Research ArticleAnalysis and Design of Short FRP-Confined Concrete-EncasedArbitrarily Shaped Steel Columns under Biaxial Loading

Wei He,1,2 Bing Fu ,3,4 and Feng-Chen An5

1School of Water Conservancy and Environment, Zhengzhou University, Zhengzhou, Henan Province, China2Zhengzhou Metro Co. Ltd., Zhengzhou, Henan Province, China3School of Civil and Transportation Engineering, Guangdong University of Technology, Guangzhou, Guangdong Province, China4Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong5College of Petroleum Engineering, China University of Petroleum, Beijing, China

Correspondence should be addressed to Bing Fu; cefubing@gdut.edu.cn

Received 25 June 2018; Revised 26 September 2018; Accepted 30 September 2018; Published 25 November 2018

Academic Editor: Mehdi Salami-Kalajahi

Copyright © 2018 Wei He et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The FRP-confined concrete-encased steel column is a new form of hybrid column, which integrates advantages of all theconstituent materials. Its structural performance, including load carrying capacity, ductility, and corrosion resistance, has beendemonstrated to be excellent by limited experimental investigation. Currently, no systematic procedure, particularly for thatwith reinforced structural steel of arbitrary shapes, has been proposed for the sectional analysis and design for such novel hybridcolumns under biaxial loading. The present paper aims at filling this research gap by proposing an approach for the rapidsection analysis and providing rationale basis for FRP-confined concrete-encased arbitrarily shaped steel columns. A robustiterative scheme has been used with a traditional so-called fiber element method. The presented numerical examplesdemonstrated the validity and accuracy of the proposed approach.

1. Introduction

Lateral confinement leads to the compressed concrete undermultiaxial compression and results in enhancements in bothductility and strength of the compressed concrete [1]. Such afeature is highly preferable for design and constructingcolumns in a region of high seismic risk, where adequate duc-tility of columns is necessary to ensure high moment redistri-bution capacity of structures and avoid collapse of structuresdue to the shaking from large earthquakes [2]. In conven-tional reinforced concrete structures, the lateral confinementto the compressed concrete is mainly provided by the trans-verse steel reinforcement in the form of either spirals orhoops. Concrete-filled steel tubular columns, which havebeen widely used in high-rise buildings, bridges, etc., are alsoutilizing the increased strength and deformability of confinedconcrete to achieve a high structural performance [3, 4]. Insuch a composite column, an outer steel tube is used toreplace longitudinal and transverse steel reinforcements in

conventional reinforced concrete columns and to providecontinuous confinement to concrete infilled. However,such a concrete-filled steel tubular column, especially ina harsh environment, is susceptible to severe corrosionproblem due to the direct exposure of the outer steel tubeto ambient environment.

Fiber reinforced polymer (FRP) composites in theform of wraps or jackets have been widely used to serveas a confining device for seismic retrofit of existing RCcolumns [5, 6]. Its wide application is mainly attributedto the superior properties of FRP composites, such as highstrength-to-weight ratio and excellent corrosion resistance.Recently, combining FRP composites with traditional con-struction materials (e.g., steel and concrete) has gainedincreasing research attention to form a hybrid column toachieve high structural performance by integrating advan-tages of the constituent materials. A successful example isthe hybrid FRP-concrete double-skin tubular columns(DSTCs), which was proposed by Teng et al. [7] and then

HindawiInternational Journal of Polymer ScienceVolume 2018, Article ID 9832894, 11 pageshttps://doi.org/10.1155/2018/9832894

received a great deal of follow-on research (e.g., [8–13]).A DSTC consists of an FRP outer tube, a steel innertube, and a layer of concrete sandwiched between them[7]. The FRP tube offers mechanical resistance primarilyin the hoop direction to confine the concrete and toenhance the shear resistance of the member; the steeltube provides the main longitudinal reinforcement andprevents the concrete from inward spalling and the steeltube from outward local buckling deformations. Optimaluse of FRP, steel, and concrete in the manner of DSTCtherefore makes it an economical and corrosion-resistantcolumn form.

Another successful example that integrates advantagesof all the constituent materials into a hybrid column isthe FRP-confined concrete-encased steel composite col-umns (FCSCs), which was first proposed by Liu et al.[14] for retrofit of existing steel columns. In their study,five notched steel columns to simulate the corroded sec-tion were encased using FRP-confined concrete to enhanceits load carrying capacity, and tests results demonstratedthe feasibility of such a retrofit technique. Karimi andhis coauthors [4, 15, 16] then introduced the FCSCs forthe new construction of a column and conducted a sys-tematic experimental study on the compressive behaviorof both short and slender FCSCs. Recently, Yu et al. [17]presented a combined experimental and theoretical studyon the behavior of FCSCs under concentric and eccentriccompression and revealed that two flanges of H-shapedsteel could provide additional confinement to infilled con-crete thus enhancing the ductility and load carrying of thecomposite columns. In order to further enhance the con-finement efficiency for FRP-confined concrete-encasedsteel composite square columns and inspired by Yu et al.[17], Huang and his coauthors [18] innovatively proposeda new form of FCSCs, in which a cross-shaped steel sec-tion was used to replace H-shaped (or I-shaped) steel.Their test results demonstrated that despite concrete infillin square section, it was effectively confined by both outertube and cross-shaped steel and justified the rationales ofthe proposed new form of composite columns. In somecases, such as corner columns of buildings, irregular crosssection or regular cross section placed asymmetrically isoften encountered [19]. Despite the fact that the structuralperformance of FCSCs has been demonstrated by a num-ber of concentric of eccentric compression tests, noapproach has been proposed for a rapid section analysisand design of FCSCs with arbitrarily shaped steel andunder biaxial loading.

Against the above background, the present paper pro-poses an approach for the rapid section analysis and pro-vides rationale basis for FRP-confined concrete-encasedarbitrarily shaped steel columns. The robust iterativescheme proposed by Chen et al. [19] has been adoptedwith a traditional so-called fiber element method, whereall the constitutive materials were treated as fiber ele-ments avoiding the integration for the stress block ofthe FRP-confined concrete. The presented numericalexamples demonstrated the validity and accuracy of theproposed approach.

2. Methodology

A proper approach is necessary for robust convergence ofthe exact location of the neutral axis. The iterative proce-dure proposed by Chen et al. [19] has been adoptedherein, in which the iterative quasi-Newton procedure isemployed within the Regula-Falsi numerical scheme forthe solution of equilibrium equations. In addition, theuse of the plastic centroidal axes of the cross section asthe reference axes of loading guarantees the convergenceof solution in the iterative process. All the constituentmaterials, including FRP-confined concrete under thecombination of axial compression and bending, are treatedas fiber elements to avoid the definition of stress block forthe FRP-confined concrete, which is difficult to analyticallydetermine. Appropriate constitutive models were used tosimulate the mechanical properties of the constituentmaterials. For instance, strain gradient along the sectionhas been taken into account in the constitutive model ofthe confined concrete, which is simulated by using a so-called variable confinement stress-strain model proposedin the Chinese code (GB50608-2010).

2.1. Basic Assumptions. The sectional analyses and design inthe present paper were conducted based on the followingbasic assumptions:

(1) Section plane remains plane after loading; thisassumption ensures that the strain at any point ofthe cross section is proportional to its distance fromthe neutral axis

(2) Failure limit state is only defined by the attainment ofthe strain εcu of the extreme compression fiber; thismeans no failure mode of steel/steel bar rupture hasbeen considered

(3) Tensile strength of confined concrete is neglected

(4) No contribution of FRP tubes to compressivestrength of composite columns has been takendirectly into account in the section analysis

2.2. Definition of the Reference-Loading Axes. If the usualdefinition of reference-loading axes (i.e., geometric cen-troid as its origin) is used, it is possible that the originof the loading may fall outside the Mx −My interactioncurve, especially when the axial load is close to the axialload capacity of columns with irregular structural steelunder concentric compression [19]. To overcome suchdivergence difficulty, Chen et al. [19] accepted the plasticcentroid of the cross section of columns as the origin ofthe reference-loading axes. Such definition of the originof the reference-loading axes ensures αm, which is theinclination of resultant bending moment resistance and isequal to arctan My/Mx and increases monotonicallyfrom 0 to 2π with increase of θn, orientation of neutralaxis from 0 to 2π. In this way, the existence and unique-ness of θn and convergence are guaranteed.

2 International Journal of Polymer Science

For any arbitrary cross section, the definition of the plas-tic centroid (i.e., the origin of loading-reference axes in thisstudy) was given in (1) and (2):

Xpc =XcAcf c/γc + XsAs f s/γs + XrAr f r/γr

XcAc f c + XsAsf s + XrAr f r, 1

Ypc =YcAcf c/γc + YsAs f /γs + YrAr f r/YrAr f r

YcAc f c + YsAs f s + YrAr f r, 2

in which Ac, As, and Ar are areas of concrete, shaped steel,and steel rebars, respectively; γc, γs, and γr are the safety ofconfined concrete, shaped steel, and steel rebars, respectively;and f c, f s, and f r are the respective characteristic strength ofconfined concrete, shaped steel, and steel rebars and set to bethe ultimate strength of confined concrete, yielding strengthof shaped steel and steel rebars, respectively.

2.3. Fiber Element Method Adopted. In the present paper, allthe structural components in the composite columns,including the confined concrete, structural steel, and steelbars, have been treated as fiber elements to calculate thestress resultants. This approach avoids the difficulty ofthe determination of the stress block for confined concreteunder bending with/without axial compression. Cross sec-tion of any shape has been first meshed to determine thestress resultants of each component. The total resultantsof each component can be obtained by a summation overall the fiber elements given by (3), (4), and (5):

Nz = 〠mc

i=1σciAci + 〠

ms

j=1σsjAsj + 〠

mr

k=1σrkArk , 3

Mx = −〠mc

i=1σciAciyci − 〠

ms

j=1σsjAsjysj − 〠

mr

k=1σrkArkyrk, 4

My = −〠mc

i=1σciAcixci − 〠

ms

j=1σsjAsjxsj − 〠

mr

k=1σrkArkxrk, 5

where Nz , Mx, and My are the stress resultants of axialcompression, moment over x-axis, and moment over y-axis, respectively; mc, ms, and mr are the numbers ofconcrete fibers, structural steel fibers, and reinforcing barfibers, respectively; σci, σsj, and σrk are the stresses of eachconcrete fiber, structural steel fiber, and reinforcing barfiber, respectively; and Aci, Asj, and Ark are areas of respec-tive concrete fiber, structural steel fiber, and reinforcingbar fiber, respectively.

2.4. Modelling of Confined Concrete. The behavior of con-crete confined by FRP jacketing has been extensivelyinvestigated, and a number of strength models have beendeveloped (e.g., [5, 20–22]). All the strength models,which give the stress-strain relationship explicitly, canbe used in the proposed approach; however, a simpleand generally accurate model is more preferable for sucha design purpose. A stress-strain model of confined

concrete in the Chinese code (GB50608-2010), whichreflects the effect of strain gradient along the section,has therefore been adopted in the present section analy-sis. In this model, the slope of the second portion of thestress-strain curve (i.e., E2) is defined as a function ofthe load eccentricity and equal to be that of the confinedmodel in Lam and Teng [5] for the concrete under theconcentric compression (i.e., a zero load eccentricity) andto be zero for the concrete under pure bending (i.e., aninfinite load eccentricity).

As shown in Figure 1, the strain-stress relationship ofconfined subjected to pure bending is formulated by (6)and (7), while that under eccentric compression is given in(8) and (9).

σcc = E1εcc −E21

4f cε2cc for 0 ≤ εcc ≤

2f cE1

, 6

σcc = f c for  2f cE1

≤ εcc ≤ εcc,ub, 7

in which σcc and εcc are the axial stress and axial strain,respectively; εcc,ub is the design ultimate compressive strainof the concrete subjected to pure bending; E1 is the elasticmodulus of unconfined concrete; and f c is the design com-pressive strength of unconfined concrete.

σcc = E1εcc −E1 − E2,ec

2

4f cε2cc for 0 ≤ εcc ≤ εt , 8

σcc = f c + E2,ecεcc for εt ≤ εcc ≤ εcc,uec, 9

in which εcc,uec is the design ultimate compressive strain ofthe concrete in sections subjected to eccentric compressionand E2,ec is the slope of the second linear portion of the

Axial strain �휀c

�휀cu

E21

Ec

�휀t�휀co

Axi

al st

ress

�휎c

fo=f′co

f′cu

In concentric compressionIn pure bending In eccentric compression

Figure 1: Stress-strain models for concrete in FRP-jacketedcomposite columns.

3International Journal of Polymer Science

stress-strain curve. Related parameters above are determinedby using the following equations:

εt =2f c

E1 − E2,ec,

E2,ec = E2d

d + ei,

E2 =f cc′ − f cεcu

,

εcc,uec = εcc,u − εcc,ubd

d + ei+ εcc,ub,

10

in which E2 is the slope of the second portion of the stress-strain curves for the FRP-confined concrete under axial com-pression; f cc′ is the ultimate strength of the confined concreteunder concentric compression; d is the diameter of theconcrete core; ei is the load eccentricity; and εcc,ub is thedesign ultimate compressive strain of the concrete subjectedto pure bending. Such a parameter should be determinedfrom tests and is defined as the smaller value of the designultimate strains obtained from the axial compression testson concrete-filled FRP tubes and hollow FRP tubes, respec-tively. Due to lack of test data, the present manuscriptassumes εcc,ub be equal to the corresponding ultimate strainof FRP-confined concrete under concentric compression.Such an assumption cannot lead to a significant loss in accu-racy and will not adversely affect the main purpose of thepresent manuscript, which is to validate the proposedapproach for the rapid section analysis of FRP-confinedconcrete-encased arbitrarily shaped steel columns. εcu, theultimate strain of FRP-confined concrete under concentriccompression, can be determined following [5] as follows:

εcuεco

= 1 75 + 12 f l,af c

εh,rupεco

0 45, 11

in which εcu and εco are the ultimate strains of the confinedconcrete under concentric compression and unconfinedconcrete, respectively; εh,rup is the ultimate hoop strain ofthe FRP reinforcement; and f l,a is the actual maximumconfining pressure given in the following:

f l,a =2EFRPtεh,rup

d, 12

where EFRP is elastic modulus of the FRP composite.As revealed by Lam and Teng [23], the ultimate hoop

strains of FRP measured in tests on FRP-confined concretecylinders are substantially below those from flat coupon ten-sile tests. As a result, a reduction factor for the determinationof hoop strains of FRP based on the ultimate rupture strain ofFRP from coupon tensile tests should be included, e.g., 0.586for CFRP and 0.624 for GFRP in the present paper.

2.5. Modelling of Shaped Steel and FRP Tubes. In the sectionalanalysis, the structural steel and/or steel bars were also

treated as fiber elements with perfectly plastic property andthe plane section assumption. The plane section assumptionis generally valid, because compression force resisted by thecolumn leads to the expansion of concrete and thereforesignificant interaction between different constituents. Thesize of elements should match that of concrete elements.The effect of FRP tubes on structural performance has beentaken into account by modifying the constitutive of the con-fined concrete with the amount of FRP tubes. This meansthat no axial loading was carried by FRP tubes and noelements in analysis represent the FRP tubes. This makessense as the fiber of tubes is hoop or predominantly hoop,mainly providing confinement to the concrete core. Neglect-ing the contribution of FRP tubes to axial load resistance istherefore reasonable and leads to a slightly conservativeprediction, which is preferable for a design purpose.

2.6. Iterative Procedure for the Sectional Analysis. For a givencomposite column, a Mx −My interaction curve under agiven axial load can be determined by the sectional analysis.The series of such Mx −My interaction curves under differ-ent axial loads can be utilized to judge whether one compos-ite column satisfies the load carrying requirement. Forexample, a point representing a design load set Nzd ,Mxd ,Myd that falls outside the interaction curves means thedesign column does not satisfy the load carrying require-ment; otherwise it is appropriate for use. Detailed iterativeprocedure for the sectional analysis of a FRP-confinedcolumn with structural steel of any shape is given inFigure 2(a). In this iterative procedure, the iterative solu-tion method for parameter dn is adopted that waspresented in Chen et al. [19], and similar approach is usedfor the eccentricity e.

2.7. Design of Required Dimensions of Hybrid Columns. It is acommon task for civil engineers to design the dimensions fora composite column with a given cross section under knownloads Nzd , Mxd , and Myd . In this task, all structural parame-ters, except the size of structural steel, are specified, and itspurpose is to determine the size of structural steel for sucha column. The iterative procedure presented in Figure 2(b)is used for this task.

3. Effect of Constitutive Model ofConfined Concrete

As mentioned in previous sections, different constitutivemodels have been proposed by researchers and adopted forthe sectional analysis for the composite columns under thecombination of axial compression and bending. Good agree-ment between analytical results and test results on hybridFRP double-skin composite columns was obtained by [24],where the constitutive models for concrete in pure bendingwere adopted for analysis. However, such simplificationmay result in too conservative results, especially for thecolumns confined by FRPs of a large amount and undereccentric compression with a small eccentricity. In thissection, sectional analyses with different constitutive models

4 International Journal of Polymer Science

for the column have been conducted to investigate thediscrepancies induced by different uses of constitutivemodels for the confined concrete under eccentric compres-sion. Three representative models for confined concrete inFigure 1 were adopted herein for the composite columnsunder eccentric compression with different eccentricities. Itis believed that the sectional analysis with the constitutivemodels for the confined concrete under pure bendingprovides the lower limit prediction (denoted as lower limitin following figures), while that from the constitutive modelfor the confined concrete under axial compression leads tothe lower limit prediction (denoted as the lower limit infollowing figures).

A composite column, which consists of a T-shaped steel,an outer FRP tube, and concrete infilled between them(Figure 3(a)), was analyzed with the mesh given in

Figure 3(b) using the proposed approach. Three differentconstitutive models illustrated in Figure 1 were used forconfined concrete infilled in the section analysis, leading tothree curves for each case, respectively, under the axial com-pression loads of 1500 kN and 2100 kN (Figure 4). The resultsfrom the section analysis indicate that bending momentscalculated with use of the lower limit model are significantlylower than those from other two models by about 33%, andsuch differences tend to be more significant in the case withthe axial compression load of 2100 kN. However, differencesbetween bending moments calculated using the upper modeland middle mode are minor and can be neglected.

Some conclusions drawn from the sectional analysisabove are as follows: (1) the discrepancies between the sec-tional analysis with the upper model and that with the middlemodel are small and can be neglected if the eccentricity is

Start

Determine the concentriccompression strength, Nz0

Nz0 < Nzd

Specify �휃n = 0

Specify ei = 0

Determine the position ofthe plastic centroid

Evaluate the resistance ofthe section: Nz, Mx, My

Nz = Nzd

Δe < error

Save Mx, My

�휃n ≥ 2π

End

Y

NAdjust dn

N

NAdjust e

NIncrease �휃n

Y

Y

Y

(a) For section analysis

Start

Determine the initial size, a0

Nz0 < Nzd

Specify �휃n = 0

Specify ei = 0

Determine the position ofthe plastic centroid

Evaluate the resistance ofthe section: Nz, Mx, My

Nz = Nzd

Δe < error

Mr = Mrd

End

Y

NAdjust dn

N

NAdjust e

NAdjust a0

Y

Y

Y

am = amdN

Adjust �휃nY

(b) For section design

Figure 2: Flowcharts for section analysis and design of composite columns under biaxial loading.

5International Journal of Polymer Science

small, and (2) the lower model may be too conservative forthe sectional analysis of the columns under eccentric com-pression with small eccentricity and high compression load.

4. Effect of Mesh Schemes

As in all analysis with the fiber element approach, elementsize plays a significant role in the accuracy and efficiencyof the analysis. Coarse mesh for the fiber element mayresult in inaccuracy in the analysis, while high computa-tional cost may be induced by the refined mesh of thefiber elements. To find an optimal mesh for subsequentcalculation, the effect of mesh on the sectional analysiswas conducted.

To achieve this aim, four different mesh schemes inFigure 5 were adopted herein (i.e., element sizes of 2mm,5mm, 20mm, and 40mm, respectively).

The sectional analysis for the column with four differ-ent mesh schemes in Figure 5 was given in Figure 6. Theresults from the sectional analysis with mesh scheme-2 arealmost the same as those from the sectional analysis withmesh scheme-1, and thought to be pretty precise. Theresults from the mesh scheme-3 are also good with minordiscrepancies compared with those from mesh scheme-3.As shown in Figure 6, large discrepancies can be foundin the results from the mesh scheme-4 due to the use ofcoarse fiber elements. It also should be mentioned herethat the computational cost of that with mesh scheme-1

3.2

5,5

R106

100

150

4,3

(a) Dimensions of the composite column (b) Mesh of the composite column

Figure 3: Dimensions and mesh of the predefined column; effect of stress-strain models of confined concrete.

−80 80−60 60−40 40−20 200−60

−40

−20

0

20

40

60

Bending moment about x-axis Mx (kN.m)

Bend

ing

mom

ent a

bout

y-a

xis M

y (k

N.m

)

fnzd = 1500.0 kN-upper limitfnzd = 1500.0 kNfnzd = 1500.0 kN-lower limit

(a) Under the axial load of 1500 kN

−40 40−30 30−20 20−10 100 50−40

−30

−20

−10

0

10

20

30

40

50

Bending moment about x-axis Mx (kN.m)

Bend

ing

mom

ent a

bout

y-a

xis M

y (k

N.m

)

fnzd = 2100.0 kN-upper limitfnzd = 2100.0 kNfnzd = 2100.0 kN-lower limit

(b) Under the axial load of 2100 kN

Figure 4: Mx −My interaction curves of the predefined column; effect of stress-strain models of confined concrete.

6 International Journal of Polymer Science

is high. The mesh scheme-2 will be used for the subse-quent sectional analysis, which combines the advantageof accuracy and computational cost.

5. Numerical Examples

Several numerical examples, including sectional analysis for agiven composite short column, check of the adequacy of pre-defined cross section, and design of required structural steel,are presented herein to demonstrate the validity and effi-ciency of the proposed approach.

5.1. Analysis of Designed Cross Section. A FRP-confinedconcrete-encased I-shaped steel column tested by Karimiet al. [15] is first analyzed in this section. The GFRP tube withfiber orienting in the circumferential direction and its thick-ness is 3.2mm. The structural steel is of I-shape and 500mmlong W150× 14 section. For details of dimension, the readerscan refer to Figure 7(a). Mesh of fiber elements was given inFigure 7(b). The Mx −My interaction curve (i.e., the isoloadcontour) of the design cross section (Figure 8) under differentaxial loads can be easily obtained by the proposed approach.As shown in Figure 8, the resistance along the x-axis is stron-ger than that along the y-axis, and the x-axis can be taken asthe strong axis in design.

(a) Mesh scheme-1 (size = about 2mm) (b) Mesh scheme-2 (size = about 5mm)

(c) Mesh scheme-3 (size = about 20mm) (d) Mesh scheme-4 (size = about 40mm)

Figure 5: Mesh schemes for the predefined column.

−40 40−30 30−20 20−10 100−40

−30

−20

−10

0

10

20

30

40

50

Bending moment about x-axis Mx (kN.m)

Bend

ing

mom

ent a

bout

y-a

xis M

y (k

N.m

)

Mesh scheme-1 Mesh scheme-2Mesh scheme-3 Mesh scheme-4

Figure 6: Mx −My interaction curve with different mesh schemes.

7International Journal of Polymer Science

To demonstrate the robust ability of the proposedapproach for different cross sections, one composite columnwith more symmetric structural steel was analyzed using thisproposed procedure. As shown in Figure 9, a cross-shapedsteel was placed in the composite column. As shown inFigure 10, the more symmetric placement of structural steelresults in a more symmetric Mx −My interaction curve (i.e.,the isoload contour); the ultimate resistance along both twoaxes was almost equal. If no strong axis is required, suchplacement of structural steel may be preferred for designers.

As mentioned above, the proposed iterative procedurecan also be conducted for the columns with structural steelof arbitrary shapes. The following example was used to

demonstrate this ability of the proposed approach. A T-shape structural steel (Figure 3) was used in the compositecolumn; its Mx −My interaction curve was obtained usingthe proposed procedure.

5.2. Checking Adequacy of Predefined Cross Section. Checkingthe adequacy of predefined cross section is a routine task forstructural engineers. For a short FRP-confined concrete-encased arbitrarily shaped steel columns, such a task canbe accomplished by comparing the design load with theMx −My interaction curve produced from the proposedsection analysis approach. If the load point falls outside theinteraction curve, the composite column cannot carry thedesign load, otherwise indicating the adequacy of thepredefined column. For example, the design task is tocheck whether a composite column with the predefinedsection in Figure 11 can carry the design load(2200.0 kN, 40.0 kN.m, and 40.0 kN.m). The checkingprocedure includes (1) spotting the bending moment pair(i.e., 40.0 kN.m, 40 kN.m) in Figure 11, (2) drawing thepart of the Mx −My interaction curve of 2200.0 kN inthe axial load near the design bending moment pair, and(3) checking whether the design bending moment pairfalls within the interaction curve or not. It is found hereinthat this bending moment pair falls outside the interactioncurve, indicating the inadequacy of the section and astronger cross section in need.

5.3. Design of Required Steel Size. The design task is to find arequired structural steel size for a composite column toaccommodate a specific load set Nzd ,Mxd ,Myd . Theiterative procedure in Figure 2(b) can be followed toaccomplish this task. For example, dimensions of thecomposite column, including the type of structural shape,were given in Figure 7. Determine the size of structural steelwhich can accommodate the load set (2400.0 kN, 71.0 kN.m,

Concrete

GFRP tube

5,5

3.24,3

150

100

R106

(a) Dimensions of the composite column (b) Mesh of the composite column

Figure 7: Dimensions and mesh of the composite column tested by Karimi et al. [15].

−80 −60 −40 −20 0 20 40 60 80 100−100−60

−40

−20

0

20

40

60

Bending moment about x-axis Mx (kN.m)

Bend

ing

mom

ent a

bout

y-a

xis M

y (k

N.m

)

fnzd = 1200.0 kN fnzd = 1500.0 kNfnzd = 1800.0 kN fnzd = 2100.0 kNfnzd = 2400.0 kN fnzd = 2700.0 kN

Figure 8: Mx −My interaction curves of the composite columntested by Karimi et al. [15] under different axial load levels.

8 International Journal of Polymer Science

and −47.9 kN.m). Following the iterative procedure inFigure 2(b), the required size of steel should be 3.316 timesthat used in Figure 11. In addition, specific dimensions forshaped steel are given in different national codes, so dimen-sions should also satisfy such requirement.

6. Conclusions

This paper presents an approach for the rapid sectionalanalysis and design of short FRP-confined concrete-encased arbitrarily shaped steel columns under biaxialloading. A robust iterative scheme has been adopted witha traditional so-called fiber element method, where all the

constituent materials were treated as fiber elementsavoiding the integration for the stress block of theFRP-confined concrete. Several numerical exampleswere presented leading to the conclusions summarizedas below:

(1) Use of the plastic centroid as the origin of reference-loading axes guarantees the convergence in the itera-tive process. Such a definition of reference-loadingaxes avoids the possibility that the origin of thereference-loading axes falls outside the Mx −My

interaction curve, even for cases with irregular struc-tural steel under eccentric compression

150

GFRP tube

150

100

5.5

100

3.2

R106

3,4

4,3 5,5 Concrete

(a) Dimensions of the composite column (b) Mesh of the composite column

Figure 9: Dimensions and mesh of the composite column with a cross-shaped steel.

−80 −60 −40 −20 0 20 40 60 80Bending moment about x-axis Mx (kN.m)

fnzd = 2300.0 kN fnzd = 2600.0 kNfnzd = 2900.0 kN fnzd = 3200.0 kNfnzd = 3500.0 kN fnzd = 3700.0 kN

−80

−60

−40

−20

0

20

40

100

80

60

Bend

ing

mom

ent a

bout

y-a

xis M

y (k

N.m

)

Figure 10: Mx −My interaction curves of the composite columnwith a cross-shaped steel under different axial load levels.

−80 −60 −40 −20 0 20 40 60 80Bending moment about x-axis Mx (kN.m)

fnzd = 1200.0 kN fnzd = 1500.0 kNfnzd = 1800.0 kN fnzd = 2100.0 kNfnzd = 2400.0 kN fnzd = 2700.0 kN

−60

−40

−20

0

20

40

60

Bend

ing

mom

ent a

bout

y-a

xis M

y (k

N.m

)

Figure 11: Mx −My interaction curves of the composite columnwith a T-shaped steel under different axial load levels.

9International Journal of Polymer Science

(2) Strain gradient along section should be properlyconsidered in the strain-stress model of confinedconcrete when conducting the section analysis ofhybrid columns under biaxial loading. Use of Lamand Teng’s model [5] for confined concrete in sectionanalysis results in an upper limit of load carryingcapacity of the hybrid columns under biaxial loading,thus leading to an unsafe prediction, especially forcases with a large eccentricity.

(3) Quick convergence has been observed in all casespresented, demonstrating the validity and accuracyof the proposed approach for the rapid section anal-ysis and design of FRP-confined concrete-encasedarbitrarily shaped steel columns.

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request.

Disclosure

Publication of the present paper has been approved byProfessor J.G. Teng at the Hong Kong Polytechnic Univer-sity, China, who supervised the second author for part ofthe work in the present paper.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the possible publication of the present paper.

Acknowledgments

The authors are grateful for the financial support receivedfrom the Nature Science Foundation of China (Project no.51608130) and China Postdoctoral Science Foundation(Project no. 2016M602441).

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10 International Journal of Polymer Science

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11International Journal of Polymer Science

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