Analysis and modelling of CFT members: Moment curvature analysis

Thin-Walled Structures 86 (2015) 157–166

Contents lists available at ScienceDirect

Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws

Analysis and modelling of CFT members: Moment curvature analysis Rosario Montuori n, Vincenzo Piluso Department of Civil Engineering, University of Salerno, Salerno, Italy

art ic l e i nf o

a b s t r a c t

Article history: Received 16 April 2014 Received in revised form 19 September 2014 Accepted 13 October 2014

The evaluation of the ultimate behaviour of Concrete Filled Tubular (CFT) members subjected to nonuniform bending moment is the main objective of this work. To this aim eight experimental tests have been performed at the Materials and Structures Laboratory of the Department of Civil Engineering of Salerno University. In particular, Square Hollow Section (SHS) CFT members have been investigated under monotonic loading conditions. The three point bending scheme is adopted for testing specimens, where an hydraulic actuator is used for the transmission of the transverse load at midspan under displacement control and an LVDT is used to measure the corresponding maximum transverse displacement. In addition, longitudinal deformations along the cross section perimeter have been measured by means of strain gauges, aiming to the experimental evaluation of moment–curvature relationship. This paper presents the structural details of the tested specimens and the corresponding experimental results. In addition, a fibre model able to predict the ultimate response of CFT members is also presented and compared with the experimental results. The presented fibre model accounts for all the effects influencing the ultimate behaviour of composite members, such as local buckling, confining effects on concrete, bi-axial stress state of the steel plate elements constituting the hollow profile and hardening of its corners due to cold forming process. The comparison between experimental and numerical results shows a good agreement pointing out the accuracy of the proposed fibre model. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Concrete Filled Tubular Moment - curvature Ductility

1. Introduction The research on composite structures has gained more and more increasing interest, because of their capacity to provide excellent performances in terms of stiffness, resistance and ductility, provided that appropriate design and detailing rules are applied. Therefore, researchers are more and more focusing their attention on the design issues concerning the behaviour of such structural typology not only with reference to building structures, but also in case of bridges. In particular, concerning building structures, CFT members can be effectively adopted for the columns obtaining significant advantages when big crosssections are required. This is the case of design procedures aimed at assuring a collapse mechanism of global type where column members have to remain in elastic range [1–3]. Moreover, in bridge design, CFT members can be effectively adopted as bridge piers, exploiting their high ductility and energy dissipation capacity.

n

Corresponding author. E-mail address: [email protected] (R. Montuori).

http://dx.doi.org/10.1016/j.tws.2014.10.010 0263-8231/& 2014 Elsevier Ltd. All rights reserved.

The issues concerning the ultimate behaviour of steel–concrete composite members are nowadays worldwide investigated [4–10]. Many researchers have focused their attention on several issues concerning this topic, such as the interaction between steel and concrete [8,9], the axial load carrying capacity [8], the response under cyclic flexural loads [6,10]) providing, in some cases, models for the prediction of the behaviour at collapse. Nevertheless, only in few cases the proposed models exhibit well defined theoretical bases, while in most cases they are obtained by means of numerical calibrations requiring regression analyses of available experimental results. Conversely, the purpose of this work is to contribute to the development of theoretical models accounting for all the effects influencing the ultimate behaviour of composite members. In particular, it has been recognised that the development of a fibre model devoted to an accurate evaluation of the moment–curvature relationship is the first step to be performed towards this research goal. Two typologies can be adopted for steel–concrete composite columns: Concrete Filled steel Tubular columns (CFT) and Concrete Encased steel Composite columns (CEC). In the first case a steel tube made of a rectangular or circular hollow section is filled with concrete, while in the second one a steel profile is encased into

158

R. Montuori, V. Piluso / Thin-Walled Structures 86 (2015) 157–166

a reinforced concrete column. However, several reasons suggest to prefer the first typology with respect to the second one. For this reason, the main topic of the planned research activity is represented by the investigation of the ultimate behaviour of CFT-columns, since they are more suitable from both the structural and the technological point of view. In fact, no other reinforcement is needed, because the tube itself acts as longitudinal and lateral reinforcement for the concrete core. In addition, the location of longitudinal steel at the perimeter of the section constitutes the most efficient use of steel, because it provides the best contribution to the inertia moment and to the flexural resistance of the section. Moreover, the continuous confinement provided to the concrete core by the steel tube enhances the core strength and ductility and prevents the concrete spalling. The concrete core gives a great contribution to the load bearing capacity under axial loads and delays the local buckling of the steel tube by preventing inward buckling. Finally, drying shrinkage and creep of concrete are smaller than those occurring in ordinary reinforced concrete structures and in CEC members. In addition, the use of CFT system is advantageous also from the technological point of view. In fact, the steel profile represents the form for concrete casting, leading to more simple erection procedures reducing both the construction costs and the time required for execution. Conversely, some difficulties in properly compacting the concrete core may lead to a gap between concrete and steel. In order to avoid the occurrence of such drawback and to ensure compactness of the member, it is suggested the use of high-quality concrete with low water-content and super plasticisers to improve the workability during the cast. On the other hand the main advantage of CEC typology is that encasing concrete protects the steel profile from atmosphere corrosion. Some studies have also shown that the concrete core improves the fire resistance of the steel tube so that the fire resistance of axially loaded CFT members is greater than that occurring in the case of the steel tube only [11]. With reference to CFT typology, the sharing of internal actions between the steel profile and the concrete core is due to the bond between the two materials. Therefore, under this point of view, CFT columns can be distinguished in Bonded Concrete Filled steel Tubular (BCFT) columns and Unbounded Concrete Filled steel Tubular (UCFT) columns. The first typology is the most usual, and is characterised by the contribution of the two materials in withstanding the external actions. In the second case, an anti-friction fluid is adopted on the inner steel walls before concrete casting, so that the role of the steel profile is limited to the confinement effect, while its contribution to the axial load carrying capacity is negligible. The effect of the bond is related to load transmission between steel and concrete, which arises due to the friction at the interface. In particular, if the axial load is applied to the entire section, the contributions of steel and concrete remain constant along the height and they are independent of the bond at the interface. Conversely, when the axial load is applied only to the steel or only to the concrete, the axial load must be transmitted by means of the contact surface between the two materials. Therefore, the bond strength between the concrete core and the steel tube influences the distribution of the axial force within the section, and, in addition, influences the structural behaviour and the axial load resistance of the CFT column. Friction develops between the concrete core and the steel tube due to the normal contact pressure generated by the lateral expansion of the concrete core when subjected to compressive loading. The magnitude of the friction force depends on the rigidity of the tube walls against pressure perpendicular to their plane. In particular, the interaction between the concrete core and the steel profile in a CFT column is due to the fact that, increasing

the axial stress, the Poisson ratio of concrete becomes, due to plasticity, even greater than that of steel. As a consequence, the lateral expansion of concrete is restrained by the steel tube, so that a radial stress state arises at the interface between the two materials, the concrete being subjected to compression stresses and the steel tube walls to tensile stresses. In particular, due to the relatively small thickness of the steel tube, the compression radial stresses are balanced by means of tensile stresses which arise along the circumference, whose value σθ can be considered uniform within the thickness. Therefore, the stress state acting in the plane of the section is responsible of two effects which influence the behaviour of concrete and steel, respectively. In fact, the solid stress state of the concrete benefits of the confinement effect and the bi-axial stress state of the steel modifies the yield stress available in the axial direction of the member [7,8]). In addition, due to the presence of the concrete core, the inward deformations of the steel tube walls are prevented. This issue is particularly important with reference to the buckling modes. In fact, local buckling of an empty steel tube is characterised by the combination of inward and outward deformations of the plates. In case of SHS (square hollow sections) and RHS (rectangular hollow sections) members, such deformations interact through the rotations at the corners due to compatibility requirements. Conversely, in CFT members only outward deformations of the plate elements constituting the steel tube are allowed. In the proposed fibre model, it is assumed that there is no slip between steel and concrete while all the remaining issues concerning the interaction between steel and concrete are accounted for: the confinement of concrete, the bi-axial stress state in the steel tube and the possible local buckling of the plate elements constituting the steel profile. All these effects are accurately treated in next sections, with particular reference to the analytical formulations adopted in the development of the proposed fibre model. Furthermore, within the research project, experimental tests have been led on CFT members subjected to non-uniform bending at the Materials and Structures Laboratory of the Department of Civil Engineering of Salerno University. The comparison between the obtained experimental results and those resulting from numerical predictions provides a preliminary validation of the proposed fibre model.

2. Fibre model development 2.1. Confined concrete modelling The behaviour of structural concrete is usually described by means of mono-axial compressive tests, i.e. by means of the relationship between the mono-axial stress and the corresponding strain. Nevertheless, in most cases concrete is actually subjected to tri-axial stress state. If stresses occurring in transversal directions are negligible, the mono-axial state is representative of the stress condition and the mono-axial resistance in compression is reliable; conversely, when transversal lateral stresses are significant, the behaviour of concrete changes significantly, both in terms of resistance and ductility [12–14]. In particular, concrete confinement is defined as the condition in which the concrete is subjected to significant transversal compressive stresses. These stresses can arise internally due to the deformations of the cross section (passive confinement), as in the case of a steel tube filled by concrete, or can be applied and controlled externally (active confinement). In the first case, lateral stresses are not uniform into the cross section, depending on the shape of the cross section, the lateral deformation of the concrete

R. Montuori, V. Piluso / Thin-Walled Structures 86 (2015) 157–166

core and the stress–strain law of the confining steel. However, the behaviour of confined concrete is the same despite of the confinement typology [15]. In the case of confined concrete significant increases of resistance, stiffness and ductility are observed with respect to unconfined concrete. In fact, it is well known that unconfined concrete (i.e. concrete without lateral reinforcement), exhibits a brittle behaviour due to micro fractures arising from tensile transversal stresses. These micro fractures increase as far as the load increases, so that a progressive worsening of the behaviour of the cross section occurs. Conversely, when lateral reinforcements are adopted, micro fractures are avoided and the material benefits of greater load carrying capacity. With reference to confinement, several analytical formulations have been proposed in the technical literature in order to obtain an appropriate constitutive law [7,9,12,16]. Mander et al. [12] model is probably the most accredited one [17]. The method was originally developed with reference to reinforced concrete circular columns confined by means of spirals or circular hoops and, successively, extended to the case of rectangular sections confined by means of stirrups and ties. However, by properly defining the lateral confining stress, the approach can be easily extended to the case of concrete filled tubes. In particular, in case of spirals or circular hoops, the confined compressive strength fcc is given by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 7:94f 1 fl ð1Þ f cc ¼ f c 2:254 1 þ  2 1:254 fc fc being fc the unconfined concrete compressive strength and fl the effective lateral confining stress. Eq. (1) can be adopted also in the case of a confining circular tube by means of an appropriate definition of the lateral confining stress fl. In particular, accounting for the hoop stress σθ arising in the steel tube walls, the equilibrium equation provides the following formula (Fig. 1): fl ¼

2σ θ t D

ð2Þ

Mander et al. have provided the solution also in the case of rectangular sections. In Fig. 2 the ratio fcc/fc between the confined strength and the unconfined one is depicted depending on the lateral confining stresses fl1 and fl2, which are determined according to the following relationships:     f l2 ¼ max f lx ; f ly ð3Þ f l1 ¼ min f lx ; f ly

159

Starting from the value of fcc, Mander et al. model is based on the relationship proposed by Popovics [18]: σc ¼

f cc x r r  1 þ xr

ð5Þ

   f cc εcc ¼ 0:002 1 þ 5 f c 1

x¼

εc εcc

r¼

Ec Ec  Esec

Esec ¼

f cc εcc

Ec ¼ 5000

εsu εcu ¼ 0:004 þ 1:4ρ f cc

qffiffiffiffi f c ðMPaÞ

ρ¼

ð6Þ

2t w 2t w þ B  2t w D 2t w ð7Þ

being σc and εc the longitudinal compressive stress and strain, respectively, of confined concrete, εcc the strain value corresponding to the achievement of the peak stress fcc, εsu the ultimate steel strain, ρ the volumetric ratio of confining steel, Ec and Esec the tangent modulus and the secant modulus of concrete, respectively. Eq. (5) provides a stress–strain constitutive law including also the softening branch, until the ultimate concrete deformation εcu is reached. Therefore, the model allows the complete description of the confined concrete behaviour. 2.2. The bi-axial stress state in the steel tube As underlined, due to the lateral expansion of compressed concrete, the steel tube of a CFT column is characterised by the simultaneous occurrence of hoop stresses σθ and longitudinal stress σv, due to the applied loads, which are responsible of a bi-axial stress state in the steel plate elements [7,8,19]. Steel tubes under bi-axial state of stress exhibit a lower yield stress in compression in the longitudinal direction. From a theoretical point of view, von Mises criterion is herein adopted. Due to the hoop stresses, it results that yielding stresses in compression and tension become different. In particular, the yield stress in compression fyc is less than the mono-axial value fy, while the yield stress in tension fyt is greater than fy. This result is easily obtained by means of the yield surface corresponding to von Mises criterion (Fig. 3): 2

σ 2ν σ θ σ ν þ σ 2θ ¼ f y

ð8Þ

being flx and fly the lateral confining stresses in the x and y directions. The equilibrium equations provide the following relationships: f lx ¼

2σ θx t D

f ly ¼

2σ θy t B

ð4Þ

Eqs. (2) and (4) require the knowledge of the hoop stresses (σθx and σθy) which depend on the interaction between the steel profile and the concrete core, as discussed in next sections.

Fig. 2. Confined strength ratio for rectangular sections (redrawn from [12]).

Fig. 1. Lateral confining stresses for CHS and RHS CFT columns.

160

R. Montuori, V. Piluso / Thin-Walled Structures 86 (2015) 157–166

By means of Eq. (11) the value of fya can be obtained: f ya ¼ f yb þ ðf yc  f yb Þ

Ac Ag

ð12Þ

This value can be substituted in Eq. f ya ¼ f yb þ ðf su  f yb Þ ðknt 2 =Ag Þ (9) providing: f yc ¼ f yb þ ðf su  f yb Þ

knt 2 Ac

ð13Þ

In addition, by means of Eq. (12), limitation (10) provides: Fig. 3. Von Mises criterion.

f yc r f yb þ being σν the longitudinal stress, and σθ the tension hoop stress (in positive value). Therefore, the determination of the yield stress values requires a preliminary estimation of the hoop stress. This is possible by means of an iterative process [7]. In particular, several σθ attempt values can be assumed (σθ ¼σθx ¼σθy for SHS columns), varying between 0 and fy. For each value, the corresponding yield longitudinal stresses fyc and fyt can be determined by solving Eq. (8) with respect to σν, so that the constitutive laws of steel in tension and in compression, for the longitudinal direction, can be obtained. Moreover, by means of Eqs. (4)–(7) the corresponding confined concrete stress–strain relationship is established. As a consequence, for each value of the hoop stress σθ, an analytical value of the moment capacity can be determined by means of a moment–curvature analysis. If experimental results are available, the actual hoop stress can be obtained as the value of σθ leading to the match between the flexural resistance numerically predicted and the one obtained from experimental evidence. Following this approach, Elremaily and Azizinamini [7] suggest an average value of σθ ¼0.1 fy with reference to CHS columns, regardless of the slenderness of the steel profile and of the concrete strength. As a consequence, by means of Eq. (8) the values fyt ¼1.046 fy and fyc ¼  0.946 fy are obtained for the longitudinal yield stress in tension and in compression, respectively. 2.3. Constitutive law of corners In addition, it is needed to underline that an important rule is played by the constitutive law adopted for the steel constituting the corners, because they are partially or totally strain-hardened due to the cold forming process. In order to account for this phenomenon, reference is made to Eurocode 3 [20] which provides an increased average yield strength fya of the whole cross-section: f ya ¼ f yb þ ðf su  f yb Þ

knt 2 Ag

ð9Þ

with the limitation: f ya r

ðf su þ f yb Þ 2

ð10Þ

where Ag represents the gross cross sectional area; k is a coefficient equal to 7 in case of roll forming is; n is the number of 901 bends in the cross section with an internal radius r r5t being t the thickness of the plate elements constituting the section; and, finally, fyb and fu are, respectively, the yield and ultimate strength of the steel. Aiming to define the corresponding yield strength of the corners, the equilibrium of the section can be considered: f ya Ag ¼ f yb ðAg  Ac Þ þf yc Ac

ð11Þ

where fyc and Ac are, respectively, the yield strength and the area of the corners.

ðf su  f yb Þ Ag Ac 2

ð14Þ

2.4. Local buckling It is well known that the behaviour of steel thin tubes is strongly influenced by the occurrence of local buckling of the plate elements constituting the cross section. In particular, local buckling can be responsible of a significant decrease of resistance, and gives rise to a softening branch of the load vs displacement curve. However, concerning concrete filled tubes, the effect of local buckling is delayed by the presence of the concrete core. In fact, in case of CFT columns the inward deformations of the thin plate walls are prevented, so that only the outward buckling is possible. The presence of the concrete core prevents the plate rotation at the corners, so that in box columns (both SHS and RHS) the buckling of each plate requires the development of yield lines at both edges. Therefore, the number of yield lines needed to develop the kinematic mechanism describing the post local buckling behaviour of the profile is greater than that needed for an empty profile. This means that the internal work engaged in the development of the kinematic mechanism is also increased, so that local buckling is delayed, even though not completely avoided. A similar phenomenon develops in the case of CHS columns, where the circular section changes in an oval one after deformation due to buckling [21]. It can be observed that the ratio D/t has a significant influence on the local buckling of steel tubes. In particular, for high values of D/t, i.e. for slender steel sections, local buckling is anticipated, so that the bond between steel and concrete at the interface is lost, and the outward instability of the plate prematurely occurs. The concrete maintains its compressive capacity until the separation is completed; thereafter, the behaviour of the member is affected by cracking and spalling phenomena. In these conditions, CFT columns cannot sustain higher loads, but they are able to maintain an high percentage of the maximum load carrying capacity, showing a ductile behaviour despite of local buckling. In particular, circular columns show a more ductile behaviour [22]. A useful approach to account for the influence of local buckling of compressed plates of box columns (both SHS and RHS) is the effective width approach. This method is based on the study of the buckling behaviour of steel plates in plastic range. In particular, starting from the formula of the elastic critical stress: σ cr:e ¼ k

π2 E 12ð1  ν2 Þðb=tÞ2

ð15Þ

being k a buckling coefficient, depending on boundary conditions and plate geometry, and E the steel modulus of elasticity, the critical stress in plastic range and the corresponding critical strain are given by: σ cr:p ¼ ησ cr:e

1 kπ 2 E εcr:p ¼ η Es 12ð1  ν2 Þðb=tÞ2

ð16Þ

being Es the steel secant modulus of elasticity. The non dimensional factor η depends on the shape of the stress–strain curve of

R. Montuori, V. Piluso / Thin-Walled Structures 86 (2015) 157–166

the material, and several proposals are provided in the technical literature [23]. In particular, the accuracy of the available formulations has been investigated by comparing the results of numerical analyses with those obtained from experimental evidence. The investigated formulations are herein listed:

 (T1) Secant modulus theory: η¼

Es E

ð17Þ

Et E

ð18Þ

 (T3) Perason (1950), Bleich (1952) and Vol’Mir, (1965) [24,25]: η¼

rffiffiffiffiffi Et E

ð19Þ

 (T4) Radhakrishnan (1956) [26]: sffiffiffiffiffi Et η¼ Es

ð20Þ

 (T5) Gerard (1962) [27]: η¼

pffiffiffiffiffiffiffiffiffi E t Es E

ð21Þ

 (T6) Weingarten et al. (1960) [28]: η¼

Es E

rffiffiffiffiffi Et E

ð22Þ

 (T7) Stowell and Bijlaard [29]: η¼

Es 0:33 þ 0:67 E

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! Et 0:25 þ 0:75 Es

ð23Þ

 (T8) Li and Reid (1992) [30]: Es 0:50 þ 0:50 η¼ E

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! Et 0:25 þ 0:75 Es

that the central portion of the plate, having a width equal to b beff, has to be neglected (i.e. σ ¼0) in the computation of axial resistance and moment capacity of the cross section. In the following section, the moment–curvature relationship will be derived for all the different formulations regarding the factor η needed to account for buckling in plastic range. Therefore, eight curves will be numerically obtained exactly by the same procedure by changing only the factor η according to Eqs. (17)–(24).

3. Moment curvature analysis

 (T2) Tangent modulus theory: η¼

161

ð24Þ

The moment–curvature relationship is the primary analytical tool to describe the non-linear structural behaviour of the cross section. In fact, the complete knowledge of such relationship allows to correctly estimate the ultimate strength and ductility supply at the section level. In order to evaluate the moment–curvature relationship a fibre model has been developed. The cross section is subdivided into several elementary areas, each one characterised by the corresponding constitutive law of the material. In particular, the elementary areas are distinguished between confined concrete and steel. Regarding confined concrete, the stress–strain constitutive law proposed by Mander et al. [12] has been adopted (Section 2.1) by assuming a lateral confining stress computed on the basis of a circumferential stress in the steel tube equal to 0.10 fy, as suggested in [7]. Concerning steel, the stress–strain constitutive law has been assumed according to the classical Ramberg-Osgood [31] relationship by adopting different values in tension and in compression to account for the bi-axial stress state in the steel tube (Section 2.2), according to von Mises yield criterion. Moreover, the influence of local buckling has been taken into account by means of the effective width approach previously described. For an assigned value of the axial force, the moment–curvature relationship is obtained step-by-step by means of an iterative process. In particular, at each step the value of the curvature χ is fixed, and the actual strain distribution is obtained iteratively by imposing the fulfilment of the translational equilibrium equation. As soon as the strain diagram is known, i.e. the stress level of each elementary area is determined, the bending moment of the section is obtained by means of rotation equilibrium equation. The procedure stops when one of the following conditions occurs:

 one of the concrete elementary areas reaches the ultimate strain;

where Et is the tangent modulus. In the post-buckling phase, the in-plane stress distribution becomes non uniform. Stress concentrations occur close to the restrained edges where the maximum stress σmax is reached, while the stress decreases as far as the distance from the edge increases. Nevertheless, by means of the effective width method the determination of the actual non uniform stress distribution is not necessary. In fact, it is substituted by an equivalent constant stress distribution σ¼σmax extending for a reduced width, namely effective width beff, which is defined as that corresponding to the condition ε¼εcr.p. Therefore, by combining Eqs. (15) and (16) with b¼beff, the following relationship is determined between the effective width and the strain level ε: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kπ 2 E ð25Þ beff ¼ t η 12ð1  ν2 ÞEs ε Eq. (25) provides the effective width in the postbuckling phase of the steel plate as a function of the strain level ε. It also means

 one of the tensile steel areas reaches the ultimate strain;  the ratio between beff and the actual width of the plate under compression is less than 5% [32]. Under the combination of axial force and bending moment, one of the plates perpendicular to the bending plane (flange plates) is completely under compression, while the two plates arte parallel to the bending plane (web plates) are partially compressed. In particular, with reference to the buckling modes, it is assumed that the compressed flange behaves as an isolated plate fully restrained at its edges, because buckling can occur outward only. The plate is subdivided into elementary areas, whose strain level ε governs local buckling. If the condition ε Zεcr.p occurs, the plate is subjected to local buckling in plastic range and the effective width method can be applied. Therefore, Eq. (25) provides the width beff to be considered, depending on the strain level, by assuming the value 6.97 for the buckling factor k, being the plate edges clamped. Conversely, the web plates are subjected to a linear strain distribution. Therefore, Eq. (16) can be applied by assuming b¼yc,

162

R. Montuori, V. Piluso / Thin-Walled Structures 86 (2015) 157–166

i.e. with reference to the compressed part of the web. When εmax 4εcr.p, the compressed portion of the web plates is subdivided into two parts: one where ε rεcr.p, with length equal to le ¼εcr.p/χ and one where ε Zεcr.p, with length equal to lb ¼yc  le, being yc the distance between the neutral axis and the top edge of the plate in compression (Fig. 4). Therefore, only the lb portion is characterised by buckling in plastic range. In order to identify the effective part of the compressed web portion, Eq. (25) is applied with ε ¼εmax. The non effective width yc  beff is assumed to be located in the central part of the lb segment as depicted in Fig. 4. If the elementary area belongs to the non-effective width its contribution is neglected. Regarding the edge conditions of the web plate in compression, it is observed that the top edge at the corner is practically clamped because inward deformations are prevented and, similarly, the same assumption can be made for the lower edge of the compressed zone due to the restraining action exerted by the part in tension below the neutral axis. Therefore, the value 6.97 can be assumed for the buckling factor k of the web plates too. It is useful to point out that, for increasing values of the curvature, local buckling provides a reduction of the effective width of the plates in compression. This is responsible, in pair with the softening branch provided by Eq. (5), of the decrease of the bending capacity of the cross section, so that the moment– curvature relationship is characterised by a softening branch.

Table 1 Geometrical and mechanical properties of tested specimens with the corresponding ultimate testing load. Specimen D  t [mm] S1 S2 S3 S4 S5 S6 S7 S8

300  7.7 300  6.1 300  7.7 300  6.1 220  7.8 220  4.9 220  4.9 220  7.8

D/t

fsu [N/ mm2]

fcu [N/ mm2]

Fu α [kN]

n

f0 [N/ mm2]

38.96 49.18 38.96 49.18 28.2 44.89 44.9 28.21

564 595 579 570 567 460 472 559

36.46 36.46 36.46 36.46 36.46 36.46 36.46 36.46

895 851 955 733 507 233 272 504

13 19.54 18.23 20 13 11 10.71 21.4

480 508 501 480 490 330 400 490

0.015 0.00419 0.00948 0.0031 0.018 0.0032 0.0206 0.0072

4. Experimental program 4.1. Specimens and testing procedure An experimental program has been developed at the Materials and Structures Laboratory of the Department of Civil Engineering of Salerno University. The tests are devoted to analyse the response of SHS columns filled with concrete under monotonic loading conditions. In the following sections, the structural details of specimens, the testing devices and the experimental results are summarised. According to the planned experimental activity, eight CFT columns made of SHS steel members have been tested. The main difference among the tested specimens is represented by the width-to-thickness ratio D/t. The measured geometrical properties and the main material properties of tested specimens are provided in Table 1. Each specimen was manufactured from a rolled steel sheet, folded and welded along one longitudinal side. At the ends of the specimen, steel square plates (20 mm thick) were welded, being one of them holed to allow concrete filling. The specimens were placed upright and filled with concrete, then waiting for its complete cure. Concrete class C30/35 has been used. In order to recover the small longitudinal shrinkage at the top of the column, high strength epoxy was used to fill the gap and let

Fig. 4. Outward buckling of steel plates.

Fig. 5. Steel stress–strain diagram for all specimens.

the end surface be uniform between concrete and steel. The length of specimens is equal to 2400 mm. Regarding material properties, three coupon specimens have been prepared for each steel tube. Coupon tests have been carried out to evaluate the average maximum tensile strength fsu, given in Table 1, and the actual stress–strain relationship, depicted in Fig. 5 all the specimens. From these figures, it can be recognised that the shape of the stress–strain relationship is significantly different from a bi-linear one, being of round house type. For this reason, in order to better describe the actual characteristics of steel of such hollow profiles, it is convenient to refer to Ramberg-Osgood [31] formulation:  n σ σ ε ¼ þα ð26Þ E f0 where σ and ε are the actual stress and strain, respectively, the term σ/E represents the elastic part of the strain, the parameters α and n describe the hardening behaviour of the material, and f0 is the conventional value of the yield stress. For each specimen, the parameters α, n and f0 have been calibrated by means of a curve fitting technique. The adopted values are reported in Table 1. Regarding the constitutive law of the steel constituting the corners, relationships (13) and (14) need to be applied. For all the specimens considered in the experimental program the value of corner yield strength provided by Eq. (13) are greater than fsu. In addition, also the upper limit provided by Eq. (14) is greater than fsu for all the tested specimens. For this reason, the constitutive law assumed of the steel of the corners has been assumed elastic perfectly-plastic with a yield strength equal to fsu. All the steel tubes were filled with the same concrete cast whose properties have been tested by taking 150 mm concrete cubes which have been subjected to compression tests, after 28 days to check the concrete class. In addition, for each specimen, concrete

R. Montuori, V. Piluso / Thin-Walled Structures 86 (2015) 157–166

cubes have been tested also the date of the main test on CFT members, revealing, obviously, a small variation of the compressive strength of concrete due to aging. However, the scatter in concrete strength for the different specimens was very small, because testing of CFT members was carried out many months after concrete casting, therefore the same unconfined concrete strength has been assumed for all the specimens. Such value of the unconfined concrete strength has been obtained as the average compression strength and it is given in Table 1 with reference to the corresponding cylindrical value. Regarding the tests on CFT members, three point bending tests have been performed. No axial load was applied to the specimens. Fig. 6 provides a schematic view of the test setup. The ends of the specimen are free to rotate in-plane due to the connection to pin jointed hinges. Therefore, a simply supported scheme is adopted to test the beam–column. The distance between the two pins, i.e. the actual length of specimens, is equal to 3000 mm. One of the two end hinges is equipped with slotted hole, in order to allow free sliding of the specimen in the longitudinal direction. The load has been applied in midspan under displacement control by means of an hydraulic actuator having 2500 kN capacity in tension and 3000 kN in compression. The connection between the specimen and the actuator is constituted by means of a rigid stub located in the middle of the specimen. The stub is realised by means of two rigid plates on the horizontal sides of the specimen, joined by means of steel bars. Therefore, the testing devices have been conceived to carry out also cyclic test, because loads both in tension and in compression can be applied to the specimen. The length of the rigid stub is equal to 460 mm. Several measuring devices have been adopted to record displacements and deformations. The measure of the displacement at force location has been carried out by means of the LVDT of the hydraulic actuator and also by means of additional external displacement transducers. Additional displacement transducers have also been located along the length of the specimen to measure its

HYDRAULIC ACTUATOR

RIGID STUB

SPECIMEN

SUPPORTS

163

deflected shape. Moreover, strain-gauges have been applied on the vertical lateral sides of the steel box, in two control sections, one on the right side and one on the left side of the rigid stub. These sections are those characterised by the occurrence of the maximum value of the bending moment. They are placed at 1250 mm distance from the center of rotation of each hinge. In particular, longitudinal strain-gauges were applied on the webs at 55 mm distance from the top and the bottom flanges of the tube, respectively. In this way, deformations close to the maximum values are measured. Finally, in order to evaluate transversal deformations, two strain gauges have been applied also transversally, one for each control section.

4.2. Test results and discussion The experimental results are herein summarised by means of measured moment–curvature relationships. They are obtained starting from the strain values measured through the strain gauges, so that they are referred to the sections adjacent to the rigid stub. Since four measures are available (two control sections with strain gauges located on the two webs), the moment– curvature relationships have been obtained considering the average values of deformations. However, it has not been possible to evaluate the whole moment–curvature relationship up to the maximum imposed midspan displacement because, due to large deformations, strain-gauges went off-scale. In Figs. 7–14 the experimental moment–curvature relationships are depicted. The maximum lateral load applied at midspan of all specimens is also reported in Table 1 (Fu). The vertical displacement has been applied up to a displacement range compatible with the actuator stroke. In all the cases, bulges formed at top flange of steel tubes, close to the rigid stub, where the bending moment assumes its maximum value. Despite of local buckling, it can be noted that the maximum lateral load is almost maintained without any loss of strength. This means that CFT columns can exhibit high global ductility with reduced loss of strength due to delayed local buckling phenomena. With reference to transversal strain-gauges, applied on the upper part of the webs to evaluate the deformations due to the interaction between the steel tube and the concrete core, even if the corresponding longitudinal stress field was in compression, they measured tensile deformations. This agrees with the phenomenon described in Section 2.2. It means that, due to the interaction between the steel tube and the confined concrete, a bi-axial stress state arises in the steel plate elements, being hoop stresses in tension.

3000

Fig. 6. Test setup.

Fig. 7. Experimental specimen S1.

vs

analytical

moment–curvature

relationships

for

164

R. Montuori, V. Piluso / Thin-Walled Structures 86 (2015) 157–166

The moment vs curvature relationships derived from experimental evidence (Figs. 7–14) will be compared in next section with the numerical results coming from the application of the proposed fibre model. Regarding the bending moment provided in these figures, it has to be observed that, with respect to the rough experimental data, the influence of friction at the base of the supporting end hinges has been accounted for. In fact, it gives rise to an unavoidable horizontal component of reaction at the supports, leading to second order effects which develop during the loading process. Such second order effects have been evaluated considering a friction coefficient equal to 0.78, according to

Fig. 8. Experimental vs analytical moment–curvature relationships for specimen S2.

literature, and accounting for the force applied by the actuator and the weight of both the specimen and the testing equipment.

5. Validation of the proposed fibre model The experimental results presented in the previous section can be used in order to investigate the accuracy of the proposed fibre model. In particular, the moment vs curvature relationship obtained by means of the proposed fibre model is compared, for each tested specimen, with the one coming from experimental evidence. Aiming to analyse the influence of the different formulations, available in technical

Fig. 11. Experimental vs analytical moment–curvature relationships for specimen S5.

Fig. 12. Experimental vs analytical moment–curvature relationships for specimen S6. Fig. 9. Experimental vs analytical moment–curvature relationships for specimen S3.

Fig. 10. Experimental vs analytical moment–curvature relationships for specimen S4.

Fig. 13. Experimental vs analytical moment–curvature relationships for specimen S7.

R. Montuori, V. Piluso / Thin-Walled Structures 86 (2015) 157–166

165

Table 2 Ratio between the maximum theoretical and maximum experimental moment for different modelling options of local buckling. η

T1 T2 T3 T4 T5 T6 T7 T8

Specimen

Mean for all specimens

1

2

3

4

5

6

7

8

0.989 0.875 0.970 1.024 0.932 0.900 0.969 0.975

0.997 0.879 0.970 1.014 0.943 0.916 0.979 0.983

0.996 0.862 0.965 1.019 0.929 0.901 0.974 0.980

1.015 0.899 0.989 1.031 0.961 0.935 0.998 1.003

1.017 0.885 0.996 1.019 0.937 0.897 0.993 1.001

1.080 0.982 1.076 1.155 1.032 0.996 1.062 1.067

1.007 0.924 0.992 1.034 0.959 0.937 0.989 0.993

1.005 0.853 0.965 1.005 0.917 0.886 0.987 0.993

1.013 0.895 0.990 1.037 0.951 0.921 0.994 0.999

Fig. 14. Experimental vs analytical moment–curvature relationships for specimen S8.

literature, to account for the occurrence of local buckling depending on the width-to-thickness ratios of the plate elements constituting the steel tube profile, the numerical prediction of the moment–curvature relationship has been carried out, for each specimen, with reference to the eight formulations presented in Section 2.4, namely T1 to T8, i.e. Eqs. (17)–(24). The comparison between the results coming from experimental evidence and those obtained by application of the proposed fibre model is given in Figs. 7–14, with reference to specimens S1 to S8, respectively. The top part of each figure provides the comparison between the experimental curve and the moment– curvature relationship obtained by means of the fibre model adopting formulations T1, T2, T3 and T4 for the η factor governing the occurrence of local buckling. Similarly, the bottom part of each figure deals with the same comparison, but referring to formulations T5, T6, T7 and T8 for local buckling prediction. From a qualitative point of view, these figures show that a good agreement between experimental results and numerical predictions provided by the fibre model is obtained in case of formulations T1, T3, T7 and T8. Conversely, the use of formulations T2, T5 and T6 leads to an underestimation of the ultimate flexural resistance and, in addition, to the premature development of the softening branch related to local buckling. Moreover, it can be noted that the use of formulation T4 always provides an overestimation of the ultimate flexural resistance and, in addition, the obtained curve is always increasing without exhibiting any significant influence of local buckling. In order to provide a quantitative comparison among the different available formulations to account for local buckling, two parameters have been considered. The first parameter is the ratio between the maximum flexural resistance evaluated with the fibre model and the experimental value. The second parameter is the ratio between the area below the theoretical moment–curvature diagram and the same area evaluated with reference to the experimental curve. The obtained results area given in Table 2 with reference to the first parameter which provides a measure of the fibre model accuracy in terms of flexural resistance. In particular, last column provides the mean value of the above parameter showing that a very good agreement between experimental evidence and fibre model predictions is obtained in case of T3, T7 and T8 formulations, leading to a mean scatter less than 1%. Regarding the second parameter, the obtained values are given in Table 3 providing the fibre model accuracy in terms of absorbed energy. Last column, again, provides the mean value of the investigated accuracy parameter outlining that T3, T7 and T8 formulations provide a very good agreement with experimental evidence, being the mean scatter less than 2%.

Table 3 Ratio between theoretical and experimental values of the area below moment vs curvature diagram for different modelling options of local buckling. η

T1 T2 T3 T4 T5 T6 T7 T8

Specimen

Mean for all specimens

1

2

3

4

5

6

7

8

1.020 0.920 0.998 1.029 0.970 0.944 1.006 1.011

1.015 0.898 0.983 1.011 0.954 0.924 0.999 1.004

1.048 0.916 1.012 1.049 0.983 0.949 1.031 1.036

1.031 0.914 0.998 1.026 0.970 0.940 1.015 1.020

1.022 0.925 1.014 1.020 0.980 0.948 1.015 1.018

1.058 0.925 1.049 1.102 0.999 0.930 1.042 1.047

1.010 0.940 0.994 1.016 0.973 0.955 1.000 1.003

1.031 0.904 1.011 1.028 0.976 0.945 1.025 1.028

1.029 0.918 1.008 1.035 0.976 0.942 1.017 1.021

The above tables and figures point out the importance of the formulation to be used for the factor η accounting for local buckling in plastic range. In fact, the range of variation of the predicted ultimate flexural resistance with respect to the experimental value is between -10% and þ4% about. Similarly, the range of variation in terms of absorbed energy, i.e. the area below the moment curvature diagram, is between  8% and þ3.5%. Such scatters can be reduced provided the formulation for computing the factor η is properly selected. Therefore, as a conclusion of the above comparisons, it is confirmed the importance of local buckling and of its prediction in the modelling of the ultimate behaviour of CFT members. The theoretical formulations, accounting for local buckling either in elastic range or in plastic range, providing the highest accuracy are those herein namely T3, T7 and T8. Under this point of view, it is interesting to note that formulation T8, due to Li and Reid [30], is able to provide a very good agreement with experimental evidence even in the case of SHS and RHS aluminium hollow members (empty), subjected to the local buckling under uniform compression [23].

6. Conclusions The main goal of the presented research activity is the development of a fibre model to predict the ultimate behaviour of concrete filled tubes. In particular, attention has been focused on the moment–curvature relationship of such composite members whose accurate evaluation is the first step forward the numerical prediction of the ultimate strength and the ductility of such members. An experimental program has been developed in order to investigate the ultimate response of steel tubes filled with concrete. The results have confirmed the high attitude of such structural members for their application in seismic resistant

166

R. Montuori, V. Piluso / Thin-Walled Structures 86 (2015) 157–166

structures, being both strength and ductility highly suitable and satisfactory for the role of dissipative zones. The accuracy of the proposed fibre model has been pointed out comparing numerical results with the ones provided by the experimental evidence. The theoretical approach has shown a very good agreement with experimental results, by correctly predicting both resistance and ductility of CFT members provided that concrete confinement, bi-axial stress state of steel and local buckling phenomena are properly accounted for. Therefore, the developed fibre model can constitute a useful tool for designing such structural members. Acknowledgements This work has been supported by Italian research grant DPC RELUIS 2005–2008. References [1] Longo A, Montuori R, Piluso V. Theory of plastic mechanism control of dissipative truss moment frames. Eng Struct 2012;37:63–75. [2] Giugliano MT, Longo A, Montuori R, Piluso V. Failure mode and drift control of MRF-CBF dual systems. Open Constr Build Technol J 2010;4:121133. [3] Longo A, Montuori R, Piluso V. Failure mode control and seismic response of dissipative truss moment frames. J Struct Eng 2012. http://dx.doi.org/10.1061/ (ASCE)ST.1943-541X.0000569 (11, November 1). [4] Iannone, F., Mastrandrea, L., Montuori, R., Piluso, V. Experimental analysis of the cyclic response of CFT-SHS members. In: Proc. of Stessa 2009, sixth international conference on behaviour of steel structures in seismic areas. , 16–20 August, Philadelphia, U.S.A.; 2008. [5] Mastrandrea, L., Montuori, R., Piluso, V. Numerical model of the ultimate behaviour of SHS-CFT columns. In: Proc. of Eurosteel 2008, conference on steel and composite structures. 3–5 September, Graz, Austria; 2008a. [6] Han LH, Yang YF. Cyclic performance of concrete-filled steel CHS columns under flexural loading. J Constr Steel Res 2005;61:423–52. [7] Elremaily A, Azizinamini A. Behaviour and strength of circular concrete-filled tube columns. J Constr Steel Res 2002;58:1567–91. [8] Shams M, Saadeghvaziri MA. Nonlinear response of concrete filled steel tubular colmuns under axial loading. ACI Struct J 1999;96(6):1009–19. [9] Susantha KAS, Ge H, Usami T. Cyclic analysis and capacity prediction of concrete-filled steel box columns. Earthquake Eng Struct Dyn 2002;31:195–216. [10] Gourley BC, Tort C, Hajjar JF, Schiller PH. A synopsis of studies of the monotonic and cyclic behavior of concrete-filled steel tube beam–columns. Struct Eng Rep N.ST 2001:01–4.

[11] Kodur VKR, Lie TT. Fire performance of concrete-filled hollow steel columns. J Fire Prot Eng 1995;7(3):89–98. [12] Mander JB, Priestley MJN, Park R. Theoretical stress–strain model for confined concrete. J Struct Eng 1988;114(8):1804–26. [13] Montuori R, Piluso V, Tisi A. Comparative analysis and criticalities of the main constitutive laws of concrete elements confined with FRP. Composites Part B 2012;43:3219–30. [14] Montuori R, Piluso V, Tisi A. Ultimate behaviour of FRP wrapped sections under axial force and bending: Influence of stress–strain confinement model. Composites Part B 2013;54:85–96. [15] Attard MM, Setung S. Stress–strain relationship of confined and unconfined concrete. (title 93-M49)ACI Mater J 1996:432–42. [16] Han LH, Yang YF, Tao Z. Concrete-filled thin-walled steel SHS and RHS beam– columns subjected to cyclic loading. Thin-Walled Struct 2003;41:801–33. [17] Montuori R, Piluso V. Reinforced concrete columns strengthened with angles and battens subjected to eccentric load. Eng Struct 2009;31:539–50. [18] Popovics S. A numerical approach to the complete strain stress–strain curves for concrete. Cem Concr Res 1973;3(5):583–99. [19] Tao, Z., Han, L.H., Zhao, X.L. Hysteretic behaviors of concrete filled steel tubular beam–columns with square section. In: Proc. of the first international conference on steel & composite structures. 14–16 June, Pusan, South Korea; 2001.p. 1717–1724. [20] CEN. E.N. 1993-1-3., Eurocode 3 – design of steel structures – Part 1–3: General rules—supplementary rules for cold-formed members and sheeting; 2006. [21] Saisho, M., Kato, M., Gao, S. Local buckling of CFT-column under seismic load. In: Proc. of 13th world conference on earthquake engineering. , 1–6 August, Vancouver, BC, Canada; 2004. [22] Shanmugam NE, Lakshmi B. State of the art report on steel–concrete composite columns. J Constr Steel Res 2001;57:1041–80. [23] Faella C, Mazzolani FM, Piluso V, Rizzano G. Local buckling of aluminum members: testing and classification. J Struct Eng 2000:353–60. [24] Bleich F. Buckling strength of metal structures. New York, NY: McGraw Hill Book Co., Inc.; 1952; 343–55. [25] Vol’Mir A.S. Stability of elastic systems, Foreign technology division, Wright Patterson Air Force Base, Ohio; 1965. [26] Radhakrishnan S. Plastic buckling of circular cylinders. J Aeronaut Sci 1956. [27] Gerard G. Plastic stability theory of orthotropic plates and shells. J Aeronaut Sci 1962. [28] Weingarten VI, Morgan EJ, Seide P. Final report of design criteria for elastic stability of thin shell structures. Space Tech Labs 1960. [29] Bijlaard PP. Theory and tests on the plastic stability of plates and shells. J Aeronaut Sci September 1949:529–43. [30] Li S, Reid SR. The plastic buckling of axially compressed square tubes. J Appl Mech 1992:59. [31] Ramberg W., Osgood W.R. Description of stress–strain curves by three parameters, technical note no. 902, National Advisory Committee For Aeronautics: Washington, DC; 1943. [32] Chen WF. Advanced analysis of steel frames. Crc Press Llc 1993.