Axial compression model for FRP confined concrete in elliptical cross sections

Most of the existing studies conducted on FRP-confined concrete considered circular and square concrete columns, while limited studies were on columns with rectangular sections. The studies have confirmed that the circular cross-sections exhibited higher confinement effectiveness, whereas in the case of non-circular cross-sections the efficiency of FRP confinement decreases with an increase of the sectional aspect ratio and there is no significant increase, particularly for columns with the aspect ratio of 2.0. As recently suggested by the researchers, to significantly increase the effectiveness of FRP-confinement for these columns is by modifying a rectangular section to an elliptical or oval section. According to the literature, most of the existing confinement models for FRP-confined concrete under axial compression have been proposed for columns with circular and rectangular cross-sections. However, modeling the axial strength and strain of concrete confined with FRP in elliptical cross-sections under compression is most limited. Therefore, this paper provides new expressions based on limited experimental data available in the literature. For a sufficient amount of FRP-confinement, the threshold value was proposed to be 0.02. Finally, the accuracy of the proposed model was verified by comparing its predictions with the same test database, together with those from the existing models. .


Introduction
It is now well-recognized that confinement of existing concrete columns in bridges and buildings using fiber-reinforced polymers (FRPs) can significantly increase the strength and ductility of the columns.Over the last 25 years, a large number of experimental tests and analytical models were focused on the axial compressive behavior of FRP-confined concrete ( Early research studies indicated that FRP confined square and rectangular sections with sharp corners provide only a little enhancement in their axial load capacities, while the confinement effectiveness increases directly with an increase in the corner radius (Wu and Zhou, 2010).Meanwhile, the curvature of the rectangular section's corners could cause stress concentration (Al-Salloum, 2007).Therefore, changing square section to circular section may minimize these stress concentration (Tsai and Lin, 2002 Yan (2005) included an experimental program involved testing 30 FRP-confined concrete columns of circular, square and rectangular sections subjected to axial compressive loads.It was concluded that the FRP jackets are not able to effectively improve the compressive behavior of square and rectangular columns exhibiting a softening behavior.Recently it has been confirmed by Isleem et al. (2018c, d) for rectangular columns of larger-sized sections that the confinement provided by the FRP wraps resulted in a significant improvement in axial strains but only a little improvement in axial strengths.In their study, the results of tests have shown that only the sufficiently confined specimens with the aspect ratio of 1.5 reached higher strengths as compared with that of the unconfined concrete cylinder, while no strength enhancement was achieved for larger-sized specimens with the aspect ratio of 2.0.All experiments showed that the stress-strain curves of the confined columns exhibited a softening behavior in their response.The key solution to reduce the corner stress concentration that causes the softening behavior for such large-sized sections and thus to improve the strength and deformability of concrete columns with light and moderate confinement level is to modify the shape of square section to circular section and the rectangular cross-section to elliptical or oval cross-section using subsequent steel or composite jacketing (Priestley et al., 1994;Priestley and Seible, 1995;Mandal and Mansur, 2001;Teng and Lam, 2002).Further, the shape modification method can also be employed when the corners of the rectangular section can no longer be rounded for fear of infringing on the minimum concrete cover for reinforcing steel bars (Parvin and Schroeder, 2008).
The most economical method of shape modification of concrete columns is adding oval precast concrete segments to the perimeter of the rectangular column with subsequent FRP wrapping (Parvin and Schroeder, 2008).For this technique of modification of section's shape to be widely used for the strengthening of rectangular columns subject to axial compression, analytical expressions for predicting the axial strength and strain of FRP reinforced concrete columns with elliptical sections are needed.Because concretes in rectangular sections confined with FRP behave differently compared with the elliptical sections, if the available models of rectangular columns are directly applied to confined elliptical columns, the strength and strain capacities may not represent the realistic behavior of the columns, and unsafe design may be performed.Therefore, this paper aims to provide expressions for the accurate predictions of the axial strength and axial strain.In addition, based on the existing test database, the threshold for sufficiently confined concrete has been proposed to be equal to 0.02.This can be an important feature of the proposed model being able to predict well the threshold confinement condition that can dictate whether the stress-strain response ascends or descends.Finally, an acceptable correlation was revealed between the predictions of the proposed model and the experimental test data.

Overview of specimen details
In order to develop new strength and strain models, the the results of experimental tests performed by Teng and Lam (2002) were used for the calibration of all expressions provided in this paper.Twenty unreinforced concrete specimens were prepared and tested under axial compression loading.The experimental tests included five groups of specimens (S1, S2, S3) divided according to their sectional aspect ratios a/b (1.0, 1.28, 1.7, 2.5) and prepared from the same batch of concrete as provided in Table 1.Each group included one specimen with a circular section and three elliptical specimens.The cross-sectional area and height of the elliptical specimens were almost equivalent to those of the circular sections.All of the specimens were 608 mm in height.The unconfined concrete strength was obtained from compressive tests on three cubes of 150 mm.Only the third and fourth groups as control specimens were considered without FRP confinement, while the other groups were confined with different numbers of layers of CFRP wraps.The variables considered in the tests were: (1) the sectional aspect ratio, (2) the batch of concrete, and (3) and the number of CFRP layers.As for the specimens' designation, S5/4L2, as an example, had a cross-sectional aspect ratio of 1.28 and was confined with two CFRP layers.The mechanical properties of the CFRP wraps are also provided in Table 1.Complete details of the tests are not provided in this paper; however, the researchers are directed to their reference.

Section details Material Properties
No. Specimen Note: a = width of a cross-section; b = depth of a cross-section; a/b = aspect ratio of a cross section; twrap = thickness of FRP composite layers; ff = maximum tensile strength of FRP composite; Ef = tensile elastic modulus of FRP composite; εfu = tensile strain of FRP at ultimate condition; fc ' = unconfined concrete strength.

Overview of experimental test results
All the confined specimens failed by the rupture of the FRP (Teng and Lam, 2002).In most cases, the rupture happened at the upper or lower quarters of the specimens.The degree of damage for specimens with smaller aspect ratios was higher than that for specimens with larger aspect ratios.In addition, results have revealed that the axial compressive strength is dependent on the amount of FRP confinement (Figure 1), which is resulted by the dependence of the FRP rupture strain on the confinement stiffness ratio (Rochette and Labossiére, 2000; Wang et al., 2012).To take this parameter's effect into account, Equation 1 was first suggested by Teng et al. (2009) for FRP-confined circular concrete columns and later modified by Pham and Hadi (2014) for rectangular columns.In this paper, the following procedure to calculate the strength confinement ratio for the test specimens listed in Table 1 was used.
The FRP rupture strain that occurred at the major axes were smaller than the strains measured at the minor axes.In this paper, the effective FRP strain (Equation 4) is related to the strain at the ultimate condition.Besides, it was reported that specimens with higher aspect ratios exhibited smaller FRP strains at peak load.Comparison of the test results of specimens with varying aspect ratios is shown in Figure 2, in which the FRP strain is found to decrease as the sectional aspect ratio increase, as it has been reported in several tests on confined concrete columns with rectangular sections (Anselm, 2005; Yan, 2005; Isleem et al., 2018a, c, d).Table 2 provides a summary of the existing models that predict the strain efficiency factor and Figure 3 provides a comparison of these models against the results.Based on these comparisons and regression of the results of the same specimens, Equation 5 is introduced.
where kε is a factor that considers the reduction in measured FRP hoop strain at rupture In which the terms a and b are the width and depth of an elliptical cross-section, respectively.Where S = suitable; U = unsatisfied.

Effective confinement pressure ratio
As shown in Table 3, three analytical models for quantifying the effectiveness of FRP confinement for specimens with non-circular cross-sections were proposed in previous studies (Yan, 2005;Shao et al., 2006;Pham and Hadi, 2014).Based on their results, it was revealed that the strength increases by the increase of the ratio of the estimated confining pressure to the unconfined concrete strength (denoted as CR Table 3) being greater than a recommended value (For example 0.3 as reported by Shao et al., 2006).Details of the models can be found in their original papers.
Table 4 lists the effective confinement ratios from the proposed MCR and existing models defined in this paper as CR (CR1,2,3 = model 1, 2, 3) for a total of 12 FRP-confined specimens selected from the paper of Teng and Lam (2002).As shown the models are not able to represent the actual results of their peak strengths.Therefore, based on the analysis of the database, the following expressions for estimating the FRP-confined peak strength and strain for elliptical columns are proposed in which the coefficient R 2 value is approximately 87 and 89 % (Figure 4).
where MCR is a non-dimensionless coefficient used to account for the contributions of FRP confinement on the enhancement in ultimate strength experienced by the specimens; ke is the FRP efficiency coefficient for elliptical sections (Figure 5).This was calculated using Equation 8 where the ' a and ' b are respectively the depth and width of a rectangular concrete block (Figure 5), referring to Campione and Fossetti (2006). -

Amount of FRP for sufficiently confined concrete
As stated in a recent by Isleem et al. (2018d), in earthquake-prone regions, a large number of existing RC columns, particularly of rectangular cross-sections, that were built based on the out-of-date codes may not have adequate lateral reinforcement to resist high seismic load levels.As a result, they are subjected to major damages causing a total collapse of the building (Ilki et al., 2008).Therefore, this discussion focuses on determining the following indicator that can ensure a sufficient confinement for the existing concrete columns.The relationship between MCR and the ratio of the test peak strength to the strength of unconfined concrete was presented in Figure 4, in which the regressed line was only based on the results provided in Table 1 due to the unavailability of relative tests in the technical literature.Based on the regressed line, when fcc ' / fc ' = 1, then the value of MCR is equal to 0.02.When the MCR is greater than the 0•02 value, then fcc ' / fc ' > 1, which means the fcc is greater than the fc ' , and as a result, the confined specimen experienced enhancement in their strength and finally an ascending stress-strain response.Conversely, when the value of MCR is less than 0•02, a second post-peak softening component occurs in the stress-strain response as reported in several tests conducted on columns in the literature (Isleem et al., 2018a).
In the next section, the overall performance of the proposed and existing models of peak strength and strain of FRP-confined rectangular columns were evaluated against the results of columns with elliptical sections (Table 1).In the current discussions, the newly proposed confining pressure ratio is further checked through comparisons between the stress-strain test results of specimens in Table 1 and analytical values obtained from the model.For more details of the selected tests, the readers are motivated to their original sources.These results were selected for the following reasons: (1) they had varying aspect ratios of cross-sections; (2) they had different strength of unconfined concrete; and (3) they had different confinement levels of FRP.In general, comparison of the test results with the predicted MCR ratios provided in Figure 6 indicates that the confinement pressure model can distinguish between the different ascending and descending responses.

Accuracy of the proposed and existing strength and strain models
The performance of existing models that are capable of predicting the confined strength and strain enhancements achieved for rectangular columns due to the FRP confinement are assessed against the results summarized in Table 1.The average absolute error (AAE) described by Equation 10is considered to establish the model accuracy.

ana exp exp
It can be generally observed from the comparisons in Figures 7 and 8 that the analytical values obtained from the proposed Equations 6 and 7 agree well with the test results compared with the models of other researchers.

Conclusions
The existing tests and analytical models conducted on the axial compressive behavior of FRPconfined concrete have been largely concerned with columns of circular sections, where the concrete exhibited higher confinement due to uniform distribution of lateral stresses.On the contrary, the stress distribution in the case of a rectangular column varies over its cross section.Generally, the efficiency of FRP wraps decreases as the cross-sectional aspect ratio increases.Significant enhancements in ultimate strengths over the strength of unconfined concrete were achieved for columns with an aspect ratio of of less than 2.0.In particular, columns with the aspect ratio equals to 2.0 were to experience a reduction in their ultimate strengths.The effectiveness of FRP confinement in terms of enhancement in ultimate strength can be significantly improved by the shape modification method.On the basis of the evaluation of existing models, it was revealed that the models available for rectangular FRP-confined specimens do not predict well the results of FRP-confined concrete in elliptical sections.Based on a regression analysis of the existing test results, a new confinement model was presented to estimate the confined strength and strain of FRP-confined concrete columns having elliptical cross-sections.Based on the proposed model and the assembled test data, the threshold value of 0.02 for sufficiently confined elliptical concrete columns was proposed.Exceeding this value dictates the post-peak curve of the axial stress-strain response exhibits hardening behavior.The proposed model better predicted the test results compared with the predictions from the existing models provided by other researchers for FRP-confined rectangular columns.

Figure 4 .Figure 5 .
Figure 4. Relationship between effective confinement pressure ratio and ratio of the confined peak strength to the unconfined concrete strength.

Figure 6 .
Figure 6.Accuracy of the proposed confinement pressure model MCR against stress-strain test responses of specimens reported in Table1.

Figure 8 .
Figure 8. Evaluation of proposed and existing models against the test results of confined axial strain, εcc.

Table 1 .
Summary of FRP reinforcement, material and mechanical properties of test specimens.

Table 2 .
Summary of published models for the FRP strain efficiency factor, kε.

Table 3 .
Published models of boundary value for sufficiently confined concrete.

Table 4 .
Comparison of published (Table3) and proposed models for specimens in Table1.
Comparison of existing models of kε with the results of tested specimens reported in Table1.