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Axial stress–strain model developed for rectangular RC columns confined with FRP wraps and anchors
Structures 23 (2020) 779–788
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Axial stress–strain model developed for rectangular RC columns confined with FRP wraps and anchors Haytham F. Isleema,b,c,d, Muhammad Tahirb,c,d, Zhenyu Wangb,c,d,
T
⁎
a
Department of Civil Engineering, Tsinghua University, Beijing, 180004, China Key Lab of Structures Dynamic Behavior and Control of the Ministry of Education, Harbin Institute of Technology, Harbin, 150090, China c Key Lab of Smart Prevention and Mitigation of Civil Engineering Disasters of the Ministry of Industry and Information Technology, Harbin Institute of Technology, Harbin, 150090, China d School of Civil Engineering, Harbin Institute of Technology, Harbin, 150090, China b
ARTICLE INFO
ABSTRACT
Keywords: Stress–strain model Rectangular RC columns Reinforced concrete Confined FRP anchors
It is widely reported that using fiber-reinforced polymer (FRP) as confinement jackets for concrete can significantly enhance its strength and ductility. The confinement effectiveness for rectangular columns has revealed significant enhancements in their strengths, in particular for well-confined columns with a cross-sectional depth of < 300 mm. However, for larger-sized columns with a section depth of larger than 300 mm and an aspect ratio of larger than 2.0, the confinement effectiveness is most limited. One of the key solutions to significantly increase the confinement effectiveness is by using a combination of FRP anchors and wraps. For good analysis and design of FRP confinement for rectangular RC columns, a model for the stress–strain behavior of rectangular reinforced concrete (RC) columns confined with FRP wraps and FRP anchors is provided in this paper. The model is mainly based on results from tests reported in recent studies, in which a series of axial compressive tests on a total of 80 FRP-confined rectangular RC columns of large size were conducted. The database included columns with aspect ratios ranging from 1.5 to 4.0. Furthermore, the depth of the section is up to 600 mm. key parameters such as the aspect ratio and size of cross section, number of layers of FRP wraps, volumetric ratio of internal hoop steel reinforcement, and cross-sectional area and configuration of anchors were all considered. All predicting expressions were validated against the test database used in the model calibration. In terms of average absolute error and mean square error, comparisons with other existing expressions were conducted, showing that the expressions of the present model reveal more accuracy. Finally, comparisons revealed the model can estimate the confinement effectiveness and predict the stress–strain relationships of FRP-confined rectangular RC columns with additional FRP anchorage.
1. Introduction It is known that many reinforced concrete columns including particularly those RC columns constructed prior to the 1970s have been reinforced with an inadequate amount of transverse steel reinforcement which provides inefficient confinement to the concrete core or lateral restraint to the longitudinal reinforcing bars. Since fiber-reinforced polymer (FRP) composites owe some of the extraordinary properties such as high strength-to-weight ratio and excellent corrosion resistance, the use of externally bonded FRP composites has significantly increased in the construction industry. Nowadays, FRP wraps as an innovative confining technique are widely used for retrofitting existing RC structures. The significance of this subject is confirmed by numerous studies provided models for the axial stress–strain behavior of FRP-confined
⁎
concrete specimens. Over the past 25 years, a large number of experimental and analytical investigations have been concentrated on FRPconfined concrete in circular cross-sections, while only limited studies considered FRP-confined concrete in rectangular cross-sections. The existing studies reveal that the efficiency of FRP confinement is much better for circular concrete sections by significantly improving the axial strength compared with limited confinement provided to rectangular concrete sections. This is due to the fact that a circular section is uniformly confined by the FRP wraps while only a part of a rectangular section is effectively confined. To understand the confinement around the perimeter of concrete in rectangular cross-sections, the following three issues should be considered in a new stress–strain model. (1) Internal steel reinforcement: In general, a review of the existing researches on investigating and modeling the axial strength and
Corresponding author. E-mail address: [email protected] (Z. Wang).
https://doi.org/10.1016/j.istruc.2019.12.020 Received 28 April 2019; Received in revised form 24 August 2019; Accepted 18 December 2019 2352-0124/ © 2019 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
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H.F. Isleem, et al.
compressive behavior of FRP-confined concrete columns reports that the existing studies have not considered the influence of internal steel reinforcement on the strength and strain enhancements of concrete columns (i.e. Isleem et al. [1,2]). Only a limited number of studies have considered RC columns. Of these limited studies, 68 FRP-confined circular and non-circular reinforced concrete columns were confined with FRP wraps and tested by Ilki et al. [3] under axial compressive loads. Generally, the confining pressure provided to the columns was derived from the confinement provided by the FRP wraps and the internal hoop steel reinforcement. In recent years, Wang et al. [4,5] have conducted a research on square RC columns confined using FRP. The results of their tests showed that the hoop steel reinforcement significantly contribute to the enhancement of their axial strength and ductility. (2) Aspect ratio and size of rectangular cross-section: One of the key parameters reported in existing studies for modeling the axial stress–strain response is the aspect ratio (defined as the ratio of the longer side, h, to the shorter side, b). Recent studies have also confirmed that the axial compressive behavior of FRP-confined columns is significantly influenced by the increase of the size of a rectangular crosssection (Wang et al. [6]). Although there are many studies available in the literature for the axial stress–strain behavior of FRP-confined concrete columns, the range of considered aspect ratio is between 1 and 1.5. Of the earlier studies on rectangular concrete sections, Abbasnia et al. [7] investigated the axial stress–strain behavior of FRP-confined unreinforced concrete prisms of sizes 120 mm × 180 mm (h/b = 1.50), and 90 mm × 180 mm (h/b = 2.0). The earlier experimental and analytical approaches of Hany et al. (2015) included concrete specimens of size 140 mm × 180 mm (h/b = 1.28), and 130 mm × 200 mm (h/b = 1.54) to investigate and propose a model for the axial stress–strain behavior. According to the recent studies conducted by Wang et al. [4,5]; the test results indicated that the axial stress–strain response of FRP-confined square columns of relatively larger size (305 mm × 305 mm) was different than that obtained from the tests of Hany et al. [8]. Furthermore, 28 large-scale rectangular RC columns of sizes 200 mm × 300 mm (h/b = 1.5) and 200 mm × 400 mm (h/ b = 2.0) confined with FRP were tested [1,2]. This study aimed to investigate the axial stress–strain behavior of rectangular RC columns of large sections confined with FRP. The results showed that the confined specimens exhibited a post-peak softening behavior with little enhancement in axial stress. The lightly-confined specimens experienced reductions in ultimate strength. Due to the effects of aspect ratio and dimensional geometry of a rectangular section, the ultimate strength enhancement was much limited for the 200 mm × 400 mm section compared with the 200 mm × 300 mm section. (3) FRP anchors: In the more recent years, a few techniques have been reported on the seismic strengthening of existing rectangular columns by anchorage of the FRP wraps along the depth of the crosssection using means of FRP anchors (i.e. Triantafillou et al. [9]; Li et al. [10]; Hany et al. [11]). For FRP-confined concrete columns of rectangular cross-sections, similar to the effect of internal confinement by transverse steel ties, anchoring the long side of the FRP wrap using FRP anchors distributed along the height and depth of specimen increases the effectively confined area by restraining the FRP wrap from bulging out at the flat sides of the rectangular section. Another advantage of using FRP anchors is increasing the total cross-sectional area of FRP, which generates additional confining pressure to the cross-section leading to a significant strength enhancement. As a result, the contribution of FRP anchors to the strength enhancement of FRP-confined rectangular RC columns should be considered in the analytical model. As a result, in the recent studies by Isleem et al. [1,2,14] it has been confirmed that with respect the unconfined concrete strength, rectangular columns of small size (i.e., the depth of a cross-section < 300 mm and the aspect ratio < 2.0) achieved a significant enhancement in confined strength, while there is no any strength enhancement for those of larger-sized cross sections [12]. Therefore, the effectiveness of FRP was proposed to be functional of the section’s depth and with an
increase of the section’s depth, the effectiveness of confinement decreases. This expression was incorporated into a new confinement pressure model used to propose expressions for predicting the stress–strain responses with bilinear and post-peak softening behaviors. The model expressions also take into account the effects of section size, aspect ratio of cross sections, and internal hoop steel reinforcement. The depth of cross-sections ranged between 100 and 500 mm and the cross-sectional aspect ratio between 1.0 and 2.0. The model in these studies was described by three curves. The first region is parabolic ascending and is dependent on the properties of concrete and internal steel reinforcement rather than the FRP. The second region is linear descending and dependent on the number of FRP layers, hoop reinforcement ratio, and sectional aspect ratio. The third region is linear and also dependent on the same factors that influence the second branch. To significantly increase the confinement effectiveness of columns with a depth of greater than 300 mm and sectional aspect ratio of greater than 2.0, it was proposed using a combination of FRP anchors and FRP wraps. First a new confinement strength model was provided using existing 44 test specimens. Based on the proposed model, the threshold values of the lightly and heavily FRP-confined rectangular columns with FRP anchorage were 0.125 and 0.25, respectively [12]. The heavily-confined concrete threshold dictates whether the post-peak curve of an axial stress–strain response of FRP-confined columns have strain softening or hardening. In this paper, based on an assessment of the existing stress–strain models against test results influenced by the presence of FRP anchors and FRP wraps, a complete stress–strain response model was proposed for FRP-confined rectangular RC columns with and without FRP anchor confinement. Experimental results from a series of tests performed on a total of 80 FRP-confined rectangular unreinforced and reinforced rectangular cross-sections with FRP wraps and anchors were used. The parameters that influence the confined strength and strain included the aspect ratio, number of FRP layers, hoop steel reinforcement, and crosssectional area and configuration of FRP anchors were all considered in their model. It was shown that the average absolute error (AAE) and the mean square error (MSE) for the proposed model are minimal compared with those of the existing expressions, indicating that the model predictions are more close to the experimental results. 2. Existing experimental tests 2.1. General To propose a new model and to validate its accuracy, results of tests provided in recent studies performed by Isleem et al. [1], Triantafillou et al. [9], and Hany et al. [11] were chosen to serve as a basis for all the model expressions in this paper. The number of specimens considered in the regression analysis is 80. To consider the influences of a wide range of column parameters on the strength and strain of confined specimens, the unconfined concrete compressive strength ranged from 18 MPa to 46.3 MPa and the cross-sectional areas ranged from 293 cm2 to 896 cm2. The side length of the rectangular section ranged between 100 mm and 600 mm and the aspect ratio between 1.5 and 4.0. The number of FRP anchors along the depth and height of specimens is up to 18, with spacing varying from 140 mm to 200 mm and width of from 66.5 mm to 960 mm. In addition, the tensile strength of FRP ranged between 1046 MPa and 4340 MPa, the modulus of elasticity ranged between 93.7 and 240 GPa, and the thickness of FRP varied from 0.13 mm to 1.0 mm per single layer. A summary of each reference is provided as follows: 2.2. Tests by Triantafillou et al. [9] In reference to the study of Triantafillou et al. [9], a total of 30 rectangular RC columns of 800 mm in height and 150 mm × 450 mm 780
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Table 1 Detail of tested FRP-confined unreinforced and RC specimens. Cross section geometry and reinforcement detail No. Specimen b h (mm) (mm) Triantafillou et al. [9] 1 II3 2 II3 3 II3 4 II4 5 II4 6 II4 7 1AlII3 8 1AlII3 9 1AlII3 10 2AlII3 11 2AlII3 12 2AlII3 13 1AhII3 14 1AhII3 15 1AhII3 16 1AhIII3 17 1AhIII3 18 1AhIII3 19 1AlII4 20 1AlII4 21 1AlII4 22 2AlII4 23 2AlII4 24 2AlII4 25 2AhII4 26 2AhII4 27 2AhII4 28 2AhIII4 29 2AhIII4 30 2AhIII4 Hany et al. [11] 31 S300 32 S313 33 S400 34 S413
Vertical reo.
Hoop reo.
Confinement detail wa.y ca.y (mm)
ra.y
la.y
nf
150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150
450 450 450 600 600 600 450 450 450 450 450 450 450 450 450 450 450 450 600 600 600 600 600 600 600 600 600 600 600 600
20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
6Φ12 6Φ12 6Φ12 8Φ12 8Φ12 8Φ12 6Φ12 6Φ12 6Φ12 6Φ12 6Φ12 6Φ12 6Φ12 6Φ12 6Φ12 6Φ12 6Φ12 6Φ12 8Φ12 8Φ12 8Φ12 8Φ12 8Φ12 8Φ12 8Φ12 8Φ12 8Φ12 8Φ12 8Φ12 8Φ12
Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150
– – – – – – 66.5 66.5 66.5 66.5 66.5 66.5 133 133 133 133 133 133 66.5 66.5 66.5 66.5 66.5 66.5 133 133 133 133 133 133
– – – – – – 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
– – – – – – 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
– – – – – – 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 2 2 2 2 2 2 2 2 2 3 3 3
140 140 140 140
210 210 210 210
10 10 10 10
– – – –
– – – –
– 560 – 840
– 1 – 1
– 3 – 3
– 260 – 260
3 3 4 4
rc (mm)
Vertical reo.
Hoop reo.
Confinement detail wa.y ca.y (mm)
ra.y
la.y
nf
Cross section geometry and reinforcement detail No. Specimen b h (mm) (mm) Hany et al. [11] 35 M200 36 M213 37 M223 38 M300 39 M313 40 M323 41 M400 42 M413 43 M423 44 L200 45 L224 46 L234 47 L300 48 L324 49 L334 50 L400 51 L424 52 L434 Isleem et al. [1] 53 R1.5H1L2 54 R1.5H2L2 55 R1.5H1L3 56 R1.5H2L3 57 R1.5H0L2 58 R1.5H0L3 59 R2.0H1L3 60 R2.0H1L4 61 R2.0H2L3 62 R2.0H2L4 63 R2.0H0L3
rc (mm)
125 125 125 125 125 125 125 125 125 100 100 100 100 100 100 100 100 100
240 240 240 240 240 240 240 240 240 300 300 300 300 300 300 300 300 300
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
– – – – – – – – – – – – – – – – – –
– – – – – – – – – – – – – – – – – –
– 320 160 – 640 320 – 960 480 – 150 100 – 300 200 – 600 400
– 1 2 – 1 2 – 1 2 – 2 3 – 2 3 – 2 3
– 3 3 – 3 3 – 3 3 – 4 4 – 4 4 – 4 4
– 245 245 – 245 245 – 245 245 – 220 220 – 220 220 – 220 220
2 2 2 3 3 3 4 4 4 2 2 2 3 3 3 4 4 4
200 200 200 200 200 200 200 200 200 200 200
300 300 300 300 300 300 400 400 400 400 400
30 30 30 30 30 30 40 40 40 40 40
6Φ16 6Φ16 6Φ16 6Φ16 – – 8Φ16 8Φ16 8Φ16 8Φ16 –
Φ8@180 Φ8@90 Φ8@180 Φ8@90 – – Φ8@200 Φ8@200 Φ8@100 Φ8@100 –
– – – – – – – – – – –
– – – – – – – – – – –
– – – – – – – – – – –
– – – – – – – – – – –
2 2 3 3 2 3 3 4 3 4 3
(continued on next page) 781
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Table 1 (continued) Cross section geometry and reinforcement detail No. Specimen b h (mm) (mm)
rc (mm)
Vertical reo.
Hoop reo.
Confinement detail wa.y ca.y (mm)
ra.y
la.y
nf
64 65 66 67 68 69 70 71 72
40 30 30 30 30 30 30 40 40
– 6Φ16 6Φ16 6Φ16 6Φ16 – – 8Φ16 8Φ16
– Φ8@180 Φ8@90 Φ8@180 Φ8@90 – – Φ8@200 Φ8@200
– – – – – – – – –
– – – – – – – – –
– – – – – – – – –
4 2 2 3 3 2 3 3 4
Cross section geometry and reinforcement detail No. Specimen b h (mm) (mm)
rc (mm)
Vertical reo.
Hoop reo.
Confinement detail wa.y ca.y (mm)
ra.y
la.y
nf
Isleem et al. [1] 73 R2.0H2L3 74 R2.0H2L4 75 R2.0H0L3 76 R2.0H0L4 77 R1.5H2L0 78 R1.5H1L0 79 R2.0H2L0 80 R2.0H1L0
40 40 40 40 30 30 40 40
8Φ16 8Φ16 – – 6Φ16 6Φ16 8Φ16 8Φ16
Φ8@100 Φ8@100 – – Φ8@90 Φ8@180 Φ8@100 Φ8@200
– – – – – – – –
– – – – – – – –
– – – – – – – –
3 4 3 4 0 0 0 0
R2.0H0L4 R1.5H1L2 R1.5H2L2 R1.5H1L3 R1.5H2L3 R1.5H0L2 R1.5H0L3 R2.0H1L3 R2.0H1L4
200 200 200 200 200 200 200 200 200
200 200 200 200 200 200 200 200
400 300 300 300 300 300 300 400 400
400 400 400 400 400 400 400 400
– – – – – – – – –
– – – – – – – –
Note: b and h = the shorter and the longer sides of a rectangular cross-section, respectively; rc = corner radius of the cross-section; reo. = reinforcement; nf = number of externally-bonded FRP composite layers; ca.y = number of FRP anchors distributed along the y- direction; wa.y = width of FRP strip used to form one anchor in the y- direction (shorter side); la.y = length of FRP anchor in the y- direction; ra.y = number of anchor rows along the y- direction.
Table 2 Yield strength of longitudinal and hoop reinforcement steel bars. Source
Triantafillou et al. [9] Hany et al. [11] Isleem et al. [1]
Longitudinal steel fyl (MPa)
Hoop steel fyh (MPa)
FRP material Ef (GPa)
ff (MPa)
tf (mm)
εfu (%)
f c' (MPa)
570 (Φ12) – 360 (Φ16)
570 (Φ8) – 373 (Φ8)
93.7 230 240
1046 3500 4340
1.000 0.130 0.167
1.12 1.50 1.81
18.0 35.0 46.3
Unconfined concrete
Note: tf = thickness of one layer of FRP wraps; ff = maximum tensile strength of FRP; Ef = tensile modulus of elasticity of FRP; εfu = ultimate tensile strain of FRP; fyl and fyh = yield strengths of longitudinal and hoop steel reinforcement, respectively; f c' = compressive strength of unconfined concrete cylinders.
and 150 mm × 600 mm in cross-sections were confined with FRP wraps and anchors and tested under axial compression. The experimental program included two series of specimens, according to their cross-sectional aspect ratios of 3.0 and 4.0, respectively. The test variables considered for the assembled 30 tests were: (1) cross-sectional aspect ratio, (2) anchors number and their weight, and (3) and the number of layers of FRP wrap. As an example for their specimens’ designation, 1AlII3 is a specimen from the first series. It had a crosssectional aspect ratio 3.0, was confined with light-weight FRP anchors (i.e. Al) distributed around the long side of the section, and confined with two layers of FRP wraps (i.e. II). Another specimen with a crosssectional aspect ratio equal to 4.0 selected from the second series is 2AhIII4 confined using heavy-weight FRP anchors (i.e. Ah) and three FRP layers (i.e. III). Table 1, of appendix A, summarized their crosssection geometries and confinement details (i.e. hoop reinforcement and anchors). The mechanical properties of the internal longitudinal and hoop steel reinforcement and the FRP used for the wraps and anchors are provided in Table 2, of appendix A, for all specimens utilized in this paper.
variables of tests were: (1) aspect ratio, (2) number of FRP layers, and (3) area and configuration of FRP anchors. These specimens are divided into three series labeled as S, M, and L based on their cross-sectional aspect ratios of 1.5, 1.92, and 3.0, respectively. The columns sizes in these three series are 140 mm × 210 mm × 420 mm, 125 mm × 240 mm × 480 mm, and 100 mm × 300 mm × 600 mm, respectively. According to their designations provided in Table 1, the following letter refers to the number of FRP layers (i.e. SL3 had three FRP layers). The last two numbers in the specimen designation correspond to the number of anchor columns and anchor rows, respectively. 2.4. Tests by Isleem et al. [1] In reference to the study of Isleem et al. [1], a total of 28 rectangular unreinforced and reinforced concrete columns having a height of 1000 mm and cross-sectional dimensions of 200 mm × 300 mm and 200 mm × 400 mm were fabricated and tested under concentric loads. Table 1 displays their identification designations, section dimensions, and confinement details. The test program included two series of specimens, according to their aspect ratios of 1.5 and 2.0, respectively. According to the designation used in Table 1 and all of the subsections of this paper, R represents the rectangular cross-section of specimens, while 1.5 and 2.0 correspond to side-aspect ratios of 1.5:1 and 2:1. The following letter H refers to the volumetric ratio of hoop reinforcement.
2.3. Tests by Hany et al. [11] The second group consisting of 22 test specimens in Table 1 was selected from the test program conducted by Hany et al. [11]. The 782
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The following number 0 denotes unreinforced concrete, whereas the numbers 1 and 2 denote hoop reinforcement of 0.33 and 0.67% volumetric ratio, respectively. Finally, the last symbol L and the number following it correspond to the number of layers of carbon fiber sheet used to form the FRP confinement. The center-to-center spacing of the steel hoops corresponding to 0.33 and 0.67% volumetric ratios were 180 and 90 mm for the R1.5 specimens and 200 and 100 mm for the R2.0 specimens, respectively. Several parameters were investigated such as (1) cross-sectional size and aspect ratio, (2) number of FRP composite layers, (3) volumetric ratio of hoop steel reinforcement.
confinement which is significantly influenced by the shape and the cross-sectional dimension. In this subsection the bilinear and post-peak softening stress–strain responses drawn in Fig. 1 are used as references for confined columns with section depth of smaller than 300 mm (i.e. Wang et al. [4,6]; Hany et al. [8]) and columns of large size (h > 300 ) (i.e. Isleem et al. [1,2,15]; Wang et al. [4]; Rocca [13]; Yan [16]). As shown, the post-peak softening model (For example, with light amount of FRP anchors) consists of three distinct curves, namely (1) A polynomial ascending curve (OA), which describes the first stage of the response, initiates from the beginning point at which ( c = 0, c = 0) and extends to the peak point when the confined concrete reaches their peak strength and strain, which are referred to as ( c = fcc , c = cc ), (2) The second descending portion of the response (AB) is a linear branch extending from the peak point to the position of a transition stress ( c = fdt , c = dt ), and (3) The final linear portion (BC) extends from the transition stress to the point which represents the ultimate condition of confined columns were the FRP wrap ruptures at an ultimate stress and corresponding strain, which are referred to as ( c = fcu , c = cu ). The three portions of the response are described as.
2.5. Overview of test results and column parameters effects Results obtained from tests on FRP-confined rectangular RC columns of larger section size have generally revealed that: (1) Confined columns of large size exhibited a significant enhancement in axial strain but only a slight increase in axial strength and finally resulted in a postpeak softening behavior followed by a small increase in the stress (i.e. Isleem et al. [1,2,14,15], Wang et al. [4,5]; Rocca [13]). The effectiveness of the confinement provided by the FRP reduces as the section’s size is increased and increases with additional layers of FRP wraps (Isleem et al. [1,2,14]). (2) The smaller-dimensioned specimens experienced larger hoop strains of FRP at failure than those recorded for the larger-dimensioned specimens (i.e. Wang et al. [4,5]; Yan [16]). (3) The increase of volumetric ratio of internal steel hoops, radius of edges of the rectangular cross-section and areas of FRP wraps and FRP anchors influenced the stress–strain envelope curve by increasing the axial strength and strain values. (4) The gain in ultimate strength of the confined columns decreases with an increase in the cross-sectional aspect ratio (i.e. Ilki et al. [3]; Abbasnia et al. [7]; Hany et al. [8], Triantafillou et al. [9]; Li et al. [10]; Isleem et al. [1,2,14,15]; Yan [16]; Wu et al. [17]). For a case in which not all the parameter effects are considered in the existing models, a more generalized stress–strain model is introduced in the following discussions.
fc = fcc A
( ) + (3 c
cc
fc = fcc + E1 ( fc = fdt + E2 (
2A)
( ) c
cc
2
+ (A
2)
( ) c
3 c
cc
cc
c
cc )
cc
c
dt
c
dt )
dt
c
cu
(1)
fcc ) ( dt where E1 = (fdt cc ) (MPa) is the slope of the linear second stress–strain portion; E2 = (fcu fdt ) ( cu dt ) (MPa) is the slope of the third portion of stress–strain response; the shape parameter A, which controls the polynomial first portion, is derived from the boundary condition of d c d c = Ec at c = 0 . On substituting the boundary condition in the first portion of Eq. (1), the A is obtained as A = Ec Ep , in which Ec = 4733 f c' (MPa) = elastic modulus of unconfined concrete (ACI 318–2014 [18]); (MPa) is the second modulus at the peak point (A) in Fig. 1. 3.2. Lateral confining stresses, flf , fla and fls
3. Axial stress–strain model
The confinement stresses provided by FRP wraps (i.e. flf ), FRP anchors (i.e. fla ) and steel hoops (i.e. fls ) are three key parameters for modeling the stress–strain response of RC columns confined using a combination of FRP wraps and anchors.
3.1. General Results of tests reveal that the shape of stress–strain response was bilinear or post-peak softening response depending on the level of FRP
3.2.1. Confinement pressure by external FRP wraps, flf The effective confinement of the FRP is first determined by the following expression used in several studies [19].
flf = kef
(2)
fw ff
where ff (MPa) = maximum tensile strength of FRP wraps; the volumetric ratio of FRP wraps is calculated by fw = 4n f t f D , in which D = 2bh b + h (mm) is the equivalent diameter for a rectangular section [20]; n f = number of FRP layers; t f = thickness of one FRP layer (mm); the shape factor, k ef , for a rectangular cross-section with rounded corners is determined by
k ef =
Acf Ag
(3)
Acf = Ag A cf = Ag b ; 2
ho = 2
Ag = bh Fig. 1. Idealization of the stress–strain response for FRP-confined rectangular columns with larger aspect ratios (i.e. h/b ranges between 2.0 and 4.0).
(b
(4
(
(b
2rc )2 + (h 3 (h
2rc ) 4
2rc )2 + (h 3 2rc )2 b 2
2rc )2 4
+ 3 ho
;
(h
2rc ) 4
(h
2rc ) h o 2
)
b 2
;
(h
2rc ) 4
(4)
) rc2
(5) 2
where the terms Acf and Ag (mm ) are respectively the effectively confined concrete area and the gross cross-sectional area of the 783
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H.F. Isleem, et al.
rectangular-sectioned column with rounded corners; ho is the depth of the overlapping zone that occurs when the cross-sectional aspect ratio is more than 3.0 [21].
k se = k sh k sv =
fla = k ea
fa
=
Aa.y
Aa.x D × bH D
+
(7)
where fa is the equivalent ratio of the FRP anchors, which can take into account the effects of the confinement in the longer and shorter sides of the cross-section; the term D is considered as D = b2 + h2 (mm); the effective confinement of FRP anchors along the column height is considered by
H = (ra
(8)
1) × sa
where c x and c y are the numbers of FRP anchors distributed along the width and depth of the cross-section [12]; the terms a x or a y is the total section area of FRP strip sheets folded to form an individual anchor (Table 1), calculated by Eq. (10) in which wa.y and ta.y (mm) respectively denote the width and thickness of FRP strip sheet.
fcc f c'
Aca A A cf = c.f.a Ag Ag
Ac.f.a = Ag
(h
2rc )(h
3(ca.y + 1)
2rc )2 (12)
cc co
cc
= 1 + 0.07
h b
2.86
flfe
2.64
+ 0.5
f c'
fla
1.28
f c'
+ 0.48
b h
1.37
fls
0.12
fc'
= 1 + 0.46
h b
2.06
flfe f c'
2.11
+ 2.40
fla fc'
0.57
+ 7.94
b h
1.5
fls
0.67
f c' (17)
where co = axial strain that corresponds to the peak stress of unconfined concrete, which is calculated by the following expression [22]. co
= 0.000937(fc' )1
4
3.4. Ultimate axial stress fcu and corresponding strain
3.2.3. Confinement pressure by steel hoops, fls Third, the effective confinement pressure provided by the steel hoops is calculated using the following expression used before in several studies [1,2,14]. se fyh
(15)
cx + cy
where (MPa) = compressive strength of unconfined concrete cylinder. The peak strain is also found to depend primarily on the confinement provided by the FRP and the internal hoop reinforcement [1,2,4,5]. Based on multi-parameter regression analysis of the database, the peak strain is given as follows, in which the coefficient (R2) value is 84%.
where Acf and Ag were defined in Eqs. (4) and (5), respectively; herein the term Ac.f.a (mm2) is the total area of concrete core effectively confined by both the FRP wraps and FRP anchors and can be calculated by using the approach introduced by Triantafillou et al. [9]. In their model (Eq. (12)), sa is taken as 0 to neglect the confinement at mid-height between anchor locations for several reasons, such as the fanned part of the anchors extends beyond the anchor locations and the anchors may be covered with a vertical strip which increases the stiffness of the FRP jacket.
fls = k se
y
fc'
(11)
2rc + 1.5ca.y sa ) + (ca.y + 1)(b
(14)
(16)
(10)
Finally, the confinement efficiency k ea of the anchors can be found by using
k ea =
scx
s 2c y
1
An important requirement for defining a complete axial stress–strain response of FRP-confined concrete is the peak point ( fcc , cc ). The experimental results obtained for FRP-wrapped columns have revealed that the RC columns experienced an enhancement in their peak strengths [1,2,4,5]. This enhancement was due to the effects of the FRP wrap as well as internal hoop steel reinforcement. Based on a regression analysis of the database in Table 1, the following expression for predicting the fcc , which takes into account the contribution of these three parameters and has an (R2) value of about 84%, is proposed.
(9)
a x = wa.x × ta.x a y = wa.y × ta.y
Ashy
x
s 2c x
1
cc
3.3. Peak axial stress fcc and corresponding strain
where sa = vertical spacing between two FRP anchors (mm); ra is the number of anchor rows along the height of test specimen (see Table 1). Referring to Eq. (7), the Aa.x and Aa.y (mm2) (Eq. (9)) are the crosssectional areas of FRP anchors perpendicular on the longer and shorter sides of the section, respectively defined as b and h.
Aa.x = (c × r × a) x Aa.y = (c × r × a) y
1
where wxi and wyi (mm) = clear distances between two longitudinal bars along the two longer and shorter directions of the rectangular sectional plane; the terms cx and cy (mm) = the concrete core dimensions to the centerline of the peripheral hoop bar; s (mm) = vertical net spacing between two hoop bars; cc = the ratio of the cross-sectional area of longitudinal steel reinforcement to the sectional area of the inner concrete core enclosed by the centerlines of steel hoop’s parameter; se is the equivalent volumetric ratio of hoop reinforcement; fyh (MPa) is the yield strength of hoops; Ash.x and Ash.y (mm2) are respectively the cross-sectional areas of hoops along longer and shorter sides of the section.
(6)
fa ff
hH
se
wxi2 + wyi2 ) 6c x c y
( )c + ( )c = Ashx scy
3.2.2. Confinement pressure by FRP anchors, fla Second, the effective confinement pressure by the FRP anchors is calculated using the model recently proposed by Isleem et al. [12].
(
1
(18)
cu
The ultimate axial stress, fcu , and corresponding strain, cu , are the most important parameters for a complete stress–strain response of FRP-confined concrete. Based on multi-parameter regression analysis of the experimental data of specimens in Table 1, Eq. (19) for predicting the fcu and Eq. (20) for cu , which are dependent on the three parameters defined in the introduction, are proposed.
(13)
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H.F. Isleem, et al.
6 5
f ccmod f ccexp
AAE = 0.95; MSE = 0.91 AAE = 1.25; MSE = 1.65 AAE = 0.56; MSE = 0.36 AAE = 0.17; MSE = 0.03 AAE = 0.13; MSE = 0.02 AAE = 0.09; MSE = 0.02 AAE = 0.10; MSE = 0.01 AAE = 0.05; MSE = 0.00
Kumutha et al. [23] ACI Committee [20] Toutanji et al. [24] Ozbakkaloglu [25] Triantafillou et al. [9] Hany et al. [8] Li et al. [10] Proposed model
4 3 2 1
Fig. 4. Performance of proposed and existing models for fcu against results of specimens in Table 1 and existing studies.
Fig. 2. Comparison of selected models for fcc with test database for the specimens in Table 1 and existing studies. (See above-mentioned references for further information.)
the current paper, Eq. (23) of k has been proposed by Wang et al. [6] based on their own observations and results from tests on FRP-confined square specimens. This expression is also used for the rectangular specimens used in Table 1, noting that b (the width of the square section) is replaced with h (the depth of the rectangular section). fe
=
=1
f c' cu co
= 0.59 + 2.0
h = 1 + 2.98 b
flfe = k ef
fw Ef
fe
2rc D
0.71
0.40
h b
0.35
3flfe + fla
2.22
f c'
8flfe + fla f c'
0.86
+ 5.45
+ 0.87
fls f c'
fls fc
0.38
b 100
0.41
(23)
The performance of proposed expressions and those reported by researchers [8–10,17,20,23–31] for predicting the peak and ultimate strength and strain enhancements due to the FRP confinement were assessed against the same test database in Table 1. The two indicators described by Eq. (24) namely (1) the average absolute error (AAE) and (2) the mean square error (MSE) are considered to establish the overall model accuracy. Clear comparisons of the analytical and experimental test results are provided in Figs. 2.,Fig. 3.,Fig. 4.,Fig. 5.. Over the wide range of parameters discussed in this study, the comparison shows that the proposed model is able to predict with good accuracy (respectively, AAE = 5% and 10%) the fcc and fcu , but shows less accuracy (AAE = 12% and 16%) in predicting the corresponding cc and cu . A large variability in the cu values was also reported by numerous researchers, such as Hany et al. [8] in which the AAE value, by comparison, was 28.6%. Generally the model revealed a better correlation compared to that of other models.
Fig. 3. Comparison of selected models for cc with test database for the specimens in Table 1 and existing studies. See above-mentioned references for further information
fcu
(22)
fu
0.24
(19)
0.42
AAE = (20)
MSE =
N i = 1 | (anai
expi) expi |
N N i = 1 ((anai
expi) expi )2
(24)
N
(21)
Recently, it has been observed the rupture strain of the FRP wraps to be governed by the strains recorded at the section corners [1,2,4,5]. The medium-sized columns experienced larger strains at failure than those recorded for the larger-sized columns. To take these effects into account, the well-known expression for estimating the effective rupture strain at is given by Eq. (22). For the analytical modeling performed in
3.5. Transition axial stress fdt and corresponding strain branch
dt
for the second
It has been reported that the extent of the descending branch of the softening stress–strain response is influenced by the amount of FRP confinement, the hoop steel reinforcement volumetric ratio, and the
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H.F. Isleem, et al.
fdt f c'
dt
= 0.83 + 3.10
=
1 + 5.1
co
flf
3.93
f c'
flf + fla f c'
+ 0.64
fls
0.15
f c'
2.48
+ 0.79
fls f c'
0.05
b h
0.54
h b
+ 0.8
fla fc'
0.24
(25)
0.19
(26)
4. Verification of proposed stress–strain model The accuracy of the compression stress–strain model proposed in this paper is verified by comparing the theoretical predictions with experimental results of specimens reported in Table 1. Typical comparisons are provided in Fig. 7. These specimens were selected due to that they have varying cross-sectional sizes, aspect ratios, numbers of layers of FRP wraps, and most importantly configurations of FRP anchors, which is a major parameter of this study. Therefore, comparisons in Fig. 7 with other test stress–strain responses are not presented due to very limited experimental tests available for FRP-confined large-sized rectangular RC columns under axial compression. Generally, inspection of the comparisons of the responses of the model with the experimental results of large-sized columns demonstrates that the model can trace the key features of stress–strain response (i.e. fcc , cc , fdt, εdt, fcu , and cu ). Also, the shapes of the curves are described well.
Fig. 5. Performance of proposed and existing models for cu against test results of RC columns. See above-mentioned references for further information
5. Conclusion Limited models consider the effects of the cross-sectional size on the axial stress–strain behavior of FRP-confined rectangular RC columns. No model that considers both bilinear and post-peak softening responses was also reported earlier for rectangular RC columns confined by a combination of FRP wraps and FRP anchors, in which the effectiveness of FRP confinement is significantly influenced by the weight and configuration of FRP anchors. Based on an analytical evaluation into the axial compressive strength and strain of FRP-confined unreinforced and RC rectangular concrete columns with large cross-sections strengthened with an additional FRP anchorage, the major issues that are in need to be fully considered are first presented in their paper. As concluded in the recent studies, the existing models are applicable to FRP-confined circular unreinforced specimens and rectangular reinforced sections with a limited variation in aspect ratio. Therefore, this paper provides a stress–strain model with a more generalized scope. The model consists of mathematical expressions for describing the different features of a complete stress–strain response of rectangular unreinforced and RC columns confined with FRP wraps and additional FRP anchors. The main test parameters of the proposed model included the size and aspect ratio of cross-section, volumetric ratio of internal reinforcement hoops, number of FRP layers, and the amount of confinement by FRP anchors. Finally, good agreement was shown between the predictions of the proposed model and the test results, confirming that the model can predict the stress–strain response of large rectangular sectioned RC columns confined with FRP wraps and different configurations and characteristics of FRP anchors. The model expressions are calibrated using limited tests on columns with FRP anchors. Since the stress–strain behavior of rectangular RC columns with FRP depends largely on the variables of this study, the model may not be applicable to different ranges of test parameters. More experimental and analytical investigations for expanding the model application may consider the influences of new test parameters (i.e. ultra-high strength concrete, configuration of hoop steel reinforcement, larger section size, configuration of FRP anchors, and compression loading pattern).
Fig. 6. Comparison of proposed expressions with experimental results: (a) for transition stress, fdt ; (b) for transition strain, dt
cross-sectional aspect ratio [1,2]. As a result, a regression analysis is performed on the test database in Table 1 to propose the following two expressions of fdt and dt with R2 of about 87% and 86%, respectively. It should be noted that these two expressions are based on limited test data because no other experimental results for rectangular RC columns of large-size are found in the literature. Fig. 6 (a) and (b), in general, show that the predictions given by the proposed expressions are in close agreement with the experimental results. 786
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Fig. 7. Proposed axial stress–strain model versus experimental results
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Declaration of Competing Interest
2017;21(1):DOI: 10.1061/(ASCE)CC.1943-5614.0000724. [12] Isleem HF, Wang ZY, Wang DY. A new model for reinforced concrete columns strengthened with fibre-reinforced polymer. Proceedings of the Institution of Civil Engineers: Structures and Building 2019; https://doi.org/10.1680/jstbu.18.00159 (Ahead of Print). [13] Rocca S. Experimental and analytical evaluation of FRP-confined large size reinforced concrete columns. Ph.D. thesis, University of Missouri-Rolla, Rolla, Mo, USA; 2007. [14] Isleem HF, Wang DY, Wang ZY. A new numerical model for polymer-confined rectangular concrete columns. Proceedings of the Institution of Civil Engineers (ICE): Structures and Buildings 2018c;https://doi.org/10.1680/jstbu.17.00103 (Ahead of Print). [15] Wang ZY, Isleem HF. Axial compressive stress-strain model developed for FRPconfined concrete with elliptical cross sections. J Compos Sci 2018;2(4):67. [16] Yan ZH. Shape modification of rectangular columns confined with FRP composites. Ph.D. thesis, University of Utah, Salt Lake County, UT; 2005. [17] Wu G, Wu ZS, Lu ZT. Design-oriented stress-strain model for concrete prisms confined with FRP composites. Constr Build Mater 2007;21:1107–21. [18] American Concrete Institute (ACI). Building code requirements for structural concrete and commentary. ACI 318-2014, Farmington Hills, MI, USA; 2014. [19] Isleem HF, Wang DY, Wang ZY. Axial stress–strain model for square concrete columns internally confined with GFRP hoops. Mag Concr Res 2018;70(20):1064–79. [20] American Concrete Institute (ACI). Guide for the design and construction of externally bonded FRP systems for strengthening concrete structure. ACI 440.2R-08, Farmington Hills, MI, USA; 2008. [21] Tan KH, Bhowmik T, Balendra T. Confinement model for FRP-bonded capsuleshaped concrete columns. Eng Struct 2013;51:51–9. [22] Tasdemir MA, Tasdemir C, Akyüz S, Jefferson AD, Lydon FD, Barr BIG. Evaluation of strains at peak stresses in concrete: A three-phase composite model approach. Cem Concr Compos 1998;20(4):301–18. [23] Kumutha R, Vaidyanatha R, Palanichamy MS. Behavior of reinforced concrete rectangular columns strengthened using GFRP. Cem Concr Compos 2007;29:609–15. [24] Toutanji H, Han M, Gilbert J, Matthys S. Behavior of large-scale rectangular columns confined with FRP composites. J Compos Constr 2010;14(1):62–71. [25] Ozbakkaloglu T. Axial compressive behavior of square and rectangular highstrength concrete-filled FRP tubes. J Compos Constr 2013;17(1):151–61. [26] Harries KA, Kharel G. Behavior and modeling of concrete subject to variable confining pressure. ACI Mater J 2002;99(2):180–9. [27] Binici B. An analytical model for stress-strain behavior of confined concrete. Eng Struct 2005;27(7):1040–51. [28] Harajli MH. Axial stress-strain relationship for FRP confined circular and rectangular concrete columns. Cem Concr Compos 2006;28:938–48. [29] Teng JG, Yu T, Wong YL, Dong SL. Hybrid FRP-concrete-steel tubular columns: Concept and behavior. Constr Build Mater 2007;21(4):846–54. [30] Konstantinidis DK, Kappos A, Izzuddin BA. Analytical stress-strain model for highstrength concrete members under cyclic loading. J Struct Eng 2007;133(4):484–94. [31] Lim JC, Ozbakkaloglu T. Design model for FRP-confined normal- and high-strength concrete square and rectangular columns. Mag Concr Res 2014;66(20):1020–35. https://doi.org/10.1680/macr.14.00059.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This study is financially supported by the National Natural Science Foundation of China (Grant Nos. 51408153, 51478143, and 51278150); the China Postdoctoral Science Foundation (Grant Nos. 2014M551252 and 2015T80354); and the Guangdong Provincial Key Laboratory of Durability for Marine Civil Engineering, Shenzhen University (GDDCE16-09). References [1] Isleem HF, Wang ZY, Wang DY, Smith ST. Monotonic and cyclic axial compressive behavior of CFRP-confined rectangular RC columns. J Compos Constr 2018a;22(4):DOI: 10.1061/(ASCE)CC.1943-5614.0000860. [2] Isleem HF, Wang DY, Wang ZY. Modeling the axial compressive stress-strain behavior of CFRP-confined rectangular RC columns under monotonic and cyclic loading. Compos Struct 2018;185(1):229–40. https://doi.org/10.1016/j. compstruct.2017.11.023. [3] Ilki A, Peker O, Karamuk E, Demir C, Kumbasar N. FRP retrofit of low and medium strength circular and rectangular reinforced concrete columns. J Mater Civil Eng 2008;20(2):169–88; https://doi.org/0.1061/(ASCE)0899-1561(2008)20:2(169). [4] Wang ZY, Wang DY, Smith ST, Lu DG. CFRP-confined square RC columns. I: experimental investigation. J Compos Constr 2012;16(2):150–60. [5] Wang ZY, Wang DY, Smith ST, Lu DG. CFRP-confined square RC columns. II: cyclic axial compression stress-strain model. J Compos Constr 2012;16(2):161–70. [6] Wang DY, Wang ZY, Smith ST. Yu T Size effect on axial stress-strain behavior of CFRP-confined square concrete columns. Constr Build Mater 2016;18:116–26. [7] Abbasnia R, Hosseinpour F, Rostamian M, Ziaadiny H. Cyclic and monotonic of FRP confined concrete rectangular prisms with different aspect ratios. Constr Build Mater 2013;40:118–25. [8] Hany NF, Hantouche EG, Harajli MH. Axial stress-strain model of CFRP-confined concrete under monotonic and cyclic loading. J Compos Constr 2015;16:DOI: 10. 1061/(ASCE)CC.1943-5614.0000557. [9] Triantafillou TC, Choutopoulou E, Fotaki E, Skorda M, Stathopoulou M, Karlos K. FRP confinement of wall-like reinforced concrete columns. Mater Struct 2015. https://doi.org/10.1617/s11527-015-0526-5. [10] Li X, Ding D, Lu J, Wang W. Axial strength of FRP-confined rectangular RC columns with different cross-sectional aspect ratios. Mag Concr Res 2017;69(19):1011–26. [11] Hany NF, Hantouche EG, Harajli MH. Generalized axial stress-strain response of rectangular columns confined using CFRP jackets and anchors. J Compos Constr
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Structures journal homepage: www.elsevier.com/locate/structures
Axial stress–strain model developed for rectangular RC columns confined with FRP wraps and anchors Haytham F. Isleema,b,c,d, Muhammad Tahirb,c,d, Zhenyu Wangb,c,d,
T
⁎
a
Department of Civil Engineering, Tsinghua University, Beijing, 180004, China Key Lab of Structures Dynamic Behavior and Control of the Ministry of Education, Harbin Institute of Technology, Harbin, 150090, China c Key Lab of Smart Prevention and Mitigation of Civil Engineering Disasters of the Ministry of Industry and Information Technology, Harbin Institute of Technology, Harbin, 150090, China d School of Civil Engineering, Harbin Institute of Technology, Harbin, 150090, China b
ARTICLE INFO
ABSTRACT
Keywords: Stress–strain model Rectangular RC columns Reinforced concrete Confined FRP anchors
It is widely reported that using fiber-reinforced polymer (FRP) as confinement jackets for concrete can significantly enhance its strength and ductility. The confinement effectiveness for rectangular columns has revealed significant enhancements in their strengths, in particular for well-confined columns with a cross-sectional depth of < 300 mm. However, for larger-sized columns with a section depth of larger than 300 mm and an aspect ratio of larger than 2.0, the confinement effectiveness is most limited. One of the key solutions to significantly increase the confinement effectiveness is by using a combination of FRP anchors and wraps. For good analysis and design of FRP confinement for rectangular RC columns, a model for the stress–strain behavior of rectangular reinforced concrete (RC) columns confined with FRP wraps and FRP anchors is provided in this paper. The model is mainly based on results from tests reported in recent studies, in which a series of axial compressive tests on a total of 80 FRP-confined rectangular RC columns of large size were conducted. The database included columns with aspect ratios ranging from 1.5 to 4.0. Furthermore, the depth of the section is up to 600 mm. key parameters such as the aspect ratio and size of cross section, number of layers of FRP wraps, volumetric ratio of internal hoop steel reinforcement, and cross-sectional area and configuration of anchors were all considered. All predicting expressions were validated against the test database used in the model calibration. In terms of average absolute error and mean square error, comparisons with other existing expressions were conducted, showing that the expressions of the present model reveal more accuracy. Finally, comparisons revealed the model can estimate the confinement effectiveness and predict the stress–strain relationships of FRP-confined rectangular RC columns with additional FRP anchorage.
1. Introduction It is known that many reinforced concrete columns including particularly those RC columns constructed prior to the 1970s have been reinforced with an inadequate amount of transverse steel reinforcement which provides inefficient confinement to the concrete core or lateral restraint to the longitudinal reinforcing bars. Since fiber-reinforced polymer (FRP) composites owe some of the extraordinary properties such as high strength-to-weight ratio and excellent corrosion resistance, the use of externally bonded FRP composites has significantly increased in the construction industry. Nowadays, FRP wraps as an innovative confining technique are widely used for retrofitting existing RC structures. The significance of this subject is confirmed by numerous studies provided models for the axial stress–strain behavior of FRP-confined
⁎
concrete specimens. Over the past 25 years, a large number of experimental and analytical investigations have been concentrated on FRPconfined concrete in circular cross-sections, while only limited studies considered FRP-confined concrete in rectangular cross-sections. The existing studies reveal that the efficiency of FRP confinement is much better for circular concrete sections by significantly improving the axial strength compared with limited confinement provided to rectangular concrete sections. This is due to the fact that a circular section is uniformly confined by the FRP wraps while only a part of a rectangular section is effectively confined. To understand the confinement around the perimeter of concrete in rectangular cross-sections, the following three issues should be considered in a new stress–strain model. (1) Internal steel reinforcement: In general, a review of the existing researches on investigating and modeling the axial strength and
Corresponding author. E-mail address: [email protected] (Z. Wang).
https://doi.org/10.1016/j.istruc.2019.12.020 Received 28 April 2019; Received in revised form 24 August 2019; Accepted 18 December 2019 2352-0124/ © 2019 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
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compressive behavior of FRP-confined concrete columns reports that the existing studies have not considered the influence of internal steel reinforcement on the strength and strain enhancements of concrete columns (i.e. Isleem et al. [1,2]). Only a limited number of studies have considered RC columns. Of these limited studies, 68 FRP-confined circular and non-circular reinforced concrete columns were confined with FRP wraps and tested by Ilki et al. [3] under axial compressive loads. Generally, the confining pressure provided to the columns was derived from the confinement provided by the FRP wraps and the internal hoop steel reinforcement. In recent years, Wang et al. [4,5] have conducted a research on square RC columns confined using FRP. The results of their tests showed that the hoop steel reinforcement significantly contribute to the enhancement of their axial strength and ductility. (2) Aspect ratio and size of rectangular cross-section: One of the key parameters reported in existing studies for modeling the axial stress–strain response is the aspect ratio (defined as the ratio of the longer side, h, to the shorter side, b). Recent studies have also confirmed that the axial compressive behavior of FRP-confined columns is significantly influenced by the increase of the size of a rectangular crosssection (Wang et al. [6]). Although there are many studies available in the literature for the axial stress–strain behavior of FRP-confined concrete columns, the range of considered aspect ratio is between 1 and 1.5. Of the earlier studies on rectangular concrete sections, Abbasnia et al. [7] investigated the axial stress–strain behavior of FRP-confined unreinforced concrete prisms of sizes 120 mm × 180 mm (h/b = 1.50), and 90 mm × 180 mm (h/b = 2.0). The earlier experimental and analytical approaches of Hany et al. (2015) included concrete specimens of size 140 mm × 180 mm (h/b = 1.28), and 130 mm × 200 mm (h/b = 1.54) to investigate and propose a model for the axial stress–strain behavior. According to the recent studies conducted by Wang et al. [4,5]; the test results indicated that the axial stress–strain response of FRP-confined square columns of relatively larger size (305 mm × 305 mm) was different than that obtained from the tests of Hany et al. [8]. Furthermore, 28 large-scale rectangular RC columns of sizes 200 mm × 300 mm (h/b = 1.5) and 200 mm × 400 mm (h/ b = 2.0) confined with FRP were tested [1,2]. This study aimed to investigate the axial stress–strain behavior of rectangular RC columns of large sections confined with FRP. The results showed that the confined specimens exhibited a post-peak softening behavior with little enhancement in axial stress. The lightly-confined specimens experienced reductions in ultimate strength. Due to the effects of aspect ratio and dimensional geometry of a rectangular section, the ultimate strength enhancement was much limited for the 200 mm × 400 mm section compared with the 200 mm × 300 mm section. (3) FRP anchors: In the more recent years, a few techniques have been reported on the seismic strengthening of existing rectangular columns by anchorage of the FRP wraps along the depth of the crosssection using means of FRP anchors (i.e. Triantafillou et al. [9]; Li et al. [10]; Hany et al. [11]). For FRP-confined concrete columns of rectangular cross-sections, similar to the effect of internal confinement by transverse steel ties, anchoring the long side of the FRP wrap using FRP anchors distributed along the height and depth of specimen increases the effectively confined area by restraining the FRP wrap from bulging out at the flat sides of the rectangular section. Another advantage of using FRP anchors is increasing the total cross-sectional area of FRP, which generates additional confining pressure to the cross-section leading to a significant strength enhancement. As a result, the contribution of FRP anchors to the strength enhancement of FRP-confined rectangular RC columns should be considered in the analytical model. As a result, in the recent studies by Isleem et al. [1,2,14] it has been confirmed that with respect the unconfined concrete strength, rectangular columns of small size (i.e., the depth of a cross-section < 300 mm and the aspect ratio < 2.0) achieved a significant enhancement in confined strength, while there is no any strength enhancement for those of larger-sized cross sections [12]. Therefore, the effectiveness of FRP was proposed to be functional of the section’s depth and with an
increase of the section’s depth, the effectiveness of confinement decreases. This expression was incorporated into a new confinement pressure model used to propose expressions for predicting the stress–strain responses with bilinear and post-peak softening behaviors. The model expressions also take into account the effects of section size, aspect ratio of cross sections, and internal hoop steel reinforcement. The depth of cross-sections ranged between 100 and 500 mm and the cross-sectional aspect ratio between 1.0 and 2.0. The model in these studies was described by three curves. The first region is parabolic ascending and is dependent on the properties of concrete and internal steel reinforcement rather than the FRP. The second region is linear descending and dependent on the number of FRP layers, hoop reinforcement ratio, and sectional aspect ratio. The third region is linear and also dependent on the same factors that influence the second branch. To significantly increase the confinement effectiveness of columns with a depth of greater than 300 mm and sectional aspect ratio of greater than 2.0, it was proposed using a combination of FRP anchors and FRP wraps. First a new confinement strength model was provided using existing 44 test specimens. Based on the proposed model, the threshold values of the lightly and heavily FRP-confined rectangular columns with FRP anchorage were 0.125 and 0.25, respectively [12]. The heavily-confined concrete threshold dictates whether the post-peak curve of an axial stress–strain response of FRP-confined columns have strain softening or hardening. In this paper, based on an assessment of the existing stress–strain models against test results influenced by the presence of FRP anchors and FRP wraps, a complete stress–strain response model was proposed for FRP-confined rectangular RC columns with and without FRP anchor confinement. Experimental results from a series of tests performed on a total of 80 FRP-confined rectangular unreinforced and reinforced rectangular cross-sections with FRP wraps and anchors were used. The parameters that influence the confined strength and strain included the aspect ratio, number of FRP layers, hoop steel reinforcement, and crosssectional area and configuration of FRP anchors were all considered in their model. It was shown that the average absolute error (AAE) and the mean square error (MSE) for the proposed model are minimal compared with those of the existing expressions, indicating that the model predictions are more close to the experimental results. 2. Existing experimental tests 2.1. General To propose a new model and to validate its accuracy, results of tests provided in recent studies performed by Isleem et al. [1], Triantafillou et al. [9], and Hany et al. [11] were chosen to serve as a basis for all the model expressions in this paper. The number of specimens considered in the regression analysis is 80. To consider the influences of a wide range of column parameters on the strength and strain of confined specimens, the unconfined concrete compressive strength ranged from 18 MPa to 46.3 MPa and the cross-sectional areas ranged from 293 cm2 to 896 cm2. The side length of the rectangular section ranged between 100 mm and 600 mm and the aspect ratio between 1.5 and 4.0. The number of FRP anchors along the depth and height of specimens is up to 18, with spacing varying from 140 mm to 200 mm and width of from 66.5 mm to 960 mm. In addition, the tensile strength of FRP ranged between 1046 MPa and 4340 MPa, the modulus of elasticity ranged between 93.7 and 240 GPa, and the thickness of FRP varied from 0.13 mm to 1.0 mm per single layer. A summary of each reference is provided as follows: 2.2. Tests by Triantafillou et al. [9] In reference to the study of Triantafillou et al. [9], a total of 30 rectangular RC columns of 800 mm in height and 150 mm × 450 mm 780
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Table 1 Detail of tested FRP-confined unreinforced and RC specimens. Cross section geometry and reinforcement detail No. Specimen b h (mm) (mm) Triantafillou et al. [9] 1 II3 2 II3 3 II3 4 II4 5 II4 6 II4 7 1AlII3 8 1AlII3 9 1AlII3 10 2AlII3 11 2AlII3 12 2AlII3 13 1AhII3 14 1AhII3 15 1AhII3 16 1AhIII3 17 1AhIII3 18 1AhIII3 19 1AlII4 20 1AlII4 21 1AlII4 22 2AlII4 23 2AlII4 24 2AlII4 25 2AhII4 26 2AhII4 27 2AhII4 28 2AhIII4 29 2AhIII4 30 2AhIII4 Hany et al. [11] 31 S300 32 S313 33 S400 34 S413
Vertical reo.
Hoop reo.
Confinement detail wa.y ca.y (mm)
ra.y
la.y
nf
150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150
450 450 450 600 600 600 450 450 450 450 450 450 450 450 450 450 450 450 600 600 600 600 600 600 600 600 600 600 600 600
20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
6Φ12 6Φ12 6Φ12 8Φ12 8Φ12 8Φ12 6Φ12 6Φ12 6Φ12 6Φ12 6Φ12 6Φ12 6Φ12 6Φ12 6Φ12 6Φ12 6Φ12 6Φ12 8Φ12 8Φ12 8Φ12 8Φ12 8Φ12 8Φ12 8Φ12 8Φ12 8Φ12 8Φ12 8Φ12 8Φ12
Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150 Φ8@150
– – – – – – 66.5 66.5 66.5 66.5 66.5 66.5 133 133 133 133 133 133 66.5 66.5 66.5 66.5 66.5 66.5 133 133 133 133 133 133
– – – – – – 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
– – – – – – 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
– – – – – – 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 2 2 2 2 2 2 2 2 2 3 3 3
140 140 140 140
210 210 210 210
10 10 10 10
– – – –
– – – –
– 560 – 840
– 1 – 1
– 3 – 3
– 260 – 260
3 3 4 4
rc (mm)
Vertical reo.
Hoop reo.
Confinement detail wa.y ca.y (mm)
ra.y
la.y
nf
Cross section geometry and reinforcement detail No. Specimen b h (mm) (mm) Hany et al. [11] 35 M200 36 M213 37 M223 38 M300 39 M313 40 M323 41 M400 42 M413 43 M423 44 L200 45 L224 46 L234 47 L300 48 L324 49 L334 50 L400 51 L424 52 L434 Isleem et al. [1] 53 R1.5H1L2 54 R1.5H2L2 55 R1.5H1L3 56 R1.5H2L3 57 R1.5H0L2 58 R1.5H0L3 59 R2.0H1L3 60 R2.0H1L4 61 R2.0H2L3 62 R2.0H2L4 63 R2.0H0L3
rc (mm)
125 125 125 125 125 125 125 125 125 100 100 100 100 100 100 100 100 100
240 240 240 240 240 240 240 240 240 300 300 300 300 300 300 300 300 300
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
– – – – – – – – – – – – – – – – – –
– – – – – – – – – – – – – – – – – –
– 320 160 – 640 320 – 960 480 – 150 100 – 300 200 – 600 400
– 1 2 – 1 2 – 1 2 – 2 3 – 2 3 – 2 3
– 3 3 – 3 3 – 3 3 – 4 4 – 4 4 – 4 4
– 245 245 – 245 245 – 245 245 – 220 220 – 220 220 – 220 220
2 2 2 3 3 3 4 4 4 2 2 2 3 3 3 4 4 4
200 200 200 200 200 200 200 200 200 200 200
300 300 300 300 300 300 400 400 400 400 400
30 30 30 30 30 30 40 40 40 40 40
6Φ16 6Φ16 6Φ16 6Φ16 – – 8Φ16 8Φ16 8Φ16 8Φ16 –
Φ8@180 Φ8@90 Φ8@180 Φ8@90 – – Φ8@200 Φ8@200 Φ8@100 Φ8@100 –
– – – – – – – – – – –
– – – – – – – – – – –
– – – – – – – – – – –
– – – – – – – – – – –
2 2 3 3 2 3 3 4 3 4 3
(continued on next page) 781
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Table 1 (continued) Cross section geometry and reinforcement detail No. Specimen b h (mm) (mm)
rc (mm)
Vertical reo.
Hoop reo.
Confinement detail wa.y ca.y (mm)
ra.y
la.y
nf
64 65 66 67 68 69 70 71 72
40 30 30 30 30 30 30 40 40
– 6Φ16 6Φ16 6Φ16 6Φ16 – – 8Φ16 8Φ16
– Φ8@180 Φ8@90 Φ8@180 Φ8@90 – – Φ8@200 Φ8@200
– – – – – – – – –
– – – – – – – – –
– – – – – – – – –
4 2 2 3 3 2 3 3 4
Cross section geometry and reinforcement detail No. Specimen b h (mm) (mm)
rc (mm)
Vertical reo.
Hoop reo.
Confinement detail wa.y ca.y (mm)
ra.y
la.y
nf
Isleem et al. [1] 73 R2.0H2L3 74 R2.0H2L4 75 R2.0H0L3 76 R2.0H0L4 77 R1.5H2L0 78 R1.5H1L0 79 R2.0H2L0 80 R2.0H1L0
40 40 40 40 30 30 40 40
8Φ16 8Φ16 – – 6Φ16 6Φ16 8Φ16 8Φ16
Φ8@100 Φ8@100 – – Φ8@90 Φ8@180 Φ8@100 Φ8@200
– – – – – – – –
– – – – – – – –
– – – – – – – –
3 4 3 4 0 0 0 0
R2.0H0L4 R1.5H1L2 R1.5H2L2 R1.5H1L3 R1.5H2L3 R1.5H0L2 R1.5H0L3 R2.0H1L3 R2.0H1L4
200 200 200 200 200 200 200 200 200
200 200 200 200 200 200 200 200
400 300 300 300 300 300 300 400 400
400 400 400 400 400 400 400 400
– – – – – – – – –
– – – – – – – –
Note: b and h = the shorter and the longer sides of a rectangular cross-section, respectively; rc = corner radius of the cross-section; reo. = reinforcement; nf = number of externally-bonded FRP composite layers; ca.y = number of FRP anchors distributed along the y- direction; wa.y = width of FRP strip used to form one anchor in the y- direction (shorter side); la.y = length of FRP anchor in the y- direction; ra.y = number of anchor rows along the y- direction.
Table 2 Yield strength of longitudinal and hoop reinforcement steel bars. Source
Triantafillou et al. [9] Hany et al. [11] Isleem et al. [1]
Longitudinal steel fyl (MPa)
Hoop steel fyh (MPa)
FRP material Ef (GPa)
ff (MPa)
tf (mm)
εfu (%)
f c' (MPa)
570 (Φ12) – 360 (Φ16)
570 (Φ8) – 373 (Φ8)
93.7 230 240
1046 3500 4340
1.000 0.130 0.167
1.12 1.50 1.81
18.0 35.0 46.3
Unconfined concrete
Note: tf = thickness of one layer of FRP wraps; ff = maximum tensile strength of FRP; Ef = tensile modulus of elasticity of FRP; εfu = ultimate tensile strain of FRP; fyl and fyh = yield strengths of longitudinal and hoop steel reinforcement, respectively; f c' = compressive strength of unconfined concrete cylinders.
and 150 mm × 600 mm in cross-sections were confined with FRP wraps and anchors and tested under axial compression. The experimental program included two series of specimens, according to their cross-sectional aspect ratios of 3.0 and 4.0, respectively. The test variables considered for the assembled 30 tests were: (1) cross-sectional aspect ratio, (2) anchors number and their weight, and (3) and the number of layers of FRP wrap. As an example for their specimens’ designation, 1AlII3 is a specimen from the first series. It had a crosssectional aspect ratio 3.0, was confined with light-weight FRP anchors (i.e. Al) distributed around the long side of the section, and confined with two layers of FRP wraps (i.e. II). Another specimen with a crosssectional aspect ratio equal to 4.0 selected from the second series is 2AhIII4 confined using heavy-weight FRP anchors (i.e. Ah) and three FRP layers (i.e. III). Table 1, of appendix A, summarized their crosssection geometries and confinement details (i.e. hoop reinforcement and anchors). The mechanical properties of the internal longitudinal and hoop steel reinforcement and the FRP used for the wraps and anchors are provided in Table 2, of appendix A, for all specimens utilized in this paper.
variables of tests were: (1) aspect ratio, (2) number of FRP layers, and (3) area and configuration of FRP anchors. These specimens are divided into three series labeled as S, M, and L based on their cross-sectional aspect ratios of 1.5, 1.92, and 3.0, respectively. The columns sizes in these three series are 140 mm × 210 mm × 420 mm, 125 mm × 240 mm × 480 mm, and 100 mm × 300 mm × 600 mm, respectively. According to their designations provided in Table 1, the following letter refers to the number of FRP layers (i.e. SL3 had three FRP layers). The last two numbers in the specimen designation correspond to the number of anchor columns and anchor rows, respectively. 2.4. Tests by Isleem et al. [1] In reference to the study of Isleem et al. [1], a total of 28 rectangular unreinforced and reinforced concrete columns having a height of 1000 mm and cross-sectional dimensions of 200 mm × 300 mm and 200 mm × 400 mm were fabricated and tested under concentric loads. Table 1 displays their identification designations, section dimensions, and confinement details. The test program included two series of specimens, according to their aspect ratios of 1.5 and 2.0, respectively. According to the designation used in Table 1 and all of the subsections of this paper, R represents the rectangular cross-section of specimens, while 1.5 and 2.0 correspond to side-aspect ratios of 1.5:1 and 2:1. The following letter H refers to the volumetric ratio of hoop reinforcement.
2.3. Tests by Hany et al. [11] The second group consisting of 22 test specimens in Table 1 was selected from the test program conducted by Hany et al. [11]. The 782
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The following number 0 denotes unreinforced concrete, whereas the numbers 1 and 2 denote hoop reinforcement of 0.33 and 0.67% volumetric ratio, respectively. Finally, the last symbol L and the number following it correspond to the number of layers of carbon fiber sheet used to form the FRP confinement. The center-to-center spacing of the steel hoops corresponding to 0.33 and 0.67% volumetric ratios were 180 and 90 mm for the R1.5 specimens and 200 and 100 mm for the R2.0 specimens, respectively. Several parameters were investigated such as (1) cross-sectional size and aspect ratio, (2) number of FRP composite layers, (3) volumetric ratio of hoop steel reinforcement.
confinement which is significantly influenced by the shape and the cross-sectional dimension. In this subsection the bilinear and post-peak softening stress–strain responses drawn in Fig. 1 are used as references for confined columns with section depth of smaller than 300 mm (i.e. Wang et al. [4,6]; Hany et al. [8]) and columns of large size (h > 300 ) (i.e. Isleem et al. [1,2,15]; Wang et al. [4]; Rocca [13]; Yan [16]). As shown, the post-peak softening model (For example, with light amount of FRP anchors) consists of three distinct curves, namely (1) A polynomial ascending curve (OA), which describes the first stage of the response, initiates from the beginning point at which ( c = 0, c = 0) and extends to the peak point when the confined concrete reaches their peak strength and strain, which are referred to as ( c = fcc , c = cc ), (2) The second descending portion of the response (AB) is a linear branch extending from the peak point to the position of a transition stress ( c = fdt , c = dt ), and (3) The final linear portion (BC) extends from the transition stress to the point which represents the ultimate condition of confined columns were the FRP wrap ruptures at an ultimate stress and corresponding strain, which are referred to as ( c = fcu , c = cu ). The three portions of the response are described as.
2.5. Overview of test results and column parameters effects Results obtained from tests on FRP-confined rectangular RC columns of larger section size have generally revealed that: (1) Confined columns of large size exhibited a significant enhancement in axial strain but only a slight increase in axial strength and finally resulted in a postpeak softening behavior followed by a small increase in the stress (i.e. Isleem et al. [1,2,14,15], Wang et al. [4,5]; Rocca [13]). The effectiveness of the confinement provided by the FRP reduces as the section’s size is increased and increases with additional layers of FRP wraps (Isleem et al. [1,2,14]). (2) The smaller-dimensioned specimens experienced larger hoop strains of FRP at failure than those recorded for the larger-dimensioned specimens (i.e. Wang et al. [4,5]; Yan [16]). (3) The increase of volumetric ratio of internal steel hoops, radius of edges of the rectangular cross-section and areas of FRP wraps and FRP anchors influenced the stress–strain envelope curve by increasing the axial strength and strain values. (4) The gain in ultimate strength of the confined columns decreases with an increase in the cross-sectional aspect ratio (i.e. Ilki et al. [3]; Abbasnia et al. [7]; Hany et al. [8], Triantafillou et al. [9]; Li et al. [10]; Isleem et al. [1,2,14,15]; Yan [16]; Wu et al. [17]). For a case in which not all the parameter effects are considered in the existing models, a more generalized stress–strain model is introduced in the following discussions.
fc = fcc A
( ) + (3 c
cc
fc = fcc + E1 ( fc = fdt + E2 (
2A)
( ) c
cc
2
+ (A
2)
( ) c
3 c
cc
cc
c
cc )
cc
c
dt
c
dt )
dt
c
cu
(1)
fcc ) ( dt where E1 = (fdt cc ) (MPa) is the slope of the linear second stress–strain portion; E2 = (fcu fdt ) ( cu dt ) (MPa) is the slope of the third portion of stress–strain response; the shape parameter A, which controls the polynomial first portion, is derived from the boundary condition of d c d c = Ec at c = 0 . On substituting the boundary condition in the first portion of Eq. (1), the A is obtained as A = Ec Ep , in which Ec = 4733 f c' (MPa) = elastic modulus of unconfined concrete (ACI 318–2014 [18]); (MPa) is the second modulus at the peak point (A) in Fig. 1. 3.2. Lateral confining stresses, flf , fla and fls
3. Axial stress–strain model
The confinement stresses provided by FRP wraps (i.e. flf ), FRP anchors (i.e. fla ) and steel hoops (i.e. fls ) are three key parameters for modeling the stress–strain response of RC columns confined using a combination of FRP wraps and anchors.
3.1. General Results of tests reveal that the shape of stress–strain response was bilinear or post-peak softening response depending on the level of FRP
3.2.1. Confinement pressure by external FRP wraps, flf The effective confinement of the FRP is first determined by the following expression used in several studies [19].
flf = kef
(2)
fw ff
where ff (MPa) = maximum tensile strength of FRP wraps; the volumetric ratio of FRP wraps is calculated by fw = 4n f t f D , in which D = 2bh b + h (mm) is the equivalent diameter for a rectangular section [20]; n f = number of FRP layers; t f = thickness of one FRP layer (mm); the shape factor, k ef , for a rectangular cross-section with rounded corners is determined by
k ef =
Acf Ag
(3)
Acf = Ag A cf = Ag b ; 2
ho = 2
Ag = bh Fig. 1. Idealization of the stress–strain response for FRP-confined rectangular columns with larger aspect ratios (i.e. h/b ranges between 2.0 and 4.0).
(b
(4
(
(b
2rc )2 + (h 3 (h
2rc ) 4
2rc )2 + (h 3 2rc )2 b 2
2rc )2 4
+ 3 ho
;
(h
2rc ) 4
(h
2rc ) h o 2
)
b 2
;
(h
2rc ) 4
(4)
) rc2
(5) 2
where the terms Acf and Ag (mm ) are respectively the effectively confined concrete area and the gross cross-sectional area of the 783
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rectangular-sectioned column with rounded corners; ho is the depth of the overlapping zone that occurs when the cross-sectional aspect ratio is more than 3.0 [21].
k se = k sh k sv =
fla = k ea
fa
=
Aa.y
Aa.x D × bH D
+
(7)
where fa is the equivalent ratio of the FRP anchors, which can take into account the effects of the confinement in the longer and shorter sides of the cross-section; the term D is considered as D = b2 + h2 (mm); the effective confinement of FRP anchors along the column height is considered by
H = (ra
(8)
1) × sa
where c x and c y are the numbers of FRP anchors distributed along the width and depth of the cross-section [12]; the terms a x or a y is the total section area of FRP strip sheets folded to form an individual anchor (Table 1), calculated by Eq. (10) in which wa.y and ta.y (mm) respectively denote the width and thickness of FRP strip sheet.
fcc f c'
Aca A A cf = c.f.a Ag Ag
Ac.f.a = Ag
(h
2rc )(h
3(ca.y + 1)
2rc )2 (12)
cc co
cc
= 1 + 0.07
h b
2.86
flfe
2.64
+ 0.5
f c'
fla
1.28
f c'
+ 0.48
b h
1.37
fls
0.12
fc'
= 1 + 0.46
h b
2.06
flfe f c'
2.11
+ 2.40
fla fc'
0.57
+ 7.94
b h
1.5
fls
0.67
f c' (17)
where co = axial strain that corresponds to the peak stress of unconfined concrete, which is calculated by the following expression [22]. co
= 0.000937(fc' )1
4
3.4. Ultimate axial stress fcu and corresponding strain
3.2.3. Confinement pressure by steel hoops, fls Third, the effective confinement pressure provided by the steel hoops is calculated using the following expression used before in several studies [1,2,14]. se fyh
(15)
cx + cy
where (MPa) = compressive strength of unconfined concrete cylinder. The peak strain is also found to depend primarily on the confinement provided by the FRP and the internal hoop reinforcement [1,2,4,5]. Based on multi-parameter regression analysis of the database, the peak strain is given as follows, in which the coefficient (R2) value is 84%.
where Acf and Ag were defined in Eqs. (4) and (5), respectively; herein the term Ac.f.a (mm2) is the total area of concrete core effectively confined by both the FRP wraps and FRP anchors and can be calculated by using the approach introduced by Triantafillou et al. [9]. In their model (Eq. (12)), sa is taken as 0 to neglect the confinement at mid-height between anchor locations for several reasons, such as the fanned part of the anchors extends beyond the anchor locations and the anchors may be covered with a vertical strip which increases the stiffness of the FRP jacket.
fls = k se
y
fc'
(11)
2rc + 1.5ca.y sa ) + (ca.y + 1)(b
(14)
(16)
(10)
Finally, the confinement efficiency k ea of the anchors can be found by using
k ea =
scx
s 2c y
1
An important requirement for defining a complete axial stress–strain response of FRP-confined concrete is the peak point ( fcc , cc ). The experimental results obtained for FRP-wrapped columns have revealed that the RC columns experienced an enhancement in their peak strengths [1,2,4,5]. This enhancement was due to the effects of the FRP wrap as well as internal hoop steel reinforcement. Based on a regression analysis of the database in Table 1, the following expression for predicting the fcc , which takes into account the contribution of these three parameters and has an (R2) value of about 84%, is proposed.
(9)
a x = wa.x × ta.x a y = wa.y × ta.y
Ashy
x
s 2c x
1
cc
3.3. Peak axial stress fcc and corresponding strain
where sa = vertical spacing between two FRP anchors (mm); ra is the number of anchor rows along the height of test specimen (see Table 1). Referring to Eq. (7), the Aa.x and Aa.y (mm2) (Eq. (9)) are the crosssectional areas of FRP anchors perpendicular on the longer and shorter sides of the section, respectively defined as b and h.
Aa.x = (c × r × a) x Aa.y = (c × r × a) y
1
where wxi and wyi (mm) = clear distances between two longitudinal bars along the two longer and shorter directions of the rectangular sectional plane; the terms cx and cy (mm) = the concrete core dimensions to the centerline of the peripheral hoop bar; s (mm) = vertical net spacing between two hoop bars; cc = the ratio of the cross-sectional area of longitudinal steel reinforcement to the sectional area of the inner concrete core enclosed by the centerlines of steel hoop’s parameter; se is the equivalent volumetric ratio of hoop reinforcement; fyh (MPa) is the yield strength of hoops; Ash.x and Ash.y (mm2) are respectively the cross-sectional areas of hoops along longer and shorter sides of the section.
(6)
fa ff
hH
se
wxi2 + wyi2 ) 6c x c y
( )c + ( )c = Ashx scy
3.2.2. Confinement pressure by FRP anchors, fla Second, the effective confinement pressure by the FRP anchors is calculated using the model recently proposed by Isleem et al. [12].
(
1
(18)
cu
The ultimate axial stress, fcu , and corresponding strain, cu , are the most important parameters for a complete stress–strain response of FRP-confined concrete. Based on multi-parameter regression analysis of the experimental data of specimens in Table 1, Eq. (19) for predicting the fcu and Eq. (20) for cu , which are dependent on the three parameters defined in the introduction, are proposed.
(13)
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H.F. Isleem, et al.
6 5
f ccmod f ccexp
AAE = 0.95; MSE = 0.91 AAE = 1.25; MSE = 1.65 AAE = 0.56; MSE = 0.36 AAE = 0.17; MSE = 0.03 AAE = 0.13; MSE = 0.02 AAE = 0.09; MSE = 0.02 AAE = 0.10; MSE = 0.01 AAE = 0.05; MSE = 0.00
Kumutha et al. [23] ACI Committee [20] Toutanji et al. [24] Ozbakkaloglu [25] Triantafillou et al. [9] Hany et al. [8] Li et al. [10] Proposed model
4 3 2 1
Fig. 4. Performance of proposed and existing models for fcu against results of specimens in Table 1 and existing studies.
Fig. 2. Comparison of selected models for fcc with test database for the specimens in Table 1 and existing studies. (See above-mentioned references for further information.)
the current paper, Eq. (23) of k has been proposed by Wang et al. [6] based on their own observations and results from tests on FRP-confined square specimens. This expression is also used for the rectangular specimens used in Table 1, noting that b (the width of the square section) is replaced with h (the depth of the rectangular section). fe
=
=1
f c' cu co
= 0.59 + 2.0
h = 1 + 2.98 b
flfe = k ef
fw Ef
fe
2rc D
0.71
0.40
h b
0.35
3flfe + fla
2.22
f c'
8flfe + fla f c'
0.86
+ 5.45
+ 0.87
fls f c'
fls fc
0.38
b 100
0.41
(23)
The performance of proposed expressions and those reported by researchers [8–10,17,20,23–31] for predicting the peak and ultimate strength and strain enhancements due to the FRP confinement were assessed against the same test database in Table 1. The two indicators described by Eq. (24) namely (1) the average absolute error (AAE) and (2) the mean square error (MSE) are considered to establish the overall model accuracy. Clear comparisons of the analytical and experimental test results are provided in Figs. 2.,Fig. 3.,Fig. 4.,Fig. 5.. Over the wide range of parameters discussed in this study, the comparison shows that the proposed model is able to predict with good accuracy (respectively, AAE = 5% and 10%) the fcc and fcu , but shows less accuracy (AAE = 12% and 16%) in predicting the corresponding cc and cu . A large variability in the cu values was also reported by numerous researchers, such as Hany et al. [8] in which the AAE value, by comparison, was 28.6%. Generally the model revealed a better correlation compared to that of other models.
Fig. 3. Comparison of selected models for cc with test database for the specimens in Table 1 and existing studies. See above-mentioned references for further information
fcu
(22)
fu
0.24
(19)
0.42
AAE = (20)
MSE =
N i = 1 | (anai
expi) expi |
N N i = 1 ((anai
expi) expi )2
(24)
N
(21)
Recently, it has been observed the rupture strain of the FRP wraps to be governed by the strains recorded at the section corners [1,2,4,5]. The medium-sized columns experienced larger strains at failure than those recorded for the larger-sized columns. To take these effects into account, the well-known expression for estimating the effective rupture strain at is given by Eq. (22). For the analytical modeling performed in
3.5. Transition axial stress fdt and corresponding strain branch
dt
for the second
It has been reported that the extent of the descending branch of the softening stress–strain response is influenced by the amount of FRP confinement, the hoop steel reinforcement volumetric ratio, and the
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fdt f c'
dt
= 0.83 + 3.10
=
1 + 5.1
co
flf
3.93
f c'
flf + fla f c'
+ 0.64
fls
0.15
f c'
2.48
+ 0.79
fls f c'
0.05
b h
0.54
h b
+ 0.8
fla fc'
0.24
(25)
0.19
(26)
4. Verification of proposed stress–strain model The accuracy of the compression stress–strain model proposed in this paper is verified by comparing the theoretical predictions with experimental results of specimens reported in Table 1. Typical comparisons are provided in Fig. 7. These specimens were selected due to that they have varying cross-sectional sizes, aspect ratios, numbers of layers of FRP wraps, and most importantly configurations of FRP anchors, which is a major parameter of this study. Therefore, comparisons in Fig. 7 with other test stress–strain responses are not presented due to very limited experimental tests available for FRP-confined large-sized rectangular RC columns under axial compression. Generally, inspection of the comparisons of the responses of the model with the experimental results of large-sized columns demonstrates that the model can trace the key features of stress–strain response (i.e. fcc , cc , fdt, εdt, fcu , and cu ). Also, the shapes of the curves are described well.
Fig. 5. Performance of proposed and existing models for cu against test results of RC columns. See above-mentioned references for further information
5. Conclusion Limited models consider the effects of the cross-sectional size on the axial stress–strain behavior of FRP-confined rectangular RC columns. No model that considers both bilinear and post-peak softening responses was also reported earlier for rectangular RC columns confined by a combination of FRP wraps and FRP anchors, in which the effectiveness of FRP confinement is significantly influenced by the weight and configuration of FRP anchors. Based on an analytical evaluation into the axial compressive strength and strain of FRP-confined unreinforced and RC rectangular concrete columns with large cross-sections strengthened with an additional FRP anchorage, the major issues that are in need to be fully considered are first presented in their paper. As concluded in the recent studies, the existing models are applicable to FRP-confined circular unreinforced specimens and rectangular reinforced sections with a limited variation in aspect ratio. Therefore, this paper provides a stress–strain model with a more generalized scope. The model consists of mathematical expressions for describing the different features of a complete stress–strain response of rectangular unreinforced and RC columns confined with FRP wraps and additional FRP anchors. The main test parameters of the proposed model included the size and aspect ratio of cross-section, volumetric ratio of internal reinforcement hoops, number of FRP layers, and the amount of confinement by FRP anchors. Finally, good agreement was shown between the predictions of the proposed model and the test results, confirming that the model can predict the stress–strain response of large rectangular sectioned RC columns confined with FRP wraps and different configurations and characteristics of FRP anchors. The model expressions are calibrated using limited tests on columns with FRP anchors. Since the stress–strain behavior of rectangular RC columns with FRP depends largely on the variables of this study, the model may not be applicable to different ranges of test parameters. More experimental and analytical investigations for expanding the model application may consider the influences of new test parameters (i.e. ultra-high strength concrete, configuration of hoop steel reinforcement, larger section size, configuration of FRP anchors, and compression loading pattern).
Fig. 6. Comparison of proposed expressions with experimental results: (a) for transition stress, fdt ; (b) for transition strain, dt
cross-sectional aspect ratio [1,2]. As a result, a regression analysis is performed on the test database in Table 1 to propose the following two expressions of fdt and dt with R2 of about 87% and 86%, respectively. It should be noted that these two expressions are based on limited test data because no other experimental results for rectangular RC columns of large-size are found in the literature. Fig. 6 (a) and (b), in general, show that the predictions given by the proposed expressions are in close agreement with the experimental results. 786
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Fig. 7. Proposed axial stress–strain model versus experimental results
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Declaration of Competing Interest
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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This study is financially supported by the National Natural Science Foundation of China (Grant Nos. 51408153, 51478143, and 51278150); the China Postdoctoral Science Foundation (Grant Nos. 2014M551252 and 2015T80354); and the Guangdong Provincial Key Laboratory of Durability for Marine Civil Engineering, Shenzhen University (GDDCE16-09). References [1] Isleem HF, Wang ZY, Wang DY, Smith ST. Monotonic and cyclic axial compressive behavior of CFRP-confined rectangular RC columns. J Compos Constr 2018a;22(4):DOI: 10.1061/(ASCE)CC.1943-5614.0000860. [2] Isleem HF, Wang DY, Wang ZY. Modeling the axial compressive stress-strain behavior of CFRP-confined rectangular RC columns under monotonic and cyclic loading. Compos Struct 2018;185(1):229–40. https://doi.org/10.1016/j. compstruct.2017.11.023. [3] Ilki A, Peker O, Karamuk E, Demir C, Kumbasar N. FRP retrofit of low and medium strength circular and rectangular reinforced concrete columns. J Mater Civil Eng 2008;20(2):169–88; https://doi.org/0.1061/(ASCE)0899-1561(2008)20:2(169). [4] Wang ZY, Wang DY, Smith ST, Lu DG. CFRP-confined square RC columns. I: experimental investigation. J Compos Constr 2012;16(2):150–60. [5] Wang ZY, Wang DY, Smith ST, Lu DG. CFRP-confined square RC columns. II: cyclic axial compression stress-strain model. J Compos Constr 2012;16(2):161–70. [6] Wang DY, Wang ZY, Smith ST. Yu T Size effect on axial stress-strain behavior of CFRP-confined square concrete columns. Constr Build Mater 2016;18:116–26. [7] Abbasnia R, Hosseinpour F, Rostamian M, Ziaadiny H. Cyclic and monotonic of FRP confined concrete rectangular prisms with different aspect ratios. Constr Build Mater 2013;40:118–25. [8] Hany NF, Hantouche EG, Harajli MH. Axial stress-strain model of CFRP-confined concrete under monotonic and cyclic loading. J Compos Constr 2015;16:DOI: 10. 1061/(ASCE)CC.1943-5614.0000557. [9] Triantafillou TC, Choutopoulou E, Fotaki E, Skorda M, Stathopoulou M, Karlos K. FRP confinement of wall-like reinforced concrete columns. Mater Struct 2015. https://doi.org/10.1617/s11527-015-0526-5. [10] Li X, Ding D, Lu J, Wang W. Axial strength of FRP-confined rectangular RC columns with different cross-sectional aspect ratios. Mag Concr Res 2017;69(19):1011–26. [11] Hany NF, Hantouche EG, Harajli MH. Generalized axial stress-strain response of rectangular columns confined using CFRP jackets and anchors. J Compos Constr
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