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 xAi 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 -
‐
15 0.435
Inactive Active Active
-
-
‐
‐
-
-
‐
‐
-
-
‐
‐
-
‐
‐
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 zză ´ 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, aconstant 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