement of stiffening ribs in two group columns differ The  cavity  construction  and  the  arrangement  of  stiffening  ribs  in  two  group  columns  differ  from each other. If the arranged stiffening ribs are weak, the confinement action of the steel tube to from each other. If the arranged stiffening ribs are weak, the confinement action of the steel tube to  the core concrete is relatively weak. In that case, the concrete near to the stiffening ribs cannot be the  core concrete is relatively weak. In that  case, the concrete near to  the  stiffening ribs cannot be  divided into the active confined region. Keeping this in view, specific criteria has been proposed to divided into the active confined region. Keeping this in view, specific criteria has been proposed to  evaluate whether the stiffening ribs are able to offer sufficient confinement ability or not. Firstly, the evaluate whether the stiffening ribs are able to offer sufficient confinement ability or not. Firstly, the  division of active and inactive confined regions is based on the straight sides of the external steel tube. division of active and inactive confined regions is based on the  straight sides of the  external steel  In case the boundary parabola intersects all the longitudinal stiffening ribs whose width to thickness tube.  In  case  the  boundary  parabola  intersects  all  the  longitudinal  stiffening  ribs  whose  width  to  ratios are relatively small, sufficient confinement ability can preliminary be verified. Secondly, if the thickness  ratios  are  relatively  small,  sufficient  confinement  ability  can  preliminary  be  verified.  boundary parabola only intersects parts of the longitudinal stiffening ribs, a parameter S describing the Secondly,  if  the  boundary  parabola  only  intersects  parts  of  the  longitudinal  stiffening  ribs,  a  contribution of stiffening ribs of the cavity side on the whole cavity is additionally used to verify the parameter S describing the  contribution of stiffening ribs of the cavity side on the whole cavity is  confinement ability. Similarly, the confinement ability of transversal stiffening ribs has been assured by additionally used to verify the confinement ability. Similarly, the confinement ability of transversal  S. The expression of parameter S can be written as follows: stiffening ribs has been assured by S. The expression of parameter S can be written as follows: 

S

Aj  100% , where Aj is the cross‐sectional area of stiffening ribs on one cavity side, bj is  bj Aaj cj

the length of cavity side, cj is the perimeter of the cavity, and Aaj is the cross‐section area of the cavity. 

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Aj S “ ˆ 100%, where A j is the cross-sectional area of stiffening ribs on one cavity side, b j is The issue of confinement ability of stiffening ribs is very complex and needs extensive research.  bj A Additionally, the samples of  experimental column are limited, so the validity of stiffening ribs can  c j aj be determined on  qualitative basis  only. The estimated results have  been shown in Table 4. The  the length of cavitya side, c j is the perimeter of the cavity, and Aaj is the cross-section area of the cavity. division of active and inactive confined concrete has been shown in Figure 10.  The issue of confinement ability of stiffening ribs is very complex and needs extensive research. Additionally, the samples of experimental column are limited, so the validity of stiffening ribs can be Table 4. The validity determination of stiffening ribs.  determined on a qualitative basis only. The estimated results have been shown in Table 4. The division of active and inactive confinedSide  concrete has been shown in Figure Side  10. Evaluating    CFT1‐P  CFT2‐P  CFT3‐P  Columns  CFT1‐H  CFT2‐H  CFT3‐H  Parameters  Number  Number  Table‐ 4. The validity of stiffening b/t  15  determination 15  15  ‐  ribs. 8.3  8.3  8.3  Columns

Longitudinal    stiffening ribs 

Evaluating Parameters

S/% 

b/t

Longitudinal S/% stiffening ribs Validity 

b/t  Validity b/t

Transverse    Transverse stiffening ribs  stiffening ribs

S/% 

S/%

Validity Validity 

1‐1  Side Number 1‐2 

3.192 

3.192  Side 1‐1 (1‐2) 

0.850 

2.530  Number 2 

0.787 

1.958  CFT2-P CFT3-P CFT1-P 2.530 

CFT1-H

0.850  CFT2-H

0.787 

0.850  CFT3-H

0.787 

2 -

15 1.948 

15 3.809  3.192 1.958 ‐  ‐  2.530 1.948 3.809 Inactive  Active  15  15  Inactive Active

15 - 3  8.3 3.809  0.526  8.3 0.526  8.3 0.526  3.192 1-1 (1-2) 0.850 0.850 0.850 4‐1 (4‐2)  0.542  0.787 0.542  0.787 0.542  2.530‐  2 0.787 3.809 0.526 0.526 Active  3 ‐  Inactive  0.526 Inactive  Inactive  4-1 (4-2) 0.542 0.542 0.542 15  ‐  Inactive ‐  Inactive ‐  Active - ‐  Inactive

1‐1  1‐2  1-1 1-2 2 

3.629  15 2.128  3.629 0.435 0.649  2.128 1.686  0.649 1.686 0.743  1.270  0.743 1.270 Inactive Active

3.629  15 2.128  3.629 2.128 1.686  1.686 1.270  1.270 Active

1-1 3  1-2 ‐ 2 3 ‐ -

2 3 3 -

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  Figure 10. The active confined concrete and inactive confined concrete.  Figure 10. The active confined concrete and inactive confined concrete.

To determine the extent to which concrete is confined actively, a local coordinate system o‐xyz  To determine the extent to which concrete is confined actively, a local coordinate system o-xyz   has  been defined, as it  has  been  shown in Figure  11. The x axis is parallel to the straight side; the  has been defined, as it has been shown in Figure 11. The x axis is parallel to the straight side;of  the y y  axis  is  perpendicular  to  the  straight  side;  and  the  z  axis  is  in  the  longitudinal  direction  the  axis is perpendicular to the straight side; and the z axis is in the longitudinal direction of the column. column. The distance either  between the adjacent effective longitudinal stiffening ribs or between  The distance either between the adjacent effective stiffening ribsplate  or between the adjacent the  adjacent  effective  longitudinal  stiffening  rib longitudinal and  the  partition  steel  is  b.  The  distance  effective longitudinal stiffening rib and the partition steel plate is b. The distance between the adjacent between  the  adjacent  effective  transversal  stiffening  ribs  is  regarded  as  H.  If  the  points  of  effective transversal stiffening ribs is regarded as H. If the points of intersection between the parabola intersection between the parabola and steel plate side are (−b/2, 0) and (b/2, 0) and the included angle  and steel the  plate side are (´b/2, 0) parabola  and (b/2,and  0) and theplate  included angle the tangent line between  tangent  line  of  the  steel  side  at  the between points  of  intersection  is  of θ,    the parabola and steel plate side at the points of intersection is θ, the transversal boundary between the transversal boundary between the active confined concrete and the inactive confined region can  the active confined concrete and the inactive confined region tan θ 2canbbe expressed by using the relation: y  f ( x )   x  tan θ .  Analogously,  the  lateral  b be  expressed  tanθ by  using  the  relation:  y “ f pxq “ ´ x2 ` tanθ. Analogously, the lateralb boundary 4 can be expressed by using the b 4 tan θ 2 h tanθ 2 h y  g ( z)   z  tan θ .  relation: y “ gpzq “ ´ z ` tanθ. boundary can be expressed by using the relation:  h 4 h 4

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  (a) 

  (b) 

  (c)

(d)

Figure  11.  Schematic  diagram  of  3D  boundary  of  inactive  and  active cofined  concrete.  (a)  Isolated  Figure 11. Schematic diagram of 3D boundary of inactive and active cofined concrete. (a) Isolated body; (b) Transverse boundary; (c) Lateral boundary; (d) Plane graph.  body; (b) Transverse boundary; (c) Lateral boundary; (d) Plane graph.

The maximum value of f(x) and g(z) may not be equal to each other. When H is greater than that  The maximum value of f (x) and g(z) may not be equal to each other. When H is greater than of  b,  g(z)max  is  greater  than  that  of  f(x)max,  and  the  transversal  boundary  differs  from  the  lateral  that of b, g(z)max is greater than that of f (x)max , and the transversal boundary differs from the lateral boundary. It shows that the confinement action in transversal cross‐section is stronger than that of  boundary. It shows that the confinement action in transversal cross-section is stronger than that of the  lateral  cross‐section.  The  transversal  confinement  action  is  sufficient;  therefore,  the  transverse  the lateral cross-section. The transversal confinement action is sufficient; therefore, the transverse boundary can be regarded as a primary boundary.  boundary can be regarded as a primary boundary. a( h  b ) h h It is logical to know that when  z   ph ´ bqh, then g(z) = f(x)max =  htan θ . Therefore, the 3D  It is logical to know that when z “ ˘ , then g(z) = f (x)max =4 tanθ. Therefore, the 3D 2 2 4 boundary of active and inactive confined concrete can be expressed as follows:  boundary of active and inactive confined a a concrete can be expressed as follows: ( h ph  b´ ) hbqh hh ( hph ´ b ) hbqh h h When    or or  ă  zză ´  ză z ă, the included angle between the tangent line  When ´ , the included angle between the tangent 22 2 2 2 2 2 2 line of the parabola f (x) and the axis at the intersection points changes along with the z value. of the parabola f(x) and the axis at the intersection points changes along with the z value. The f(x)max  The f (x) can be determined by using the relation g(z). The parabola f (x) intersects xz plane at can  be  max determined  by  using  the  relation  g(z).  The  parabola  f(x)  intersects  xz  plane  at  points    points (´b/2, 0, z) and (b/2, 0, z); therefore, the transversal boundary can be modified to be included (−b/2,  0,  z)  and  (b/2,  0,  z);  therefore,  the transversal boundary can  be modified to  be  included  as:  4 2 ` gpzq. as: y “ f pxq “4 ´ 2 gpzqx b y  f ( x)   a g ( z ) x 2  g ( z ) .  a 2 b ph ´ bqh ph ´ bqh When ´ ă z ă , the transversal boundary cannot be modified and the ( h 2 b ) h ( h  b )2h When   ,  the  boundary  cannot  be  modified  and  the  z tanθtransversal  b 2 2 “´ expression is considered as: y “ f pxq x2 ` tanθ. b 4 tan θ 2 b as: As a result, the final 3D boundary y  f ( x) can  be written x  tan θ .  expression is considered as:  b 4 a $ ph ´ bqh 4 h As a result, the final 3D boundary can be written as:  ’ 2 ’ ´ 2 gpzqx ` gpzq, ´ ă z ă ´ ’ ’ ’ 2 b a a2 &  ph ´ bqh tanθ b ( h ph b)´ h bqh 4 h 2 2 y “ tpx, zq “ (6) ´ ),  g ( z´   ză  z ă  b 2xg (`z )4x tanθ, ’ 2 2 ’ a 2 2 ’ b  ’ ph ´ bqh ’ 4 h %  2 ă z(ă h 2b)h b ` gpzq, (h  b2)h  tan´θb2 2gpzqx z y  t ( x, z )   x  tan θ ,    (6)

b

4

2

2

To simplify the estimation, aconstant value of θ = 45˝ is applied [26]. The volume of the inactive  ( h  b) h 4 h process, the distance 2 confined concrete can be calculated  2using  g ( z ) , (7). In the estimation z g ( z ) xEquation  by 2 as an effective length b between the two adjacent active stiffening ribs can2be regarded  transversal H. If there are no active transversal stiffening ribs, the whole length of column can be regarded as To  simplify  the  estimation,  a  constant  value  of  θ  =  45°  is  applied  [26].  The  volume  of  the  an effective length. inactive  confined  concrete  can  be  calculated  by  using  Equation  (7).  In  the  estimation  process,  the  ” ı a distance between the two adjacent active transversal stiffening ribs can  be  regarded as an  effective  a 2´ 2 btanθ 2h p2h ` bq hph ´ bq s b tanθ hph ´ bq h h b b length  If  there  be  Vin “ H. tpx, zqdxdz,are  ´ no  ă z active  ă , ´ transversal  ă x ă “ stiffening  ribs,  the  ` whole  length  of  column  can  (7) 2 2 2 2 6 18 regarded as an effective length. 

Vin 



2 2 h h b b b tan θ h(h  b) b tan θ  2h  (2h  b) h(h  b)  t( x, z )dxdz,  0.85, the yielding strength of steel tube plate, due to local buckling, can be assessed by using Equation (11). fa 1.2 0.3 “ ´ 2 ď 1.0 (11) fy Ri Ri During estimation, ν = 0.283 [13], even though the stiffening ribs in columns CFT1-H–CFT3-H are inactive in the division of active and inactive region of confined concrete, they are active in considering local buckling of the steel tube plate. The estimated results have been shown in Table 5, and the issue of local buckling for all the columns in this article does not need to be considered. By neglecting the issue of local buckling, Sakino et al. [28] have suggested that the hooping stress f sr and the longitudinal stress f a can be assumed as 0.19f y and ´0.89f y respectively, whereas Architectural Institute of Japan (AIJ) [29] standard suggested that the hooping stress f sr and the longitudinal stress f a can be assumed as 0.21f y and ´0.89f y, respectively. By considering the non-uniformity of the special-shaped CFT coupled with multiple cavities in this paper, the smaller values of f sr = 0.19f y was applied in the estimation process. To simplify the estimation, the stress path of steel tube plate was simplified into a straight line, as has been shown in Figure 12.

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Table 5. Estimated results of related parameters of confined concrete. Columns

CFT1-P

CFT2-P

CFT3-P

CFT1-H

CFT2-H

CFT3-H

ke R fl fl ξ1 ξ2 ξ3 ξ η f c0 εc0 Ec f cc εcc r Fut /kN Fut /Fut εcc /εut

0.461 (0.498) 0.346 3.563 (0.648) 1.641 (0.323) 0.8309 0.0255 0.1991 1.0555 3.800 38.84 1772 32831 49.12 (51.01) 3565 (3881) 1.725 (1.718) 27312 1.041 1.075

0.856 0.255 4.692 4.017 0.8476 0 0.8992 1.7468 2.796 38.84 1772 32831 61.41 4654 1.673 33343 1.038 1.149

0.856 (0.498) 0.255 4.692 (0.648) 4.015 (0.323) 0.8503 0.0316 0.9021 1.7840 2.855 38.84 1772 32831 61.40 (62.87) 4709 (4901) 1.659 (1.705) 33760 1.008 0.834

0.699 0.465 2.473 1.729 0.4638 0.1290 0.7075 1.3003 2.353 23.32 1531 26269 33.55 3111 1.696 15350 1.037 0.990

0.702 0.465 2.473 1.735 0.3361 0 0.5126 0.8487 1.528 31.90 1671 30753 42.57 2523 2.213 17470 1.004 1.011

0.699 0.465 2.473 1.729 0.3391 0.0943 0.5172 0.9506 1.720 31.90 1671 30753 42.53 2629 2.109 18343 1.045 0.939

Note: The values as shown in brackets are meant for multi-confined concrete by steel tube and effective reinforcement. Materials 2016, 9, 86  13 of 19 

  Figure 12. Curve of hooping and longitudinal stresses of steel tube. 

Figure 12. Curve of hooping and longitudinal stresses of steel tube. 3.4. Determination of Equivalent Tensile Force Fb of Partition Steel Plate at Peak Point 

3.4. Determination of Equivalent solid partition steel plate is similar to that of steel tube. The radial stress  Tensile Force Fb of Partition Steel Plate at Peak Point The stress status of a  Thecan be neglected, as it is relatively small in comparison with longitudinal and hooping stresses. It is  stress status of a solid partition steel plate is similar to that of steel tube. The radial stress assumed  that  the  partition  steel  plate  can  be in a  plane stress  status  and accords  with  von Mises  can be neglected, as it is relatively small in comparison with longitudinal and hooping stresses. It is yield criterion. Both sides of the steel plate are constrained by concrete. Therefore, the issue of local  assumed that the partition steel plate can be in a plane stress status and accords with von Mises yield buckling can be avoided. The estimation method of longitudinal and hooping stresses can be used to  criterion. Both of the steel plate are by concrete. Therefore, issue of local refer  to sides the  steel  tube  plate  when  R constrained