FRP-confined circular columns subjected to eccentric loading

Abstract

In this article, a new model for eccentrically loaded FRP-confined circular concrete columns is presented. The cross-section is built up by 2D finite elements (where the variation of the stresses and strains is 3D), and the concrete modeled by a confinement-sensitive (non-associated) material law. The numerical solution of this model showed agreement with the experimental results available in the literature. It is shown that the maximum axial stresses in the cross-section – despite of the big differences in the radial- and hoop confinement stresses – are about the same for concentric and eccentric loadings. It is also shown that the recommended stress–strain curves available in the literature may lead to non-conservative results for eccentric loading, and – based on our numerical calculations – new stress–strain curves are recommended.

Keywords

FRP-confined circular concrete column eccentric

Introduction

Load-bearing capacity of axially loaded concrete and reinforced concrete columns can be improved using lateral confinement. In the past two decades, the use of fiber-reinforced polymer (FRP) confinement has significantly increased due to its high corrosion resistance, high ultimate stress, and because it is easy to use. FRP confinement is most frequently applied for circular and rectangular cross-sections (with rounded edges).

The load internal forces develop in the column, which may result in failure and the internal forces are as follows

axial force due to concentric axial load (Figure 1(a)),

axial force and bending moment due to eccentric axial (or axial and horizontal) load (Figure 1(b)), and

shear force due to horizontal load (Figure 1(c)).

Concentrically loaded columns are discussed in a companion article,¹ where the literature is also summarized and not reiterated here. We recall, however, that depending on the stiffness and strength of the confinement, there are three types of stress–strain responses for confined concrete, which are shown in Figure 2.

Relatively few documented experiments on eccentrically loaded FRP-confined circular columns can be found in the literature. Hadi² presented data for unreinforced concrete columns with unidirectional confinement, where the fibers are arranged in the hoop direction and the FRP provides no axial resistance. Hadi^3,4 published data for reinforced concrete columns with unidirectional (hoop) confinement. Fam and Rizkalla⁵ ran experiments where FRP tubes (with both axial and circumferential resistances) were filled with concrete. Their results are summarized in Figure 3. The diagrams show the normalized force and moment, where the normal force is divided by the axial resistance of theunconfined cross-section and the moment by the maximum moment resistance of the unconfined cross-section.

There are a few further experimental results on eccentrically loaded FRP-confined columns, where the confinement did not play an important role: the confinement is insufficient¹ for the glass fiber-reinforced polymer-confined specimens of Hadi;^2,3 the eccentricity is quite high⁶^–⁸ and hence the confinement hardly influenced the load-carrying capacity (as it is stated by Bisby and Ranger⁹). These results are also shown in Figure3, but they are not considered for further investigations.

There are two types of models of confined columns:¹ design-oriented models, where the Bernoulli–Navier hypothesis (plane cross-section) is combined with an axial stress–strain curve of concrete (which contains the effect of confinement); and analysis-oriented models, which are based on a triaxial material model for concrete. The axial stress–strain curves of design-oriented models are built on the axial strength and failure strain (f_cc, ε_cc, Figure 2) of concentrically loaded confined columns.¹

For example, according to Lam and Teng:¹⁰

(1)

(2)

We also show the formula used in Eurocode 2:¹¹

(3)

Here, f_c0 is the uniaxial compressive strength of concrete, f_cc the compressive strength of confined concrete, f_l_,a the actual confining stress at failure, ε_cc the axial strain at maximal strength of confined concrete, ε_c0 the axial strain at maximal strength of unconfined concrete, and ε_fu the hoop strain at ultimate state of the confining FRP. Note that f_l_,a is smaller than the confining strength f_l, which belongs to the failure strength of FRP (Equation (7)).

The ratio of f_l_,a and f_l is the ‘strain efficiency factor’,¹⁰ which is denoted by κ_ε:

(4)

The simplest model (also used in Eurocode 2¹¹) applies a similar stress–strain curve as for unconfined concrete, where the strength is replaced by f_cc. This curve is identified as ‘Eurocode 2’ in Figure 4. Lam and Teng¹⁰ recommended a parabolic–linear curve for confined concrete:

(5)

This curve (Figure 4) was recommended by Bisby and Ranger⁹ and also by Rocca et al.¹² for eccentric loading.

Based on the three stress–strain curves of Figure 4, we calculated the normalized N–M failure envelopes (capacity diagrams) of confined columns, which were experimentally investigated. Three typical curves are shown in Figure 5. Note that according to Bisby and Ranger,⁹ the confining effect of steel stirrups should be neglected; however, for the sake of better comparability, we took the effect of steel stirrups into account by increasing the confining stress in the area inside the stirrups. Bisby and Ranger⁹ and Rocca et al.¹² suggested that due to the ‘strain efficiency factor,’ the top part of failure envelope must be cut off by a horizontal line and the maximum normal force is reduced by the factor κ_ε (Equation (4)). This is the reason for the formation of the plateaus shown in Figure 5(a) and (b). In the case presented in Figure 5(c), κ_ε ≈ 1. (The calculation of concrete-filled composite tubes with design-oriented expressions was complex. The tube enhances the load-bearing capacity in two ways: (1) it has axial resistance and (2) through circumferential confinement, it increasesthe concrete (axial) strength. The influence of these effects and hence the failure load depends on theratioof the axial and the circumferential strain. Unfortunately, the design-oriented expressions do not predict the circumferential strains. In the calculation, we have chosen this ratio in such a way that the experimental data for concentric load could be obtained, and then, this ratio was applied for eccentric loading as well.) Some of the results are acceptable (the calculated curves are close to the experimental results); however, in some cases, the models seem to overestimate the effect of the confinement at high eccentricity.

Analysis-oriented concrete models for concentric loading were discussed in Ref.¹ and not reiterated here. We found only one article in the literature which applied an analysis-oriented model for the eccentrically loaded FRP-confined concrete columns. Parvin and Wang¹³ used the MARC™ non-linear finite element software and applied the built-in Mohr–Coulomb yield criterion with isotropic hardening rule for concrete. They demonstrated the applicability of the three-dimensional (3D) model for confined columns; however, only qualitative comparisons with experiments were made.

The third typical internal force of columns is the shear force (Figure 1(c)). This internal force is not considered in this article.

Problem statement

As we stated in the ‘Introduction’ section, there are models for concentrically and eccentrically loaded FRP-confined columns; however, all the existing models fail to properly predict the behavior of eccentrically loaded confined columns. Authors also admit⁹ that further research in this area is needed.

Our aim in this article is to develop a new model for eccentrically loaded (Figure 1(b)) FRP-confined concrete or reinforced concrete columns (Figure 6). With the aid of this model, we wish to predict the experimental data and to explain the behavior of confined columns.

Approach

Confined columns can be analyzed by design-oriented or analysis-oriented models. While for concentric loading, both approaches are feasible, for eccentric loading design-oriented models – which are based on the confined concrete strength (e.g., Equations (1)–(3)) – seem unacceptable, because the effect of confinement is different for concentric- and for eccentric loading. This isillustrated in Figure 7. Under concentric loading, the in-plane stresses in the hoop and radial directions are identical (Figure 7(a)). For pure bending – assuming a linearly elastic material law – the highest confining stress in the hoop direction is three times bigger than in the radial direction (Figure 7(b)). In addition, shear stresses arise between the concrete and the confining FRP. For eccentric loading, the stress state (again for linearly elastic behavior) is between the previous two cases, as it is illustrated in Figure 7(c). (When concrete starts yielding, the behavior is similar; however, the calculation is more complex.)

Because of the significant difference in the radial and hoop confining stresses, we decided to use an ‘analysis-oriented’ model, which is based on a new, sophisticated 3D concrete material law proposed by Papanikolaou and Kappos.¹⁴

The new model

A two-dimensional (2D) finite element model was developed for the calculation of the cross-section. Note that the finite element mesh is 2D; however, the strains and stresses are 3D. It is assumed that the axial strain varies linearly through the cross-section (and the non-linearly varying axial stress is calculated by the FE code).

The concrete was modeled by the confinement-sensitive plasticity constitutive model of Papanikolaou and Kappos,¹⁴ which was also discussed in a companion article.¹ A triangular finite element mesh was applied (Figure 8).

The FRP confinement was calculated by the laminated plate theory assuming a linearly elastic behavior until failure. It was modeled by boundary elements along the circumference (Figure 8).

Axial rebars and stirrups – if present – were calculated by an elastic–plastic material law. The effect of stirrups was taken into account by assuming continuous layer (with zero axial resistance), and applying a boundary layer within the cross-section (Figure 8(b)).

Due to the high non-linearity of the concrete material law, an incremental calculation was implemented; the block diagram is shown in Figure 9.

If the eccentricity is high, the neutral axis will be inside the cross-section. At the tension-part of the cross-section, where the axial strains are positive, we assumed that the concrete is cracked and has no axial strength (however, the in-plane stiffnesses are non-zero and hence the cross-section remains approximately circular). As we stated above, shear stresses arise at the concrete surface. These shear stresses can lead to principal tensile stresses, which can cause rupture in concrete even in the compressed region of the cross-section. This effect was also considered in our model.

As an example of the numerical calculation, a strain–force diagram (and the corresponding N-M curve) is shown in Figure 10, assuming that the neutral axis is fixed. The calculation is terminated, when the FRP breaks. (The behavior of concrete (plastic hardening, softening, etc.) is explained in a companion paper¹).

Calculation of capacity diagrams (failure envelopes)

For a given cross-section, the strain–force (and the corresponding force–moment) diagrams (Figure 10) were calculated for different load paths. For each load path, the position of the neutral axis was fixed. The envelope of all N–M curves is identical to the capacity diagram (or failure envelope), as it is shown in Figure 11.

The capacity diagram depends on the load path. We obtain different envelopes if – for example – we assume fixed neutral axis or fixed eccentricity of the force. According to our calculations, these differences are small. An example of calculation with different load paths is shown in Figure 12.

An axial stress distribution at failure is shown in Figure 13(a). The axial stress varies slightly perpendicular to the plane of eccentricity; the stresses – due to the shear stresses – are smaller at the edges. The average stresses ( $({\bar{σ}}_{x} = \int_{- b}^{b} σ_{x} d z / (2 b))$ ) are presented in Figure 13(b).

The average stress curves at different eccentricitiesare given in Figure 14(a) for a high-stiffness confinement and in Figure 14(b) for a low-stiffness confinement.

Verification

The available experimental data were compared to our numerical results and also to the diagrams based on design-oriented models (Figure 4).

We recall that¹ – for concentric loading – instead of the confining stress due to the FRP failure strength, only a reduced stress must be taken into account:

(6)

where

(7)

σ_f is the failure strength of FRP (measured on coupon tests), t the thickness of FRP, d the diameter of the cross-section (Figure 6), and f_l_,a the radial stress measured at the failure. For concentric loading, the ‘strain efficiency factor’ κ_ε is typically between 0.6 and 1.0.¹⁰ This is due to the uneven hoop stresses in FRP because of the axial cracks in concrete.

Unfortunately, for eccentric loading, considerably few experimental data on κ_ε are available. The results of Bisby and Ranger⁹ showed that for eccentric loading, the reduction is much less (κ_ε is closer to unity) than for concentric loading. This statement can be explained by investigating the stresses presented in Figure 7, which shows that under eccentric loading, the hoop compressions are higher than for concentric loading, which reduces the likelihood of axial concrete cracks.

Because of the lack of reliable data, we simply apply the following κ_ε values (Figure 15):

(8)

where κ_ε0 is the strain efficiency factor for concentric loading and ρ the curvature (Figure 15(b)). ρ = 0 is the case of concentric loading and ρ_β = ε_cc/d the case when the neutral axis is at the edge of the cross-section. Accordingly, only for the cases of small eccentricity, κ_ε plays an important role. We must admit, however that further experiments are needed to clarify the effect of κ_ε under higher eccentricities.

The comparison of experimental results and numerical calculations are shown in Figure 16. The values of κ_ε0 are given in the figures. Only in one case (Figure 16(d)), κ_ε0 was measured (κ_ε0 = 0.49), in all the other cases, we have chosen its value in such a way that f_cc matches the data for concentric loading. The results of the new model including the effect of the ‘strain efficiency factor’ through Equation (8) are shown by solid lines. The results without this correction are plotted by dashed lines.

In Figure 16(f), it seems that for concentric loading, our model significantly overestimates the failure load. Due to the fact that all the other points are reasonable and considering the calculated load–strain curve (Figure 17), there is an other explanation: this case is a “low-stiffness confinement,”¹ which means that there is a local maximum on the force–strain curve. This local maximum agrees well with the failure load measured in the experiment. It is possible that the increasing branch was not measured.

Discussion

In the article, a new model was presented to calculate the stress–strain curve of eccentrically loaded FRP-confined, circular concrete columns based on a sophisticated (confinement-sensitive plasticity constitutive) concrete model. Our results agree with the experimental data.

In the “Calculation of capacity diagrams” section, the average stress curves at failure were presented (assuming κ_ε = 1.0). Based on these curves (Figure 14), the following observations can be made:

the maximum axial stress of eccentrically loaded columns roughly agrees with the axial stress of concentrically loaded columns.

the average axial stress decreases rapidly as we move away from the most compressed part of the cross-section, much faster than it is predicted by the (linear) diagram of Lam and Teng.¹⁰

The first observation can be explained by the concrete material model. In a companion article,¹ we presented lower and higher limits (f_cc,min and f_cc,max) for the axial strength for the case when the confining stresses are identical in the principal directions (σ₁ = σ₂ = σ_l). The curves are shown in Figure 18(a). We recall¹ that the failure stress f_cc is between f_cc,min and f_cc,max.

We reran this calculation by assuming that σ₂ = 2σ₁ (Figure 7). It was found that f_cc,min and f_cc,max are slightly higher, but there is no significant difference (Figure 18(b)). This is the reason that the maximum stresses are roughly the same for concentric and eccentric loadings. Based on the above two observations, new stress–strain curves can be recommended for the simple ‘design-oriented’ model, as it is discussed below.

Possible material law for the design-oriented model

In Figure 19, we again show the simple Eurocode 2 stress–strain diagram and that proposed by Lam and Teng.¹⁰

In addition, three possible stress–strain diagrams are shown, two have a parabolic first part followed by a bilinear curve (P1 and P3) and one has two parabolic parts (P2). For the simplified diagram P1, the breakpoint between the linear parts is at ε_cu; and the breakpoint for curve P3 (ε_c,P3) is calculated as follows:

(9)

We have calculated the capacity curves of the previous cases, and it was found that the Eurocode 2,¹¹ the Lam and Teng¹⁰ and the P1 curves overestimate the failure load for eccentric loading, while both P2 and P3 seem reasonable. Examples are shown in Figure 20. (Note that the use of approximate formulas is straightforward only for unidirectional confinement, as it was discussed for concrete-filled tubes in the ‘Introduction’ section.) In all approximate diagrams, we used the strain efficiency factor (κ_ε), as given in Equation (8).

Footnotes

Funding

This work was supported by the OTKA foundation [grant number K-77803].

Nomenclature

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