Numerical Analysis of the Axial-Flexural Behavior of CFST Columns with Active Transverse Prestressing

: This paper presents a numerical study on the vertical (axial) and lateral (ﬂexure) behavior of CFST (Concrete-Filled Steel Tube) columns with active hoop prestress achieved by bolting together two steel half-tubes. Twelve prototype CFST column specimens differing in the prestressing force (three levels) and in the gravity loading ratio (four levels) are analyzed; they are selected to represent typical ground columns of mid-rise buildings. Their structural behavior is simulated with a nonlinear model implemented in Abaqus; concrete and steel behavior are described with a damage-plasticity and a plasticity model, respectively. The concrete-steel interaction is represented by a hard (compression-only) surface-to-surface contact model. The calculations involve three consecutive loading steps: (i) transverse prestress, (ii) axial force, and (iii) lateral loading (shear force and bending moment). The calculation results show that the axial-ﬂexural capacity of the prototype CFST columns is adequate. However, the hoop prestress beneﬁt on axial compressive performance is not outstanding because the tube transverse Poisson expansion impairs the concrete conﬁnement. The beneﬁt in the ﬂexural performance is smaller, due to the lack of sectional lateral expansion during bending. Preliminary studies on mid-rise buildings equipped with the prototype CFST columns show that their gravity and wind capacities are largely enough; conversely, their seismic strength is sufﬁcient only for moderate earthquakes.


Introduction
CFST (Concrete-Filled Steel Tube) is a composite construction technology increasingly used for columns and beam-columns of buildings and other constructions.CFST is basically an outer steel tube filled with either plain or reinforced concrete.Nowadays, it is widely accepted that this solution has several relevant structural and non-structural advantages.The structural gains have been verified both with testing [1][2][3][4][5][6] and numerical simulation [7][8][9][10].Regarding structural issues, the main benefit of CFST is the confinement effect of the concrete by the steel tube; therefore, transverse (hoop) prestressing might provide even greater and adaptive confinement.In this sense, one approach is using expansive concrete [11]; but a more efficient strategy is an active (adaptive) prestressing generated with mechanical means [12].Paper [12] numerically analyzes the performance of axially-compressed circular concrete-filled steel tubes having external hoop and longitudinal pre-stressing; the results show that properly selected prestressing forces can provide a relevant enhancement in compressive capacity, both in terms of strength and ductility.The work [12] deals only with single short columns under centered compression; nonetheless, most actual columns also undergo relevant bending.Therefore, this paper takes a step forward, considering the coupled axial and bending behavior of CFSTs with active hoop prestressing.The authors have some experience in similar fields, as one of them has carried out research on column steel jacketing for seismic retrofit of RC frame columns [13][14][15].
This paper numerically analyzes the structural behavior of half parts (segments) of 12 prototype building CFST columns that have active hoop prestressing and undergoes axial and lateral forces.These specimens differ in the prestressing force and gravity load ratio are selected to represent typical ground columns of mid-rise buildings.The loading process is represented by three consecutive steps: transverse prestress, axial compression, and shear (lateral) force; this sequence is intended to roughly reproduce the situation in real buildings.From these results, tentative sectional interaction charts are derived.Finally, the vertical and lateral behavior of arrays of building columns (representing stories of building frames) is investigated to obtain relevant preliminary conclusions for the overall structural performance of the building.
The objective of the research is to investigate the structural capacity of CFST columns with active transverse prestressing, trying to gain knowledge of their structural behavior.The research focuses on their use in mid-rise buildings; this is relevant to obtain more resistant, stiffer and more ductile columns, in addition to the non-structural advantages inherent to this technology.This research is part of a wider research effort aimed at promoting active hoop prestressing in CFST columns [12][13][14][15], mainly for buildings undergoing wind and seismic effects.Upcoming research activity will involve retrofit strategies, experimental testing, numerical parametric studies, and implementation in actual full-scale buildings [16].
This research has a number of limitations: it deals only with new construction (i.e., no retrofit), the analysis of individual CFST columns extends only to mid-rise buildings, the numerical model does not account for rheological (concrete creep and shrinkage) and buckling effects (through geometrical imperfections), and, mainly, the research has a rather introductory character.This last consideration is relevant and should be kept in mind, given the high complexity of the problem addressed.

Analyzed Column Segments
As discussed in Section 1, the analyzed structural elements (specimens) represent the lower half segment of a building column (comprised between two consecutive floors); Figure 1b displays such a full column, and Figure 1c represents the aforementioned segment.In Figure 1b, the column undergoes axial compression (N, noticeably, the upper and lower forces are alike as the column weight is neglected), shear (V), and two equal bending moments (M).As both moments are alike, in Figure 1c, there is no bending moment in the top section of the segment (inflection point in the column middle section, where the curvature is reversed).These demanding internal forces are related through the following second-order moment equilibrium condition (Figure 1b): In Equation (1), ∆ is the lateral deflection of the column middle section and L is the segment length (Figure 1b).Appendix A contains a list of symbols.
The column segments are considered to be clamped at their lower end (i.e., all their displacements and rotations are restrained), and free at the upper one; however, in this top section, the Navier-Bernouilli condition (cross sections remain planar and orthogonal to the column axis) should hold.These boundary conditions are well suited for firststory columns, provided they are connected to a rigid foundation.For story columns, the situation is only slightly different, as the bending flexibility of the lower column and framed beams may allow some rotation in the segment lower section; hence, ∆ will be higher (Equation (1) still applies).Therefore, although this study refers mainly to first-story columns, their conclusions can be broadly extended to story columns.
Buildings 2023, 13, x FOR PEER REVIEW 3 of 28 higher (Equation (1) still applies).Therefore, although this study refers mainly to firststory columns, their conclusions can be broadly extended to story columns.Figure 1a,c present a plan view and an elevation of the column, respectively; they show that the column consists of a two-halved steel coating and a confined plain concrete (unreinforced) core.Each steel half consists of a curved central part and two external short vertical flat flanges, intended to hold the prestressing bolts (transversally).Therefore, the concrete core is transversally prestressed through the steel coating; the concrete-steel interaction stress (p0) is assumed to be uniformly distributed along the core perimeter, as displayed in Figure 1d.Noticeably, simple equilibrium equations of proper steel coating parts (Section 4.6) can relate the tensile hoop stress in the steel tube (and the forces in the bolts) to p0.
In all the specimens, the core diameter is D = 500 mm, the steel coating thickness is t = 8 mm, the steel yield point is fy = 355 MPa, the characteristic value of the concrete compressive strength is fck = 30 MPa (C30), and the column segment height is L = 1.5 m.These values have been selected to correspond to the lower stories of most mid-rise buildings (broadly speaking, between 12 and 25 floors).Apart from these common values, the analyzed cases (specimens) are distinguished by the active hoop prestress (p0) and the initial axial compression (N).The demanding axial force and maximum bending moment are normalized with respect to their code sectional resistance values for p0 = 0 (without transverse prestressing): In Equation (2), NR is the resisting axial force, MR is the pure moment strength (i.e., without any shear force) corresponding to zero axial force, and ν and µ are the normalized values of N and M, respectively.In this paper, ν0 refers to the initial value of ν (i.e., for V = 0, see Section 3).
NR is obtained, according to the Chinese regulation [17,18], as: Figure 1a,c present a plan view and an elevation of the column, respectively; they show that the column consists of a two-halved steel coating and a confined plain concrete (unreinforced) core.Each steel half consists of a curved central part and two external short vertical flat flanges, intended to hold the prestressing bolts (transversally).Therefore, the concrete core is transversally prestressed through the steel coating; the concrete-steel interaction stress (p 0 ) is assumed to be uniformly distributed along the core perimeter, as displayed in Figure 1d.Noticeably, simple equilibrium equations of proper steel coating parts (Section 4.6) can relate the tensile hoop stress in the steel tube (and the forces in the bolts) to p 0 .
In all the specimens, the core diameter is D = 500 mm, the steel coating thickness is t = 8 mm, the steel yield point is f y = 355 MPa, the characteristic value of the concrete compressive strength is f ck = 30 MPa (C30), and the column segment height is L = 1.5 m.These values have been selected to correspond to the lower stories of most mid-rise buildings (broadly speaking, between 12 and 25 floors).Apart from these common values, the analyzed cases (specimens) are distinguished by the active hoop prestress (p 0 ) and the initial axial compression (N).The demanding axial force and maximum bending moment are normalized with respect to their code sectional resistance values for p 0 = 0 (without transverse prestressing): In Equation ( 2), N R is the resisting axial force, M R is the pure moment strength (i.e., without any shear force) corresponding to zero axial force, and ν and µ are the normalized values of N and M, respectively.In this paper, ν 0 refers to the initial value of ν (i.e., for V = 0, see Section 3).
N R is obtained, according to the Chinese regulation [17,18], as: In Equation (3), A c is the concrete core area, α is a dimensionless parameter that accounts for the positive confinement effect of concrete (α depends on the concrete strength [17,18], for C30 concrete, α = 2), and θ is the ratio between steel and concrete individual strengths: In Equation (4), A s is the steel tube area and f y is the steel yield point.In the specimens analyzed in this paper, θ = 2 π250×8×355 π250 2 ×30 = 0.7576; therefore, θ ≤ 1/(α − 1) 2 and, thus, the first relation in Equation (3) applies: Analogously to ν, the normalized moment is denoted by µ and is defined as the ratio between the demanding (M) and resisting (M R ) moments (Equation ( 2)); the corresponding strength is also determined according to the Chinese regulation [17,18] as: As previously announced, different values of p 0 and ν 0 are considered for the analyzed specimens; the values adopted are discussed below.As for p 0 , three levels are taken: 0 (no hoop prestress), 2 MPa (moderate prestress), and 5 MPa (intense prestress).About ν 0 , 4 loading intensities are taken: 0.3 (slightly loaded column), 0.5 (intensely loaded column), 0.7 (extremely loaded column), and 0.9 (overloaded column).Table 1 displays such values of the common and variable geometric and mechanical parameters.  1 shows that 3 × 4 = 12 CFST specimens are analyzed in this paper.

Loading Steps of the Column Segments
As briefly discussed in Section 1, the active hoop prestress (p 0 ), axial compression (N), and lateral force (V) are applied consecutively; these three load steps are intended to reproduce the situation of columns in real buildings, as N corresponds to gravity loads and V to lateral actions.Noticeably, this sequence (p 0 , N and V) corresponds to new construction only; for retrofit, N should come before p 0 .Section 3.1 describes the relationship between the position of columns within the building and their loading steps; Section 3.2 graphically illustrates the loading steps considered.

Loading Steps in Actual Building Columns
The loading steps of building columns depend on the vertical (gravity) and lateral actions (whether seismic or wind-generated).Regarding the lateral effects, Figure 2a displays a prototype mid-rise unbraced frame building undergoing horizontal forces.
Figure 2a shows that the horizontal force F b (base shear) is the sum (resultant) of all forces pushing on the stories of a given frame; it is located at a height d above ground level.F b mainly generates two opposite axial forces ∆N in the external columns of the frame (Figure 2b), while the effect on the axial forces in the inner columns can be ignored.F b and ∆N are related by the following equilibrium condition:  Figure 2a shows that the horizontal force Fb (base shear) is the sum (resultant) of all forces pushing on the stories of a given frame; it is located at a height d above ground level.Fb mainly generates two opposite axial forces ΔN in the external columns of the frame (Figure 2b), while the effect on the axial forces in the inner columns can be ignored.Fb and ΔN are related by the following equilibrium condition: In Equation (7), B is the horizontal separation between both extreme columns.Equation (7) shows that the ratio between Fb and ΔN is rather constant, depending mainly on the geometric characteristics of the building.Regarding the particular case depicted in Figure 2a, it can be reasonably assumed that the external shear force Fb is distributed uniformly between all the columns of the frame: V = Fb/5 (V is the column shear force due to the lateral effect, Figure 2b).For mid-rise buildings, ordinarily d ≈ 2 B; for instance, in a 20-story building (Figure 2a, about 60 m high) assuming that the seismic forces are triangularly distributed, the height of the resultant force is d = 40 m, and a common value of the building length is B = 20 m (d = 2 B).Therefore, in this paper, it is assumed that ΔN and V are related by: In Equation ( 8), the positive and negative signs correspond to the right and left columns, respectively (Figure 2b).It should be kept in mind that Equation (8) refers merely to a feasible and representative case.
These considerations show that, in the last loading step (seismic), the situation of the internal and external columns is different: in the internal ones, only the lateral forces increase, while in the external ones, both axial and lateral forces vary.Therefore, two sets of loading steps are considered in this paper: Step 1 corresponds to p0, and steps 2 and 3 to N and V, respectively.

•
External columns.Steps 1 and 2 are the same as in the internal columns; regarding the last loading step, it is known as 3', and involves variations of both N and V (these variations are represented by ΔN and V, respectively).In Equation (7), B is the horizontal separation between both extreme columns.Equation (7) shows that the ratio between F b and ∆N is rather constant, depending mainly on the geometric characteristics of the building.Regarding the particular case depicted in Figure 2a, it can be reasonably assumed that the external shear force F b is distributed uniformly between all the columns of the frame: V = F b /5 (V is the column shear force due to the lateral effect, Figure 2b).For mid-rise buildings, ordinarily d ≈ 2 B; for instance, in a 20-story building (Figure 2a, about 60 m high) assuming that the seismic forces are triangularly distributed, the height of the resultant force is d = 40 m, and a common value of the building length is B = 20 m (d = 2 B).Therefore, in this paper, it is assumed that ∆N and V are related by: ∆N ≈ ±10 V In Equation (8), the positive and negative signs correspond to the right and left columns, respectively (Figure 2b).It should be kept in mind that Equation (8) refers merely to a feasible and representative case.
These considerations show that, in the last loading step (seismic), the situation of the internal and external columns is different: in the internal ones, only the lateral forces increase, while in the external ones, both axial and lateral forces vary.Therefore, two sets of loading steps are considered in this paper: Step 1 corresponds to p 0 , and steps 2 and 3 to N and V, respectively.

•
External columns.Steps 1 and 2 are the same as in the internal columns; regarding the last loading step, it is known as 3', and involves variations of both N and V (these variations are represented by ∆N and V, respectively).

Graphical Description of Loading Steps
For further clarity, Figure 3 graphically describes such loading paths for the internal and external columns.

Graphical Description of Loading Steps
For further clarity, Figure 3 graphically describes such loading paths for the internal and external columns.Figure 3 shows that loading step 1 (purple) has three possible horizontal (coaxial) branches: no load (i.e., without prestress), or reaching the p0 = 2 MPa or p0 = 5 MPa points.Then, loading step 2 consists of three parallel horizontal branches (red) starting from these points, respectively; they reach the points corresponding to ν0 = 0.3, 0.5, 0.7 or 0.9 and continue until failure.Finally, loading steps 3 and 3' develop into 12 vertical (blue, step 3) and inclined (grey, step 3') parallel branches that are maintained until failure.Noticeably, unlike step 3, both branches of step 3' (upward and downward) are not symmetric; the upward one (termed next as 3' + , + sign in Equation ( 8)) corresponds to the right external column in Figure 2b, while the downward branch (termed next as 3' − , − sign in Equation ( 8)) refers to the left column in that Figure .The loading steps depicted in Figure 3 aim to reproduce the situation of actual new buildings (or even other constructions): step 1 corresponds to the initial prestressing (before most of the gravity loads are applied), step 2 represents the effect of gravity loads, and steps 3 and 3' roughly reproduce the effect of lateral forces.Regarding this last effect, such forces can be extremely important (mainly in high seismicity regions); hence, steps 3 and 3' are prolonged until collapse.It should be noted that steps 3 and 3' for ν0 = 0.7 and 0.9 are only included for comparison purposes, as highly loaded columns are not expected in seismic areas; for this reason, these branches are indicated with dashed lines in Figure 3.

Numerical Model of the Column Segments
The structural behavior of the specimens presented in Section 2 is described with a model implemented in the finite element program ABAQUS 6.14-1 [19] using an implicit (standard) formulation.This software has been chosen as it is highly reputable and suitable for accurately reproducing complex structural behaviors; in this sense, it has been considered previously by several authors for the simulation of CFSTs [8,12,20].
The next subsections describe the main characteristics of the model.Figure 3 shows that loading step 1 (purple) has three possible horizontal (coaxial) branches: no load (i.e., without prestress), or reaching the p 0 = 2 MPa or p 0 = 5 MPa points.Then, loading step 2 consists of three parallel horizontal branches (red) starting from these points, respectively; they reach the points corresponding to ν 0 = 0.3, 0.5, 0.7 or 0.9 and continue until failure.Finally, loading steps 3 and 3' develop into 12 vertical (blue, step 3) and inclined (grey, step 3') parallel branches that are maintained until failure.Noticeably, unlike step 3, both branches of step 3' (upward and downward) are not symmetric; the upward one (termed next as 3' + , + sign in Equation ( 8)) corresponds to the right external column in Figure 2b, while the downward branch (termed next as 3' − , − sign in Equation ( 8)) refers to the left column in that Figure .The loading steps depicted in Figure 3 aim to reproduce the situation of actual new buildings (or even other constructions): step 1 corresponds to the initial prestressing (before most of the gravity loads are applied), step 2 represents the effect of gravity loads, and steps 3 and 3' roughly reproduce the effect of lateral forces.Regarding this last effect, such forces can be extremely important (mainly in high seismicity regions); hence, steps 3 and 3' are prolonged until collapse.It should be noted that steps 3 and 3' for ν 0 = 0.7 and 0.9 are only included for comparison purposes, as highly loaded columns are not expected in seismic areas; for this reason, these branches are indicated with dashed lines in Figure 3.

Numerical Model of the Column Segments
The structural behavior of the specimens presented in Section 2 is described with a model implemented in the finite element program ABAQUS 6.14-1 [19] using an implicit (standard) formulation.This software has been chosen as it is highly reputable and suitable for accurately reproducing complex structural behaviors; in this sense, it has been considered previously by several authors for the simulation of CFSTs [8,12,20].
The next subsections describe the main characteristics of the model.

Finite Element Mesh and Boundary Conditions
The finite element mesh includes the column segment itself and two fictitious (dummy) rigid steel plates (600 mm × 600 mm × 30 mm).Such plates are fixed to both segment ends to reproduce the adequate boundary conditions (bottom end clamped and top end free, although keeping the planar section condition, Section 2); in this respect, the bottom plate is fixed, while the top one is free to rotate and to experience vertical and horizontal displacements.
Steel and concrete are discretized with 8-node 24-DOFs solid hexahedral (brick) elements C3D8R (Continuum, 3D, 8-node, Reduced integration) [19]; reduced integration refers to considering a single integration point in the center of the element.This element is chosen because of its accuracy (despite the reduced integration), and because shear self-locking under bending load is not likely to occur.In opposition, this element is too soft, due to the so-called hour-glassing data problem (caused by the aforementioned reduced integration); therefore, for bending, this problem can be overcome by discretizing the component (steel tube) thickness with several layers of elements (Figure 4a).  Figure 4a shows that the bolts and flanges (Figure 1a,c) are not modeled, just the tube itself.This approach relies mainly on three reasons: (i) neither the flanges nor the bolts are expected to participate significantly in the axial load-carrying mechanism, (ii) modeling these elements would importantly increase the number of degrees of freedom, among other relevant complications, and (iii) the representation of the hoop prestressing by tightening the bolts is highly complex and cumbersome task, while its simulation with a uniaxial transverse temperature decrease in the steel (Sections 4.6, 4.7 and 5.2) is highly convenient and easy.Figure 4a also shows that, as previously discussed, to avoid the hourglassing effect the tube thickness is discretized into three layers of elements (being 2.67 mm thick each), and Figure 4d shows that two layers of elements are utilized in the plate thickness (15 mm thick each).

Concrete Constitutive Law
The structural behavior of in-tube (encased) concrete is totally different from the one The mesh sizes are determined through convergence studies to achieve accurate simulations with minimal increase in computational cost; the basic element size is selected as 35 mm. Figure 4a shows that the bolts and flanges (Figure 1a,c) are not modeled, just the tube itself.This approach relies mainly on three reasons: (i) neither the flanges nor the bolts are expected to participate significantly in the axial load-carrying mechanism, (ii) modeling these elements would importantly increase the number of degrees of freedom, among other relevant complications, and (iii) the representation of the hoop prestressing by tightening the bolts is highly complex and cumbersome task, while its simulation with a uniaxial transverse temperature decrease in the steel (Sections 4.6, 4.7 and 5.2) is highly convenient and easy.Figure 4a also shows that, as previously discussed, to avoid the hour-glassing effect the tube thickness is discretized into three layers of elements (being 2.67 mm thick each), and Figure 4d shows that two layers of elements are utilized in the plate thickness (15 mm thick each).

Concrete Constitutive Law
The structural behavior of in-tube (encased) concrete is totally different from the one of ordinary (uncoated) RC members, mainly due to the important confinement; as expected, the greatest effect has been found in circular columns [21].For that reason, the average (mean) uniaxial compressive concrete behavior is described in this paper by a law that is specific to circular CFSTs [22].Regarding tensile behavior, despite that concrete tensile stresses are anticipated (Section 2), no CFST-specific tensile constitutive law has been reported (no intense confinement is expected in that situation); therefore, the tensile behavior is modeled according to the general purpose law in the Chinese regulation [23].
The aforementioned compressive stress-strain law consists of two segments (branches): the initial (growing) and the following (either growing or descending).These segments are described by Equations ( 9) and ( 10), respectively.
In Equations ( 9) and ( 10), σ c and ε c are the compressive concrete stress and strain, respectively.In the first growing segment (ε c ≤ ε 0 , Equation ( 9)), σ 0 is the peak stress, and ε 0 is the corresponding strain; A and B are dimensionless coefficients, given by A = 2 − K and B = 1 − K, where K = 0.1 θ 0.745 , θ is the ratio between the steel and concrete capacities (Equation ( 4)).
In this study, the following major values are determined: σ 0 = 43.95MPa, σ 0t = 2.91 MPa, ε 0 = 0.003451, and σ 0 A/ε 0 = 43.95× 1.919/0.003451= 24435 MPa.By using these data, Figure 5 displays the concrete compressive (Figure 5a) and tensile (Figure 5b) curves.Figure 5 shows that the compressive behavior is highly ductile (due to the confining effect of the steel tube), while the tensile one is quite fragile.Concrete 3-D behavior modeling is discussed in this paragraph.Regarding initial behavior, the concrete Poisson ratio is νc = 0.18.Going into plastic behavior, the yield surface is described by parameters Kc and σb0/σ0; Kc is the ratio between the biaxial and triaxial isobaric compressive strengths, and σb0/σ0 is the ratio between the equi-biaxial to the uniaxial compressive maximum stress.In this paper, Kc = 2/3 [24] and σb0/σ0 = 1.16 [25].After yielding, the flow rule is characterized by the dilation angle (θ) and the eccentricity; the dilation angle is θ = 20° [26], and the eccentricity is set to 0.1 [25].This rather unusually small dilation angle aims to compensate for the fact that the confinement effect (in terms of ductility) is already taken into consideration by the highly ductile uniaxial concrete compressive constitutive law (Equation ( 8) and Figure 5a).Finally, the viscosity coefficient is used to adjust the constitutive law to improve the convergence efficiency in the softening stage; the assumed value is close to 0 [25].

Concrete Damage Plasticity Model
The 3-D concrete loading-unloading behavior is represented by a CDPM (Concrete Damage Plasticity Model) [24,27,28]; as its name states, it combines plasticity and damage.The damage variables evolution is set according to a formulation specific to CFST [9].In that work, two damage variables are defined: compressive (dc) and tensile (dt); they range between 0 (no damage) and 1 (destruction), and their evolution is set with respect to the crushing (ε ) and cracking (ε ) strains, respectively.These strains are defined as the total strain minus the correspondent elastic (recovered) strain in the absence of damage (when dc = dt = 0) [28]: In Equation (12), ε and ε are the compressive and tensile plastic strains, respectively.These equations can be solved to provide the damage variables: In Equation (13), bc is the ratio between the plastic compressive and the crushing strains  = ; bt is the ratio between the plastic tensile strain and the cracking strains: In this study, it is assumed that bc = 0.7 and bt = 0.82 [9,29].Figure 6 displays the plots of damage variables (compressive and tensile) vs. strain.Concrete 3-D behavior modeling is discussed in this paragraph.Regarding initial behavior, the concrete Poisson ratio is ν c = 0.18.Going into plastic behavior, the yield surface is described by parameters K c and σ b0 /σ 0 ; K c is the ratio between the biaxial and triaxial isobaric compressive strengths, and σ b0 /σ 0 is the ratio between the equi-biaxial to the uniaxial compressive maximum stress.In this paper, K c = 2/3 [24] and σ b0 /σ 0 = 1.16 [25].After yielding, the flow rule is characterized by the dilation angle (θ) and the eccentricity; the dilation angle is θ = 20 • [26], and the eccentricity is set to 0.1 [25].This rather unusually small dilation angle aims to compensate for the fact that the confinement effect (in terms of ductility) is already taken into consideration by the highly ductile uniaxial concrete compressive constitutive law (Equation ( 8) and Figure 5a).Finally, the viscosity coefficient is used to adjust the constitutive law to improve the convergence efficiency in the softening stage; the assumed value is close to 0 [25].

Concrete Damage Plasticity Model
The 3-D concrete loading-unloading behavior is represented by a CDPM (Concrete Damage Plasticity Model) [24,27,28]; as its name states, it combines plasticity and damage.The damage variables evolution is set according to a formulation specific to CFST [9].In that work, two damage variables are defined: compressive (d c ) and tensile (d t ); they range between 0 (no damage) and 1 (destruction), and their evolution is set with respect to the crushing (ε ch c ) and cracking (ε ck t ) strains, respectively.These strains are defined as the total strain minus the correspondent elastic (recovered) strain in the absence of damage (when d c = d t = 0) [28]: In Equation ( 12), ε pl c and ε pl t are the compressive and tensile plastic strains, respectively.These equations can be solved to provide the damage variables: In Equation ( 13), b c is the ratio between the plastic compressive and the crushing ; b t is the ratio between the plastic tensile strain and the cracking strains: In this study, it is assumed that b c = 0.7 and b t = 0.82 [9,29].Figure 6
In this study, fy = 355 MPa (Table 1) and Es = 210 GPa; then εe = 0.00135, εe1 = 0.00203, εe2 = 0.02029, and εe3 = 0.20286.From these parameters, Figure 7 displays the five-segment curve described by Equation ( 14).Regarding the 3-D elastic behavior, the steel Poisson ratio is νs = 0.29.On the 3-D plastic behavior yield, the Von Mises yield criterion is employed, and the post-yield behavior is described with an isotropic strain hardening model.Regarding the 3-D elastic behavior, the steel Poisson ratio is ν s = 0.29.On the 3-D plastic behavior yield, the Von Mises yield criterion is employed, and the post-yield behavior is described with an isotropic strain hardening model.

Concrete-Steel Contact
The concrete-steel interaction at the interface between the inner core and the outer tube is described using the Abaqus surface-to-surface contact model; steel is the master surface and concrete is the slave one.This model is the most suitable, commonly utilized, and straightforward formulation to reproduce the contact and separation between two surfaces.The other contact models implemented in Abaqus are oriented for specific problems.The hard contact (instead of softened one) relationship is considered; this approach is compression-only (thus, bonding is neglected), and penetration is prevented.The sliding between both materials is simulated with the Mohr-Coulomb formulation using a penalty function; the friction coefficient is constant, being equal to 0.25 [30,31].A constant friction coefficient is considered since the problem analyzed (sliding between the concrete core and the steel jacket) is basically static.

Loads Application
This subsection describes how the loads involved in the three loading steps (Figure 3) are actually applied to the structure (Figure 4).Sections 4.6.1 and 4.6.2refer to prestress (step 1) and actual loading (subsequent steps 2, 3 and 3'), respectively.

Step 1
In step 1, the interaction initial contact pressure between concrete and steel (p 0 , Figure 1d) is represented by a fictitious temperature reduction (∆T) of the steel tube.This approach is used because probably it is one of the most correct and straightforward ways to reproduce the confining effect of concrete (Figure 1a,c).However, the steel temperature reduction in all directions would generate unrealistic longitudinal shear stresses in the concrete-steel interface (the shortening of the steel tube would attempt to shorten the concrete core); therefore, only temperature reduction in the transverse tangential direction is considered in this study.This is achieved with Abaqus' ability to characterize thermal expansion using non-isotropic coefficients (that is, with different values in any direction), this coefficient is taken as zero in the tube longitudinal direction.
Given that the steel behavior is guessed to be linear in this step, the required temperature reduction can be obtained from a simple formulation: the horizontal steel tension (σ s ) and concrete compression (p 0 ) are related by the equilibrium condition p 0 D = 2 σ s t (as announced in Section 2), and the steel linear constitutive law states that σ s = α s ∆T E s , where α s and E s are the steel thermal expansion coefficient and deformation modulus, respectively.By selecting α s = 10 −5 • C −1 , it follows that, for p 0 = 2 and 5 MPa, ∆T = −29.76• C and −74.41 • C, respectively.These results have been verified by comparing them with the numerical simulation using Abaqus.Noticeably, the abovementioned equilibrium condition p 0 D = 2 σ s t shows that for p 0 = 5 MPa (the highest value), σ s = 156.25 MPa; this corroborates that, as previously guessed, in this step the steel behavior is linear.

Steps 2, 3 and 3'
In loading steps 2, 3 and 3', the external axial (N) and shear (V) forces are applied to the upper dummy steel plate (Figure 4) as concentrated forces; as that plate is highly rigid, the effect of those forces on the real CFST tube is smoothened out sufficiently.
Finally, regarding the column self-weight, it is neglected, as it is clearly lower than the external axial load.

Numerical Calculation
This subsection briefs the nonlinear analyses carried out.
Geometric nonlinearity (second-order) is accounted for, since the building interstory drift (Figure 1b) for severe seismic inputs can be important.Regarding mechanical nonlinearity, it corresponds to the constitutive laws and other issues described in Sections 4.2-4.5.The ensuing nonlinear calculations are then performed incrementally (Newton-Raphson method) using virtual time.In steps 1 and 2, the temperature (T) and axial force (N) are imposed, respectively.On the contrary, in steps 3 and 3', the forces cannot be imposed, as a negative slope in the V-∆ plot is expected, both due to the descending branches of the concrete constitutive law (Figure 5) and the simultaneous variation in N and V in step 3' (Figure 3).Therefore, in steps 3 and 3', the displacements are imposed instead.The initial time increment is 0.01 s, and the maximum number of iterations per step is 10,000.If there is no convergence, the time step is automatically reduced to its minimum value of 0.00001 s.The convergence criterion is based on both force and displacement; the corresponding bound ratios are 0.005 and 0.01, respectively.The convergence error is 0.005.

Numerical Results of the Column Segments Analyses
This section presents and discusses the results of the loading steps described in Section 3; Section 5.1 through Section 5.4 refer to steps 1, 2, 3 and 3', respectively.The numerical analyses are performed using the model described in Section 4. It should be noted that this model for p 0 = 0 (that is, with active transverse prestressing) cannot currently be calibrated with experiments, since to date the proposed CFST columns with active transverse prestressing have not yet been tested.Therefore, nowadays the only possible corroborations are rough comparisons with simplified formulations (manual calculations), and deep conceptual interpretations of the results obtained.Conversely, the model for p 0 = 0 (without active transverse prestressing) can be calibrated with existing tests and code empirical expressions.All these validations are performed: conceptual interpretations and comparisons with hand calculations (for p 0 = 0) are presented mainly in this section, and calibration with tests and code expressions (for p 0 = 0) are included in Sections 5 and 6.Furthermore, regarding active transverse prestressing (p 0 = 0), it has roughly the same effect as in regular CFST columns, although it occurs before axially loading the column.The behavior of ordinary confined concrete cores has been widely tested [1][2][3][4] and can be reasonably extrapolated to the conditions in this paper.In any case, experimental campaigns are always useful.

Loading Step 1 (Transverse Prestress)
In step 1 (concrete core prestress), the steel radial (interaction) compression generates horizontal shortening in concrete; in turn, such strain tends to produce axial (vertical) concrete elongation.That deformation is restrained by the steel; as a result, vertical compression and tension arise in the concrete core and the steel tube, respectively.Then, a simple manual calculation (based on a linear elastic approach) was performed by imposing the equality of the axial vertical strains of concrete and steel (Navier-Bernouilli hypothesis).This calculation shows that the vertical tensile stress of steel (σ sz0 ) is equal to 15.69 and 39.23 MPa for p 0 = 2 and 5 MPa, respectively.These results are corroborated by the calculations with Abaqus: Figure 8 displays the sectional distribution of stress S33 (σ z ) in concrete (σ cz ) and steel (σ sz ).Furthermore, the points in Figure 9c,d that correspond to the start of loading step 2 provide the same confirmations for concrete and steel, respectively.
Figure 8 shows that the vertical normal stresses are fairly uniformly distributed along the section (both in the concrete and steel parts); this confirms the proper reproduction of the Navier-Bernouilli hypothesis (planar sections remain planar).Regarding the longitudinal (vertical) distributions of the σ cz and σ sz stresses, they are also rather uniform, except for some local effects near the two dummy upper and lower end plates (Figure 4). Figure 8 shows that the vertical normal stresses are fairly uniformly distributed along the section (both in the concrete and steel parts); this confirms the proper reproduction of the Navier-Bernouilli hypothesis (planar sections remain planar).Regarding the longitudinal (vertical) distributions of the σcz and σsz stresses, they are also rather uniform, except for some local effects near the two dummy upper and lower end plates (Figure 4).

Loading Step 2 (Axial Load)
Figure 9 displays the most meaningful and relevant results of load step 2. Figure 9a presents the plots of normalized force vs. axial shortening (ν-w), Figure 9b describes the variation of the concrete-steel interaction contact stress (p) with the normalized axial force (ν), and Figure 9c,d exhibit, also vs. ν, the vertical axial stress of concrete (σcz) and steel (σsz), respectively.Before analyzing in depth the results in Figure 9, the initial slope of the ν-w plots in Figure 9a is compared with manual linear calculations.Such calculations are performed according to the combination theory described in [32] (analogously to the hand calculation in Section 5.1), and provide a vertical axial stiffness of 5994 kN/mm; this result is satisfactory since it fits that of Figure 9a (6033 kN/mm).Likewise, Figure 9a shows that, as expected, the force capacity for the non-prestressed case (p0 = 0) corresponds approximately to ν = 1; this fit should be interpreted as a correct calibration of the numerical model described in Section 4 with the empirical expression (Equations ( 3)-( 5)) proposed by the Chinese regulation [17,18].These coincidences point out the reliability of

Loading Step 2 (Axial Load)
Figure 9 displays the most meaningful and relevant results of load step 2. Figure 9a presents the plots of normalized force vs. axial shortening (ν-w), Figure 9b describes the variation of the concrete-steel interaction contact stress (p) with the normalized axial force (ν), and Figure 9c,d exhibit, also vs. ν, the vertical axial stress of concrete (σ cz ) and steel (σ sz ), respectively.
Before analyzing in depth the results in Figure 9, the initial slope of the ν-w plots in Figure 9a is compared with manual linear calculations.Such calculations are performed according to the combination theory described in [32] (analogously to the hand calculation in Section 5.1), and provide a vertical axial stiffness of 5994 kN/mm; this result is satisfactory since it fits that of Figure 9a (6033 kN/mm).Likewise, Figure 9a shows that, as expected, the force capacity for the non-prestressed case (p 0 = 0) corresponds approximately to ν = 1; this fit should be interpreted as a correct calibration of the numerical model described in Section 4 with the empirical expression (Equations ( 3)-( 5)) proposed by the Chinese regulation [17,18].These coincidences point out the reliability of the performed simulations; then, the remarks provided by Figure 9 are discussed in the following paragraphs.In this sense, the predictable and compatible.
This paragraph presents, based on Figure 9, general remarks on the column segment structural behavior in step 2. Figure 9a shows that the hoop prestress (p 0 ) provides a 0 is, the longer the initial near-linear branch is; moreover, the maximum force capacity is greater for higher p 0 .However, this effect is only moderate, since the force for w = 8 mm is approximately 7.5% larger for p 0 = 5 MPa than for p 0 = 0. Analogously, Figure 9b provides similar positive remarks, as in the case (p 0 = 0) the steel tube confines the concrete core for approximately ν ≥ 0.4 (when most needed); in the prestressed cases (p 0 = 2 and 5 MPa) the prestressing is maintained continuously (except in the final collapse, for ν > 1), although it is significantly and progressively reduced as collapse approaches (mainly for p 0 = 5 MPa).In other words, such a significant decrease in concrete transverse compressive stress (p) apparently precludes further increases in axial force capacity.This circumstance supports the representation of the prestressing effect as a temperature reduction in the steel jacket only in its horizontal direction (that is as if it consisted of a series of independent rings, Section 4.6); noticeably, its representation by a constant external pressure (p 0 ) would lead to significant errors (often, on the unsafe side), as Poisson effects could not be taken into account.
Beyond the overall comments in the previous paragraph, deeper interpretations are provided below.To do this, the initial behavior (linear elastic) is analyzed; given that in loading step 2 the principal stresses directions inside the concrete core coincide with the axes in Figure 1 (more precisely, all the horizontal axes are principal), the initial values of the radial strains in concrete (throughout the core) and steel (in the inner surface of the tube) are given by: Equations ( 15) represent the constitutive stress-strain laws for concrete and steel, which are assumed to be linear isotropic materials.These laws allow a deep conceptual interpretation of results in Figure 9; this is considered important, as constitutes a certain qualitative verification of the calculations performed, and contributes to one of the main objectives of this research (gain knowledge of the structural behavior of CFST columns).
In Equation ( 15), subindexes r and t refer to radial and tangential directions, respectively; subindex 0 in σ cz0 and σ sz0 corresponds to their initial values (Figure 8).Again in Equation ( 15), ε cr and ε sr represent the strain variation (shortening is positive) with respect to the initial situation (i.e., when p = p 0 ), and σ cr , σ ct and σ cz are the actual concrete stresses (compression is positive).The top Equation ( 15) (for concrete), has been derived by taking into consideration that, inside the core, σ cr = σ ct = p; in the bottom Equation ( 15) (for steel), σ sr = p (compression) and σ st = p (D/2t) (tension).Notice that, since Equation ( 15) is linear, they require that p can take positive (compression) and negative (tension) values; Figure 9b shows that p > 0 except for p 0 = 0 and ν < 0.4.
Although Equation ( 15) is derived for linear elastic behavior, the same relations still hold for higher values of the normalized axial load; however, in that case, E c , E s , ν c and ν s should be replaced with the corresponding nonlinear secant values.Therefore, Equation ( 15) can be utilized to analyze, in an approximated and unquantified way (that is, conceptually), the initial trends displayed in Figure 9.To perform this operation, it is highlighted that, except when concrete and steel are separated (this only occurs in the initial segment of the non-prestressed case, Figure 9b) their tangential axial strains are equal (ε ct = ε st ); this relation provides the following expression for p: Equation ( 16) is utilized next in the interpretation of the plots in Figure 9.That Figure shows that all the plots can be divided into three segments (branches): initial (linear), intermediate (moderately nonlinear), and final (more intensely nonlinear).Those segments are separated by colored symbols and , respectively; since they correspond to the same values of ν in Figure 9a-d, it is presumable that these three segments can receive common interpretations.Accordingly, the next three paragraphs provide explanations for these three segments, respectively.

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Initial segments.The initial segments exhibit different characteristics in the nonprestressed and prestressed cases.In the non-prestressed case (p 0 = 0), initially (that is, for small values of ν), the steel tube separates from the concrete core (i.e., ε ct < ε st ), and p is still zero (Figure 9b) because it cannot be negative (Section 4.5); then, slightly before ν reaches 0.4, concrete and steel resume contact (ε ct = ε st ) and p begins to take positive values.In the prestressed cases (p 0 = 2 and 5 MPa), the situation is different, as the initial value of the interaction contact stress (p) is nonzero and, thus, can decrease from the very beginning without taking negative values.These circumstances can be explained (both for the non-prestressed and prestressed cases) because the initial steel Poisson ratio is higher than that of concrete and, thus, the numerator in Equation ( 16) is positive.For the latter segments (intermediate and final) Equation ( 16) becomes less applicable, as the behavior is nonlinear.

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Intermediate segments.Figure 9 shows three different tendencies in these segments: the linear behavior terminates (indicated by the colored symbols ), p increases (Figure 9b), and the vertical axial stress is progressively transferred from concrete to steel (Figure 9c,d).The first and last trends can be explained by the onset of a relevant concrete nonlinear behavior (Figure 5a), and the growth of p is due to an increase in the concrete Poisson ratio.Regarding this last issue, concrete crushing generates significant volume expansion, resulting in an unusually high apparent Poisson ratio ν c (greatly exceeding 0.5).

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Final segments.In Figure 9, the end of the intermediate segments (indicated by the colored symbols ) correspond to high values of ν (close to 0.8), where the steel stiffness decreases drastically (Figure 7) and, hence, most of the load is taken by the concrete (Figure 9c,d).Unsurprisingly, this steel yielding causes higher axial flexibility (Figure 9a) and less confinement (Figure 9b).

Loading Step 3 (Shear Load)
Figures 10 and 11 display the force (V) vs. displacement (∆) plots of step 3; in Figure 10 only the constitutive models in Sections 4.2 and 4.4 are considered, while in Figure 11 the concrete damage model in Section 4.3 is also contemplated.In Figures 10 and 11, the four sets of load paths for step 3 (ν 0 = 0.3, 0.5, 0.7 and 0.9, Figure 3) are plotted; however, it should be kept in mind that, in seismic areas, values of ν 0 exceeding 0.5 are not to be presumed (indeed, cases for ν 0 = 0.7 and 0.9 are included for comparison purposes only).
The maximum imposed displacements of the plots in Figures 10 and 11 are selected based on two considerations: (i) they must not exceed 100 mm (in real buildings, higher displacements are not expected, as they would correspond to an interstory drift ratio near 6.67%, largely exceeding even the loosest seismic design code prescriptions), and (ii) values clearly beyond the collapse points (peak of the shear force) are not totally reliable in any numerical simulation.From these considerations, the maximum displacements chosen for ν 0 = 0.3, 0.5, 0.7 and 0.9, are 100, 100, 60 and 18 mm, respectively.The same values are selected for plots in Figure 13, Figure 14 (step 3') and Figure 16 (global lateral analysis of a given building).
Figure 10 shows that, as expected, the most loaded columns (i.e., with larger ν 0 ) exhibit less lateral force capacity; more precisely, the resistances for ν 0 = 0.3 and 0.5 are rather similar, while those for ν 0 = 0.7 and 0.9 are clearly smaller, particularly the latter.This trend can be explained by the combination of two quite opposite tendencies: the axial preload intensity (ν 0 ) impairs the column capacity to resist lateral forces (V), but moderate values of ν 0 can provide a rather beneficial effect, similar to that of axial prestressing in RC beams (Figure 16).Regarding the influence of the active prestress (p 0 ), it is more intense for greater values of ν 0 ; the percentages of increase for p 0 = 5 MPa compared to p 0 = 0 are approximately 4.6% (ν 0 = 0.3), 8.4% (ν 0 = 0.5), 16.3% (ν 0 = 0.7) and 48.9% (ν 0 = 0.9).This tendency can be explicated because in the initial situation (i.e., for V = 0), the difference between the cases for p 0 = 0, 2 and 5 MPa is greater for higher values of ν 0 ; this circumstance is highlighted by Figure 9a.In other words, apparently, the benefit of the active transverse prestressing is not generated during step 3, but it rather is a continuation of the advantage produced through step 2. Figure 10 shows that, as expected, the most loaded columns (i.e., with larger ν0) exhibit less lateral force capacity; more precisely, the resistances for ν0 = 0.3 and 0.5 are rather similar, while those for ν0 = 0.7 and 0.9 are clearly smaller, particularly the latter.This trend can be explained by the combination of two quite opposite tendencies: the axial preload intensity (ν0) impairs the column capacity to resist lateral forces (V), but moderate  Figure 11 reveals similar trends to Figure 10; regarding the influence of the prestress, the percentage increases for p0 = 5 MPa with respect to p0 = 0 are approximately 5.8% (ν0 = 0.3), 11.4% (ν0 = 0.5), 34.2% (ν0 = 0.7) and 335% (ν0 = 0.9).About this tendency, the same explanation as in Figure 10 (based on Figure 9a) still holds.On the other hand, a comparison between Figures 10 and 11 shows that, unsurprisingly, the consideration of the concrete damage model (Section 4.3) has a significant impact.More precisely, in Figure 11 there is a reduction in the maximum capacity of lateral force; also unsurprisingly, this reduction is higher for larger values of ν0 (as the damage is higher).Finally, Figure 11d shows that in the most axially loaded specimen (ν0 = 0.9), the initial stiffness of the nonprestressed case is smaller than that of the prestressed ones.This indicates that the lack of active prestressing corresponds to the initial nonlinear behavior (i.e., for V = 0) of the analyzed structural element; the absence of this effect in Figure 10d shows that it only occurs when the concrete damage model is taken into consideration.
For greater clarity, Figure 12 displays plots of compressive and tensile vertical stresses for concrete (σcz and σtz, respectively) and steel (σsz) at the extreme left and right fibers (points) of the bottom section of the analyzed column segment; those points are represented in Figure 12 as  and , respectively.These magnitudes are plotted in Figure 12 against the lateral demanding force (V). Figure 11 reveals similar trends to Figure 10; regarding the influence of the prestress, the percentage increases for p 0 = 5 MPa with respect to p 0 = 0 are approximately 5.8% (ν 0 = 0.3), 11.4% (ν 0 = 0.5), 34.2% (ν 0 = 0.7) and 335% (ν 0 = 0.9).About this tendency, the same explanation as in Figure 10 (based on Figure 9a) still holds.On the other hand, a comparison between Figures 10 and 11 shows that, unsurprisingly, the consideration of the concrete damage model (Section 4.3) has a significant impact.More precisely, in Figure 11 there is a reduction in the maximum capacity of lateral force; also unsurprisingly, this reduction is higher for larger values of ν 0 (as the damage is higher).Finally, Figure 11d shows that in the most axially loaded specimen (ν 0 = 0.9), the initial stiffness of the nonprestressed case is smaller than that of the prestressed ones.This indicates that the lack of active prestressing corresponds to the initial nonlinear behavior (i.e., for V = 0) of the analyzed structural element; the absence of this effect in Figure 10d shows that it only occurs when the concrete damage model is taken into consideration.
For greater clarity, Figure 12 displays plots of compressive and tensile vertical stresses for concrete (σ cz and σ tz , respectively) and steel (σ sz ) at the extreme left and right fibers (points) of the bottom section of the analyzed column segment; those points are represented in Figure 12 as Figure 11 reveals similar trends to Figure 10; regarding the influence of the prestress, the percentage increases for p0 = 5 MPa with respect to p0 = 0 are approximately 5.8% (ν0 = 0.3), 11.4% (ν0 = 0.5), 34.2% (ν0 = 0.7) and 335% (ν0 = 0.9).About this tendency, the same explanation as in Figure 10 (based on Figure 9a) still holds.On the other hand, a comparison between Figures 10 and 11 shows that, unsurprisingly, the consideration of the concrete damage model (Section 4.3) has a significant impact.More precisely, in Figure 11 there is a reduction in the maximum capacity of lateral force; also unsurprisingly, this reduction is higher for larger values of ν0 (as the damage is higher).Finally, Figure 11d shows that in the most axially loaded specimen (ν0 = 0.9), the initial stiffness of the nonprestressed case is smaller than that of the prestressed ones.This indicates that the lack of active prestressing corresponds to the initial nonlinear behavior (i.e., for V = 0) of the analyzed structural element; the absence of this effect in Figure 10d shows that it only occurs when the concrete damage model is taken into consideration.
For greater clarity, Figure 12 displays plots of compressive and tensile vertical stresses for concrete (σcz and σtz, respectively) and steel (σsz) at the extreme left and right fibers (points) of the bottom section of the analyzed column segment; those points are represented in Figure 12 as  and , respectively.These magnitudes are plotted in Figure 12 against the lateral demanding force (V). Figure 11 reveals similar trends to Figure 10; regarding the influence of the prestress, the percentage increases for p0 = 5 MPa with respect to p0 = 0 are approximately 5.8% (ν0 = 0.3), 11.4% (ν0 = 0.5), 34.2% (ν0 = 0.7) and 335% (ν0 = 0.9).About this tendency, the same explanation as in Figure 10 (based on Figure 9a) still holds.On the other hand, a comparison between Figures 10 and 11 shows that, unsurprisingly, the consideration of the concrete damage model (Section 4.3) has a significant impact.More precisely, in Figure 11 there is a reduction in the maximum capacity of lateral force; also unsurprisingly, this reduction is higher for larger values of ν0 (as the damage is higher).Finally, Figure 11d shows that in the most axially loaded specimen (ν0 = 0.9), the initial stiffness of the nonprestressed case is smaller than that of the prestressed ones.This indicates that the lack of active prestressing corresponds to the initial nonlinear behavior (i.e., for V = 0) of the analyzed structural element; the absence of this effect in Figure 10d shows that it only occurs when the concrete damage model is taken into consideration.
For greater clarity, Figure 12 displays plots of compressive and tensile vertical stresses for concrete (σcz and σtz, respectively) and steel (σsz) at the extreme left and right fibers (points) of the bottom section of the analyzed column segment; those points are represented in Figure 12 as  and , respectively.These magnitudes are plotted in Figure 12 against the lateral demanding force (V)., respectively.These magnitudes are plotted in Figure 12 against the lateral demanding force (V). Figure 12a,b correspond to ν 0 = 0.3, and Figure 12c,d to ν 0 = 0.9.In all these Figures, the negative sign refers to compression.

Loading Step 3' (Shear Load with Variation of Axial Force)
Analogously to Figure 11 (for step 3), Figures 13 and 14 display the force (V) vs. displacement (∆) plots of steps 3' + and 3' − , respectively.To understand the plots of Figures 13 and 14, it should be kept in mind that in Figure 13 they correspond to a simultaneous increase in V and N (positive sign in Equation ( 8)), while in Figure 14 the opposite situation occurs (N decreases as V increases, negative sign in Equation ( 8)).The red bullets refer to the corresponding points.

Loading Step 3' (Shear Load with Variation of Axial Force)
Analogously to Figure 11 (for step 3), Figures 13 and 14 display the force (V) vs. displacement (Δ) plots of steps 3' + and 3' − , respectively.To understand the plots of Figures 13 and 14, it should be kept in mind that in Figure 13 they correspond to a simultaneous increase in V and N (positive sign in Equation ( 8)), while in Figure 14 the opposite situation occurs (N decreases as V increases, negative sign in Equation ( 8)).The red bullets refer to the corresponding points.Figure 13 presents a regular and anticipated behavior; a comparison with Figure 11 shows that the aforementioned increase in the axial force has led to a significant decrease in the shear force capacity.Unsurprisingly, the higher the initial force (ν0), the more intense the reduction.Figure 13d indicates that the initial nonlinear behavior shown by Figure 11d extends also to the case p0 = 2 MPa. Figure 13 presents a regular and anticipated behavior; a comparison with Figure 11 shows that the aforementioned increase in the axial force has led to a significant decrease in the shear force capacity.Unsurprisingly, the higher the initial force (ν 0 ), the more intense the reduction.Figure 13d indicates that the initial nonlinear behavior shown by Figure 11d extends also to the case p 0 = 2 MPa.
To better understand the results displayed in Figure 14, the final approximate values (i.e., for the maximum imposed displacement) of the axial force in the left columns are presented next (Equation ( 8)): N = −1295 kN (Figure 14a), N = +727 kN (Figure 14b), N = +2607 kN (Figure 14c), and N = +4327 kN (Figure 14d); negative values indicate tension.In general terms, the results in Figure 14 are consistent and likely.Comparison with Figure 13 indicates an important increase in shear capacity, mainly for highly loaded columns (except, perhaps, for ν 0 = 0.9); in this sense, unlike Figures 11 and 13, the shear capacity is higher for the specimen with the greatest initial compression (Figure 14d).On the other hand, the influence of the active hoop prestress has almost disappeared (apart from ν 0 = 0.9); this circumstance can be explained because the axial forces (N) are near zero or even negative (tensile).Finally, the ductility is significantly higher than in any other comparable situation (Figures 11 and 13); seemingly, it is due to the virtual absence of second-order effects (or even a positive effect for tensile forces).Regarding this trend, it should be kept in mind that, as discussed previously, plots in Figure 14c,d are deliberately interrupted for consistency with the corresponding plots in Figures 11 and 13. obtained similarly as in Figure 11.Notice that second-order effects make the maximum values of shear force V do not always coincide with the maxima of moment M.
respectively.Figure 9a (loading step 2, Figure 3) corresponds to µ = 0 (no bending); the ordinate considered is the ultimate value of ν (for w = 8 mm).In Figure 11 (loading step 3, Figure 3), obviously the ordinate values are ν = 0.3, 0.5, 0.7 and 0.9; the abscissae are obtained from the second-order equilibrium Equation (1) using the maximum (ultimate) value of Δ in the corresponding V-Δ plots in Figure 11.In Figures 13 and 14 (loading steps 3'+ and 3'−, respectively), the values of ν are determined by integration of the normal (axial) vertical stresses of concrete and steel along the column section; the abscissae are obtained similarly as in Figure 11.Notice that second-order effects make the maximum values of shear force V do not always coincide with the maxima of moment M. Figure 15 shows that, as expected, the influence of p0 is higher for large values of ν (and small values of µ, then) than in the opposite situation (ν ≈ 0); more precisely, in that case, the influence of p0 is practically negligible.Figure 15 also shows that the section capacity is not highly sensitive to the loading path, as the chart points arising from load steps 3 and 3' (for the same value of p0) are either close to each other or belong to one of the rather smooth ν-µ interaction diagrams plotted in Figure 15.
In the diagram for p0 = 0 (without transverse prestressing), the axial capacity for centered compression (µ = 0) corresponds to ν ≈ 1; this coincidence can be considered a good fit between the numerical results obtained and the expression in the Chinese code [17,18] (Equations ( 2)-( 5)).In other words, such a match can be understood as a calibration of the numerical model (Section 4) with the empirical expression proposed by the Chinese regulations.Additionally, the calculated centered axial capacity has been rather satisfactorily compared with the approximation in [33].This paper estimates the axial capacity of the analyzed columns as 12,720 kN; the agreement with the value in Equation ( 5) is quite acceptable, as buckling effects have not been considered in this paper.Noticeably, [33] uses an algorithm based on ANN (Artificial Neural Network) from a large suite of experimental results.When ν = 0, then µ ≈ 0.7.This last apparent inconsistency can be explained by the influence of the shear force on the moment strength, since µ refers to the pure moment strength MR (Equation ( 2)).
Sectional interaction diagrams ν-µ (normalized axial force vs. normalized bending moment).Symbols 23 of 28 of p0 (0, 2 or 5 MPa).These charts are generated by joining the thirteen epresenting the sectional structural capacity) of the plots in Figures 9a, ch points are represented in Figure 15 with symbols , ,  and , re 9a (loading step 2, Figure 3) corresponds to µ = 0 (no bending); the ed is the ultimate value of ν (for w = 8 mm).In Figure 11 (loading step 3, sly the ordinate values are ν = 0.3, 0.5, 0.7 and 0.9; the abscissae are second-order equilibrium Equation (1) using the maximum (ultimate) orresponding V-Δ plots in Figure 11.In Figures 13 and 14 (loading steps ctively), the values of ν are determined by integration of the normal esses of concrete and steel along the column section; the abscissae are as in Figure 11.Notice that second-order effects make the maximum rce V do not always coincide with the maxima of moment M. ows that, as expected, the influence of p0 is higher for large values of ν of µ, then) than in the opposite situation (ν ≈ 0); more precisely, in that e of p0 is practically negligible.Figure 15  Figure 15 shows that, as expected, the influence of p 0 is higher for large values of ν (and small values of µ, then) than in the opposite situation (ν ≈ 0); more precisely, in that case, the influence of p 0 is practically negligible.Figure 15 also shows that the section capacity is not highly sensitive to the loading path, as the chart points arising from load steps 3 and 3' (for the same value of p 0 ) are either close to each other or belong to one of the rather smooth ν-µ interaction diagrams plotted in Figure 15.
In the diagram for p 0 = 0 (without transverse prestressing), the axial capacity for centered compression (µ = 0) corresponds to ν ≈ 1; this coincidence can be considered a good fit between the numerical results obtained and the expression in the Chinese code [17,18] (Equations ( 2)-( 5)).In other words, such a match can be understood as a calibration of the numerical model (Section 4) with the empirical expression proposed by the Chinese regulations.Additionally, the calculated centered axial capacity has been rather satisfactorily compared with the approximation in [33].This paper estimates the axial capacity of the analyzed columns as 12,720 kN; the agreement with the value in Equation ( 5) is quite acceptable, as buckling effects have not been considered in this paper.Noticeably, [33] uses an algorithm based on ANN (Artificial Neural Network) from a large suite of experimental results.When ν = 0, then µ ≈ 0.7.This last apparent inconsistency can be explained by the influence of the shear force on the moment strength, since µ refers to the pure moment strength M R (Equation ( 2)).

Global Structural Behavior of a Representative Mid-Rise Building
This section uses the results in Sections 5 and 6 to expose a brief preliminary assessment of the structural capacity of a representative mid-rise unbraced frame building (such as the one in Figure 2a) equipped with CFST columns.This study consists of analyzing the performance of a given story in two different situations: gravity loads only (load step 2, Section 5.2), and lateral pushing forces as well (load steps 3 and 3', Sections 5.3 and 5.4); these cases are discussed in Sections 7.1 and 7.2, respectively.In both situations, the vertical and lateral story behavior is modeled as a parallel combination of the column specimens analyzed in Sections 2-5.

Story Vertical Behavior
The load distribution between the columns of a typical story of a mid-height building is almost independent of their axial stiffness and is more related to slab flexural behavior.Therefore, the story vertical capacity can be approximately estimated by adding those of each individual column (Figure 9a); in other words, the remarks exposed from that Figure (mainly in terms of axial force capacity) can be applied almost directly to the building's overall behavior under gravity loads.Given these considerations, the obvious final conclusion is that a proper design of the CFST columns can lead to a satisfactory vertical load capacity of the building story.

Story Lateral Behavior
This subsection presents a numerical study of the joint shear behavior of the five columns of a given story of the 2-D prototype building frame in Figure 2a.The behavior of the left (façade) column corresponds to loading step 3' − (Section 5.4, Figure 14), the three inner columns are associated with loading step 3 (Section 5.3) and the right (façade) column is consistent with loading step 3' + (Section 5.4, Figure 13).Cases ν 0 = 0.7 and ν 0 = 0.9 are omitted, since they are highly unfeasible in seismic regions, as discussed after Figure 3.
Before performing the announced numerical study, the demanding shear forces during wind or seismic excitations in actual situations are estimated next in order to assess the relevance of the building's lateral strength.In both cases, the dynamic effect is represented by static forces, and their approximate values are derived from simplified code-type calculations.

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Wind.The horizontal shear force on the bottom story of a 2-D frame ranges approximately between 180 kN (12 stories, covered situation, span length 5 m, story height 3 m) and 1125 kN (25-story building, exposed situation, span length 8 m, story height 4.5 m).

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Earthquakes.The equivalent static lateral force method shows that, for typical columns of mid-rise buildings (building fundamental period lying in the design spectrum plateau), the base shear force (V) ranges approximately between 0.4 a g W (rock-like soil, ordinary use, high response modification factor) and a g W (soft soil, essential use, moderate response modification factor), where W is the supported seismic weight (equal to the sum of the demanding axial forces N of each column) and a g is the seismic acceleration (corresponding to rock and 475 years return period) referred to the gravity acceleration (also commonly known as PGA, Peak Ground Acceleration).These considerations show that, for ν 0 = 0.3, in a high seismicity area (a g = 0.4 g) the demanding base shear force for a 5-column frame like the one depicted by Figure 2a ranges between 3600 and 9000 kN; in a low seismicity area (a g = 0.1 g), such bounds become 900 and 2250 kN, respectively.Obviously, for ν 0 = 0.5, these values must be multiplied by 5/3; the ranges are 6000-15,000 kN (for a g = 0.4 g) and 1500-3750 kN (for a g = 0.1 g).It should be kept in mind that these lateral forces must be combined with the vertical ones that correspond to the seismic weight.
The results of the announced numerical study are presented and discussed next.Figure 16 displays the shear force (V) vs. lateral displacement (∆) plots of the bottom 5-column 2-D frame story of the building depicted in Figure 2a.Assuming that the beams are infinitely rigid in their axial direction, these story V-∆ plots are approximately obtained by adding those of the individual column segments (Figures 11, 13 and 14 for the central, right and left columns in Figure 2a, respectively).Notice that ∆ represents the displacement of a half-column segment, while the interstory drift is actually 2 ∆ (Figure 1b); in earthquake engineering, this parameter is considered a highly reliable index of structural damage.
are infinitely rigid in their axial direction, these story V-Δ plots are approximately obtained by adding those of the individual column segments (Figures 11, 13 and 14 for the central, right and left columns in Figure 2a, respectively).Notice that Δ represents the displacement of a half-column segment, while the interstory drift is actually 2 Δ (Figure 1b); in earthquake engineering, this parameter is considered a highly reliable index of structural damage.Figure 16 shows that the story shear capacity is rather sufficient to resist moderate seismic events (approximately corresponding to PGA = 0.2 g); it should be kept in mind that, as the considered demanding seismic forces have been divided by a rather high response modification factor, significant damage (both structural and nonstructural) is to be expected.Figure 16 also points out that the benefit provided by the active hoop prestressing is only moderate; this circumstance is predictable, given the information provided by Figures 11,13 and 14.Regarding ductility, plots in Figure 16 seem to indicate a quite satisfactory displacement ductility.About wind, the story shear capacity is largely enough in all the cases analyzed.

Potential Applications of This Research
The results of this paper can contribute to the development of several applications.The main utility is the fostering of transversely prestressed CFST columns for mid-rise buildings.However, other close developments are also feasible, namely CFST columns with hoop prestressing for short and high-rise buildings and bridges, among others.

Research Performed
This paper assesses numerically the vertical and lateral structural performances of CFST (Concrete-Filled Steel Tube) columns that are actively prestressed transversely by bolting together two steel half-tubes; the study refers to new construction only.Representative prototype CFST column specimens (segments) are analyzed; they differ in their prestressing force and gravity loading ratio.The nonlinear structural static behavior of the column specimens is simulated with a model implemented in Abaqus.Figure 16 shows that the story shear capacity is rather sufficient to resist moderate seismic events (approximately corresponding to PGA = 0.2 g); it should be kept in mind that, as the considered demanding seismic forces have been divided by a rather high response modification factor, significant damage (both structural and nonstructural) is to be expected.Figure 16 also points out that the benefit provided by the active hoop prestressing is only moderate; this circumstance is predictable, given the information provided by Figures 11,13 and 14.Regarding ductility, plots in Figure 16 seem to indicate a quite satisfactory displacement ductility.About wind, the story shear capacity is largely enough in all the cases analyzed.

Potential Applications of This Research
The results of this paper can contribute to the development of several applications.The main utility is the fostering of transversely prestressed CFST columns for mid-rise buildings.However, other close developments are also feasible, namely CFST columns with hoop prestressing for short and high-rise buildings and bridges, among others.

Research Performed
This paper assesses numerically the vertical and lateral structural performances of CFST (Concrete-Filled Steel Tube) columns that are actively prestressed transversely by bolting together two steel half-tubes; the study refers to new construction only.Representative prototype CFST column specimens (segments) are analyzed; they differ in their prestressing force and gravity loading ratio.The nonlinear structural static behavior of the column specimens is simulated with a model implemented in Abaqus.

Conclusions for Individual CFST Columns
The obtained results provide the following major conclusions on the CFST column performance (Sections 5 and 6):

•
Vertical loads (axial force).As expected, the strength of CFST columns for verticalcentered axial compression is significant (Section 5.2).The benefit of the active hoop prestress is relevant, although not outstanding; this circumstance can be explained by the tube transverse expansion (due to the Poisson effect) that impairs the concrete core confinement (although not canceling it totally).
• Lateral forces (bending).Also as expected, the bending strength of CFST columns is significant (Sections 5.3 and 5.4).The benefits of the active hoop prestress are smaller than for the axial load; this trend is seemingly due to the lack of sectional global lateral expansion during bending.This gain is apparently not generated during lateral loading but is rather a continuation of the benefit produced while axial loading is applied.

•
Axial force-bending moment interaction.The strength for eccentric axial loads is pretty high; for moderate compression, the moment capacity is clearly higher than for pure bending (Section 6).The influence of the active transverse prestress is higher for large axial compression; for pure bending and for axial tension, that influence is practically negligible.

Conclusions for Mid-Rise Buildings
The results on the CFST columns are extended to make available preliminary estimates on the gravity, wind and seismic performance of mid-rise (roughly, between 12 and 25 floors) unbraced frame buildings equipped with such elements (Section 7):

•
Gravity loads.Given the notable axial capacity of the columns, a proper design can make available a satisfactory vertical strength.

•
Lateral forces (wind and seismic).The story shear capacity provided by CFST columns is sufficient for moderate seismic ground motions (approximately, Peak Ground Acceleration 0.2 g).This conclusion refers to a rather high response modification factor; therefore, the columns would be significantly damaged, and, thus, high ductility is required.Regarding this last issue, the displacement ductility is rather satisfactory.Finally, about wind, the story shear strength is largely enough.

Future Research
Further research includes more accurate and complex numerical analyses (including concrete rheology -creep and shrinkage-and column buckling), consideration of retrofit strategies, experiments on the proposed active transverse prestressing technique, wide numerical parametric studies, and implementation in actual full-scale buildings.It is expected that the results of the imagined experiments will allow the calibration of the numerical models used.As well, the economic implications of the proposed technology will be investigated.

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uildings 2023, 13, x FOR PEER REVIEW 5 of 28 (a) Frame side view (b) Seismic forces in the left and right ground columns

Figure 2 .
Figure 2. Building frame undergoing a seismic lateral force (F b ).

Figure 4
displays several views of the finite element mesh; Figure4ccontains a general view, and Figure4a,b,d present detailed views of the steel tube, the concrete core and a steel plate, respectively.

Figure 10 .
Figure 10.Loading step 3. V-∆ curves for central columns in Figure 2a.Analysis without considering the concrete damage model (Section 4.3).

Figure 11 .
Figure 11  reveals similar trends to Figure10; regarding the influence of the prestress, the percentage increases for p0 = 5 MPa with respect to p0 = 0 are approximately 5.8% (ν0 = 0.3), 11.4% (ν0 = 0.5), 34.2% (ν0 = 0.7) and 335% (ν0 = 0.9).About this tendency, the same explanation as in Figure10(based on Figure9a) still holds.On the other hand, a comparison between Figures10 and 11shows that, unsurprisingly, the consideration of the concrete damage model (Section 4.3) has a significant impact.More precisely, in Figure11there is a reduction in the maximum capacity of lateral force; also unsurprisingly, this reduction is higher for larger values of ν0 (as the damage is higher).Finally, Figure11dshows that in the most axially loaded specimen (ν0 = 0.9), the initial stiffness of the nonprestressed case is smaller than that of the prestressed ones.This indicates that the lack of active prestressing corresponds to the initial nonlinear behavior (i.e., for V = 0) of the analyzed structural element; the absence of this effect in Figure10dshows that it only occurs when the concrete damage model is taken into consideration.For greater clarity, Figure12displays plots of compressive and tensile vertical stresses for concrete (σcz and σtz, respectively) and steel (σsz) at the extreme left and right fibers (points) of the bottom section of the analyzed column segment; those points are represented in Figure12as  and , respectively.These magnitudes are plotted in Figure12against the lateral demanding force (V).Figure12a,b correspond to ν0 = 0.3, and Figure12c,d to ν0 = 0.9.In all these Figures, the negative sign refers to compression.

9 Figure 11 .
Figure 11.Loading step 3. V-Δ curves for central columns in Figure 2a.Analysis by considering the concrete damage model (Section 4.3).

9 Figure 11 .
Figure 11  reveals similar trends to Figure10; regarding the influence of the prestress, the percentage increases for p0 = 5 MPa with respect to p0 = 0 are approximately 5.8% (ν0 = 0.3), 11.4% (ν0 = 0.5), 34.2% (ν0 = 0.7) and 335% (ν0 = 0.9).About this tendency, the same explanation as in Figure10(based on Figure9a) still holds.On the other hand, a comparison between Figures10 and 11shows that, unsurprisingly, the consideration of the concrete damage model (Section 4.3) has a significant impact.More precisely, in Figure11there is a reduction in the maximum capacity of lateral force; also unsurprisingly, this reduction is higher for larger values of ν0 (as the damage is higher).Finally, Figure11dshows that in the most axially loaded specimen (ν0 = 0.9), the initial stiffness of the nonprestressed case is smaller than that of the prestressed ones.This indicates that the lack of active prestressing corresponds to the initial nonlinear behavior (i.e., for V = 0) of the analyzed structural element; the absence of this effect in Figure10dshows that it only occurs when the concrete damage model is taken into consideration.For greater clarity, Figure12displays plots of compressive and tensile vertical stresses for concrete (σcz and σtz, respectively) and steel (σsz) at the extreme left and right fibers (points) of the bottom section of the analyzed column segment; those points are represented in Figure12as  and , respectively.These magnitudes are plotted in Figure12against the lateral demanding force (V).Figure12a,b correspond to ν0 = 0.3, and Figure12c,d to ν0 = 0.9.In all these Figures, the negative sign refers to compression.
Figure 11  reveals similar trends to Figure10; regarding the influence of the prestress, the percentage increases for p0 = 5 MPa with respect to p0 = 0 are approximately 5.8% (ν0 = 0.3), 11.4% (ν0 = 0.5), 34.2% (ν0 = 0.7) and 335% (ν0 = 0.9).About this tendency, the same explanation as in Figure10(based on Figure9a) still holds.On the other hand, a comparison between Figures10 and 11shows that, unsurprisingly, the consideration of the concrete damage model (Section 4.3) has a significant impact.More precisely, in Figure11there is a reduction in the maximum capacity of lateral force; also unsurprisingly, this reduction is higher for larger values of ν0 (as the damage is higher).Finally, Figure11dshows that in the most axially loaded specimen (ν0 = 0.9), the initial stiffness of the nonprestressed case is smaller than that of the prestressed ones.This indicates that the lack of active prestressing corresponds to the initial nonlinear behavior (i.e., for V = 0) of the analyzed structural element; the absence of this effect in Figure10dshows that it only occurs when the concrete damage model is taken into consideration.For greater clarity, Figure12displays plots of compressive and tensile vertical stresses for concrete (σcz and σtz, respectively) and steel (σsz) at the extreme left and right fibers (points) of the bottom section of the analyzed column segment; those points are represented in Figure12as  and , respectively.These magnitudes are plotted in Figure12against the lateral demanding force (V).Figure12a,b correspond to ν0 = 0.3, and Figure12c,d to ν0 = 0.9.In all these Figures, the negative sign refers to compression.
also shows that the section hly sensitive to the loading path, as the chart points arising from load  correspond to Figure 9a (step 2), Figure 11 (step 3), Figure 13 (step 3' + ) and Figure 14 (step 3' − ).

Table 1 .
Geometric and mechanical properties of the analyzed column segments (specimens).