Engineering Structures 79 (2014) 70–85

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Engineering Structures

journal homepage: www.elsevier .com/locate /engstruct

Finite element modelling and dilation of FRP-confined concrete columns

http://dx.doi.org/10.1016/j.engstruct.2014.07.0450141-0296/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Address: 1401 N. Pine Street, Rolla, MO, USA.E-mail address: elgawadym@mst.edu (M.A. ElGawady).

Osama Youssf a, Mohamed A. ElGawady b,⇑, Julie E. Mills a, Xing Ma a

a University of South Australia, Adelaide, Australiab Missouri University of Science and Technology, MO, USA

a r t i c l e i n f o

Article history:Received 23 January 2014Revised 30 July 2014Accepted 31 July 2014

Keywords:FRP-confined concreteK–C plasticity modelConcrete shear dilationFinite element analysisLS-DYNA

a b s t r a c t

Concrete dilation is one of the main parameters that controls the stress–strain behaviour of confined con-crete. Several analytical studies have been carried out to predict the stress–strain behaviour of concreteencased in fibre-reinforced polymer (FRP), which is crucial for structural design. However, none of thesestudies have provided a simple formula to determine the dilation parameter that is always required in thefinite element (FE) material modelling of concrete. This paper presents a simple empirical model predict-ing the confined concrete dilation parameter within the theoretical framework of a Karagozian and Casetype concrete plasticity model. A set of 105 FRP-confined specimens with different unconfined concretestrengths (f 0c) and confinement moduli (E1) was analysed using the LS-DYNA program. The model predic-tions of the confined ultimate strength (f 0cc), confined ultimate axial strain (Ecc) and confined ultimatehoop strain (Eh) were compared with the corresponding experimental database results for each specimen.In addition, the model axial and hoop stress–strain curves of each specimen were developed and com-pared with the corresponding experimental ones. The proposed model was able to predict stress–straincurves of the test specimens quite well .The proposed model was able to predict f0cc with mean errors (M)and standard deviations (SD) of 2.6% and 10.7%, respectively. Similarly, the model predicted Ecc with Mand SD values of 0.3% and 29.0%, respectively. Finally, the model was less successful in predicting Eh withM and SD values of 13.7% and 26.3%, respectively.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, external confinement of concrete columns byfibre reinforced polymer (FRP) has become increasingly popular.This includes FRP-wrapping of existing columns (e.g. ElGawadyet al [1]) and concrete encased in FRP tubes for new column con-struction (e.g. ElGawady et al. [2], ElGawady and Sha’lan [3]). TheFRP-confinement is able to increase the concrete ductility becauseof the high tensile strain capacity of the FRP tubes in the hoopdirection which increases the axial strain capacity of the confinedcolumns.

The accuracy of a confinement constitutive model depends onhow well it captures the interaction between the concrete dilation(which depends on concrete material characteristics), the lateralpressure (which depends on confinement material characteristics),and the amount of plastic volumetric strain. Numerous studieshave established analytical constitutive models for FRP-confinedconcrete. Ozbakkaloglu et al. [4] conducted a comparative studyof 68 constitutive models for circular concrete cross sections

confined using FRP. The average absolute error (M) and standarddeviation (SD) in predicting the ultimate confined strengths (f0cc),using design-oriented models, were 18.6% and 18.9%, respectively.Using analysis-oriented models, the M and SD in predicting f0cc

increased slightly to 22.2% and 19.5%, respectively. However, thepredictions of ultimate confined axial strains (Ecc) were more chal-lenging. The M and SD in predicting Ecc using design-oriented mod-els were 53.0% and 57.1%, respectively, while they increased to130% and 173%, respectively, using analysis-oriented models.

The behaviour of concrete as a pressure sensitive material can bemodelled using the theory of plasticity. In order to provide an accu-rate prediction of the behaviour of passively-confined concrete(FRP-confined concrete), a plasticity model needs to have the follow-ing three features [5]: (1) a yield criterion that reflects the effect ofthe third deviatoric stress invariant; (2) a confinement-dependenthardening/softening rule; and (3) a confinement-dependent flowrule, in which the dilation parameter is related not only to the con-fining pressure but also to the rate of confinement increase. Druckerand Prager (D–P) type plasticity models implemented in finiteelement (FE) codes have been widely used in the literature to predictthe behaviour of FRP-confined concrete (e.g. [5–13]). However, aconventional D–P model does not include all three featuresmentioned above [5].

Fig. 1. Effect of dilation parameter on axial stress–strain curve.

Fig. 2. Negative dilation parameter for high confinement.

Fig. 3. Description of non-associative, partial-associative, and associative flow rules[15].

O. Youssf et al. / Engineering Structures 79 (2014) 70–85 71

The concrete plasticity flow rule relates to the angle of the plas-tic-deformation-rate vector to the yield surface. This angle can be90 degrees or less [14]. The plastic flow that develops along a linenormal to the yield surface is known as an associative flow rule. Ifan associative flow rule is used for the concrete model, too muchshear dilation tends to occur [15]. In the D–P model, manyresearchers (e.g. Lan [16], Fang [17], Mahfouz et al. [18]) adoptedan associative flow rule which led to an overestimation of thelateral expansion of the confined concrete. Mirmiran et al. [7],Shahawy et al. [19] and Wong et al. [20] used a non-associativeflow rule. This led to good predictions of the axial stress–straincurves, but not the volumetric responses. In these cases the lateralresponses of the confined concrete were predicted reasonably wellbut not with high accuracy [5].

While the dilation parameter is important in determining theaxial capacity and volumetric response of FRP-confined concrete,only a few studies have attempted to quantify appropriate valuesfor the dilation parameter. Yu et al. [5], Jiang et al. [8], and Jiangand Wu [6] demonstrated that the concrete axial plastic strain dur-ing the loading and the rate of confinement increment affect theshear dilation of concrete. They provided analytical models thatpresented the variation of the concrete dilation parameter as afunction of the plastic strain and the confinement lateral stiffnessratio. These models could be used as a subroutine in FE programs.However, these models were not able to provide close predictionsof FRP-confined concrete behaviour, and they were also cumber-some and complicated for use in practical design situations.

Rousakis et al. [21] and Karabinis et al. [9] developed a dilationparameter for D–P plasticity models that is a function of concretecompressive strength (f0c) and confinement modulus (E1). The con-finement modulus is defined as the ratio of confinement pressure(fl) to hoop strain (Eh). [22]. Confinement pressure, fl, can be calcu-lated using the formula:

f l ¼2f ttf

Dð1Þ

where ft is the ultimate tensile strength of the FRP material, tf is theconfinement thickness, and D is the concrete core diameter. An ana-lytical model for the dilation parameter in D–P plasticity model as afunction of f0c and E1 was provided by Rousakis et al. [21].

1.1. Shear dilation parameter

The concrete material model of Karagozian and Case (K–C) is aplasticity-based concrete material model implemented in the LS-DYNA software package [23,24]. This model has the capability ofauto-generating the required model parameters based solely onthe concrete f0c. It also incorporates many important features ofconcrete behaviour such as tensile fracture energy. The K–C con-crete model overcomes the shortcomings of the D–P plasticitymodel [25]. It is a three invariant model that uses three shear fail-ure surfaces: the initial yield surface, the maximum yield surface,and the residual yield surface. In addition, the model adopts avariable flow rule and shear dilation values which can take intoconsideration the confinement effects [25].

In the K–C model, one approach to determine the appropriateplastic volumetric strain is to control the shear dilation parameter(x). This parameter is the fraction associativity defined as the ini-tial ratio of the plastic volumetric strain increment to that wouldoccur if the plastic flow were fully associated in the hydrostaticplane [26]. The value of x was recommended to vary between0.0 for non-associative flow and 1.0 for associative flow[15,27,28]. However, guidelines for selecting such shear dilationparameters are scarce. Noble et al. [27] recommended a value ofx ranging between 0.5 and 0.7. The value of this parameter signif-icantly affects the slope of the plastic behaviour zone of the

FRP-confined concrete stress–strain curve. Fig. 1 shows the effectof this dilation parameter value on the predicted axial stress–strainbehaviour of 41.6 MPa confined concrete using 1 layer of carbonFRP. As shown in the figure, the increase of dilation parameterincreases the plastic behaviour slope. In this figure, a zero valueof x matched well with the experimental results. However, in highconfinement cases (e.g. 6 layers of FRP), the appropriate values ofx could be negative values as shown in Fig. 2. Positive dilation ofconcrete represents volume expansion. So, negative dilation

Table 1Experimental database used to develop the model.

Specimen No. f0c (MPa) D (mm) H (mm) H/D Fibre type tf (mm) Ef (GPa) Ef (%) E1 (MPa) E1/f0c Source

1 33.6 150 300 2.0 Carbon 0.38 105 1.5 533 16 [22]2 33.6 0.38 105 1.5 533 163 33.6 0.76 105 1.5 1066 324 33.6 1.14 105 1.5 1600 475 43.7 0.38 105 1.5 533 16 43.7 0.76 105 1.5 1066 27 43.7 1.14 105 1.5 1600 368 55.2 0.76 105 1.5 1066 199 38.0 0.68 241 1.5 2182 57 [36]10 38.0 1.02 241 1.5 3273 8611 38.0 1.36 241 1.5 4364 11512 45.9 Glass 0.17 80 1.7 181 413 45.9 0.34 80 1.7 353 814 45.9 0.51 80 1.7 544 1215 37.7 Carbon 0.11 260 1.5 381 1016 44.2 0.11 260 1.5 381 917 44.2 0.22 260 1.5 762 1718 47.6 0.33 250 1.5 1102 2319 35.0 0.17 250 1.5 551 16 [34]20 35.0 0.33 250 1.5 1102 3121 35.0 0.50 250 1.5 1653 4722 38.5 Glass 1.27 22 2.3 369 1023 38.5 2.54 22 2.3 738 1924 45.9 0.51 80 1.7 544 12 [36]25 41.1 Carbon 0.17 250 1.5 550 13 [35]26 38.9 0.33 247 1.5 1086 2827 39.6 Glass 0.34 80 2.2 363 9 [37]28 39.6 0.51 80 2.2 544 1429 39.0 Aramid 0.40 120 2.5 640 16 [39]30 39.0 0.40 120 2.5 640 1631 39.0 0.60 120 2.5 960 25

32 39.0 0.60 120 2.5 960 2533 34.3 Glass 2.63 18 2.0 634 18 [32]34 34.3 2.44 19 2.0 618 1835 34.3 Carbon + Glass 2.83 13 1.4 490 14

Table 2Experimental database used to validate the model.

Specimen No. f0c (MPa) D (mm) H (mm) H/D Fibre type tf (mm) Ef (GPa) Ef (%) E1 (MPa) E1/f0c Source

36 33.6 150 300 2 Carbon 0.38 105 1.5 533 16 [22]37 33.6 0.76 105 1.5 1066 3238 33.6 0.76 105 1.5 1066 3239 33.6 1.14 105 1.5 1600 4740 33.6 1.14 105 1.5 1600 4741 43.7 0.76 105 1.5 1066 2442 43.7 0.76 105 1.5 1066 2443 43.7 1.14 105 1.5 1600 3644 43.7 1.14 105 1.5 1600 3645 55.2 0.76 105 1.5 1066 1946 55.2 0.76 105 1.5 1066 1947 55.2 1.14 105 1.5 1600 2948 55.2 1.14 105 1.5 1600 2949 55.2 1.14 105 1.5 1600 2950 38 0.68 241 1.5 2182 57 [36]51 38 1.02 241 1.5 3273 8652 38 1.36 241 1.5 4364 1553 45.9 Glass 0.17 80 1.7 181 454 45.9 0.34 80 1.7 353 855 37.7 Carbon 0.11 260 1.5 381 1056 44.2 0.22 260 1.5 762 1757 47.6 0.33 250 1.5 1102 2358 47.6 0.33 250 1.5 1102 2359 35 0.17 250 1.5 551 16 [34]60 35 0.17 250 1.5 551 1661 35 0.33 250 1.5 1102 3162 35 0.33 250 1.5 1102 363 35 0.5 250 1.5 1653 464 35 0.5 250 1.5 1653 4765 38.5 Glass 1.27 22 2.3 369 1066 38.5 2.54 22 2.3 738 1967 41.1 Carbon 0.17 250 1.5 550 13 [35]

72 O. Youssf et al. / Engineering Structures 79 (2014) 70–85

(b)(a)

Constrained in X and Y

Constrained in X, Y and Z

Constrained in X and Y

Fig. 4. FE Geometry of the confined concrete core: (a) isometric view and (b)elevation view.

Fig. 5. FE dilation parameter versus E1/f0c.

Table 2 (continued)

Specimen No. f0c (MPa) D (mm) H (mm) H/D Fibre type tf (mm) Ef (GPa) Ef (%) E1 (MPa) E1/f0c Source

68 41.1 0.17 250 1.5 550 1369 38.9 0.33 247 1.5 1086 2870 38.9 0.33 247 1.5 1086 2871 39.6 Glass 0.34 80 2.2 363 9 [37]72 39.6 0.51 80 2.2 544 1473 26 160 320 2 Carbon 1.00 34 1.4 425 16 [38]74 26 3.00 34 1.4 1275 4975 41.6 100 200 2 0.13 230 2.1 598 14 [29]76 41.6 0.26 230 2.1 1196 2977 41.6 0.39 230 2.1 1794 4378 53.5 0.13 230 2.1 598 1179 53.5 0.26 230 2.1 1196 2280 53.5 0.39 230 2.1 1794 3381 39.2 0.13 230 2.1 598 1582 39.2 0.26 230 2.1 1196 3083 39.2 0.39 230 2.1 1794 4684 62.5 0.13 230 2.1 598 985 62.5 0.26 230 2.1 1196 1986 62.5 0.39 230 2.1 1794 2987 38.5 200 320 1.6 0.23 240 1.6 561 14 [33]88 38.5 0.35 240 1.6 842 2289 38.5 0.35 240 1.6 842 2290 35.7 0.12 240 1.6 280 891 35.7 0.12 240 1.6 280 892 35.7 0.23 240 1.6 561 1693 35.7 0.23 240 1.6 561 1694 35.7 0.35 240 1.6 842 2495 35.7 0.35 240 1.6 842 2496 35.7 0.35 240 1.6 842 2497 31.0 76 305 4 Glass 0.24 73 2.1 458 15 [30]98 31.0 Carbon 0.22 230 1.5 1334 4399 31.0 0.33 373 0.8 3237 104100 35.0 152 435 2.8 Glass 0.80 32 1.4 335 10 [31]101 35.0 1.60 34 1.5 713 20102 35.0 2.40 36 1.6 1133 32103 35.0 Carbon 0.11 367 0.9 529 15104 35.0 0.23 390 0.9 1177 34105 35.0 0.55 415 0.9 2995 85

O. Youssf et al. / Engineering Structures 79 (2014) 70–85 73

represents volume contraction. At a given axial strain, increasingthe confinement thickness decreases the specimen hoop strain[29]. This reflects expansions and contractions in the total volumefor the low and high confined concrete, respectively.

In LS-DYNA, the confined concrete dilation parameter mainlydepends on the flow rule of the constitutive model being used[15]. The flow rule can be the non-associative, partial associativeor associative flow rule. A description of the various flow rule typesis shown in Fig. 3 [15]. In this figure, h is the flow angle, and hn isthe flow angle at which no shear dilation occurs.

This research develops a simple, yet efficient, empirical equa-tion that takes into consideration the effect of confinement to

determine the dilation parameter ‘‘x’’ in the K–C model. A data-base consisting of the experimental constitutive stress–strain rela-tionship of 35 FRP confined specimens was collected and modelledusing the LS-DYNA program. For each specimen, the best-fit xvalue was selected based on the equal energy concept and then aregression analysis was carried out to develop an equation for xas a function of f0c and E1. Then, the developed expression for thedilation parameter was implemented in LS-DYNA and was usedto predict the f0cc, Ecc, and Eh of an additional 70 FRP-confined spec-imens, as well as the general behaviour of the stress–strain curves.

74 O. Youssf et al. / Engineering Structures 79 (2014) 70–85

2. Experimental database

A large number of circular FRP-confined concrete test resultshave been reported in the literature. However, most of the pub-lished data has not reported the full stress–strain curve but ratherhas reported some critical values such as f 0c, f 0cc, Ecc, and Eh. Duringthe current study and to calibrate the dilation parameter, only pub-lished data that included full stress–strain curve data was consid-ered. Hence, a database including 105 test results was collectedand reported in Tables 1 and 2. In these tables, D is the specimendiameter, H is the specimen height, tf is the FRP thickness, Ef is

Fig. 6. FE versus experimental results for low and high confinement modu

Table 3Predictions for the data used to develop the model.

Specimen No. f0cc EXP. f0cc Model Error (%) Ecc EXP. Ecc M

1 49 48 2 12,760 11,92 48 48 0 12,760 11,93 73 82 �11 21,850 26,04 94 116 �24 29,410 33,45 54 62 �13 8893 11,66 84 76 10 16,010 12,77 95 86 9 17,190 13,28 77 87 �14 13,580 12,79 110 104 5 24,977 19,710 136 121 11 30,662 24,911 161 136 15 36,499 22,112 48 48 0 3011 36513 55 56 �1 12,357 12,414 66 64 3 18,796 13,215 49 47 4 9966 10,616 50 53 �5 8029 11,217 65 63 3 11,357 12,418 85 77 9 16,542 13,519 52 51 1 12,550 12,220 69 64 8 16,640 13,121 92 88 4 23,980 19,822 52 53 �2 13,010 17,123 76 69 8 24,300 19,924 6 64 2 15,226 13,225 55 57 �5 10,690 12,126 79 68 13 20,450 13,627 56 52 7 19,220 16,928 64 61 3 21,810 18,129 68 68 0 22,920 20,130 67 69 �3 23,070 20,131 88 81 7 30,420 21,632 88 82 7 30,940 21,633 51 53 �4 18,353 19,634 58 52 11 26,263 20,335 52 45 13 15,297 11,5

* No data available.

the FRP modulus of elasticity, Ef is the FRP rupture strain, and E1

is the confinement modulus calculated using Eq. (2).

E1 ¼2Ef tf

Dð2Þ

The data in the tables were collected from Toutanji [30], Safi et al.[31]. Xiao and Wu [22], Zhang et al. [32], Karabinis and Rousakis[33], Lam and Teng [34], Lam et al. [35], Jiang and Teng [36], Tenget al. [37], Benzaid et al. [38], Ozbakkaloglu and Akin [39], andYoussf et al. [29]. Of the total database, 35 specimens that hadslenderness ratios (H/D) of 2.0 were used for the calibration

lus: (a) volumetric strain–axial stress and (b) axial strain–hoop strain.

odel Error (%) Eh EXP. Eh Model Error (%) Source

85 6 10,880 8675 20 [22]85 6 11,060 8675 2249 �19 10,500 13,631 �3091 �14 9600 14,075 �4751 �31 5059 4042 2063 20 10,000 7429 2695 23 8118 6169 2400 6 7529 7745 �301 21 9583 7769 19 [36]60 19 9160 5966 3565 39 8454 6197 276 �21 4172 2024 5147 �1 18,252 12,239 3317 30 19,712 10,858 4565 �1 10,871 12,083 �1167 �40 8255 10,279 �2455 �10 10,660 8522 2095 18 10,301 7820 2476 2 10,120 9115 10 [34]85 21 9571 7076 2686 17 8834 8524 310 �31 14,410 15,716 �924 18 17,310 12,680 2717 13 15,749 10,858 31 [36]11 �13 10,500 9551 9 [35]46 33 11,290 7935 3071 12 * – – [37]15 17 * – –90 12 * – – [39]91 12 * – –15 29 * – –16 30 * – –63 �7 * – – [32]77 22 * – –22 25 * – –

Table 4Predictions for the data used to validate the model.

Specimen No. f0cc EXP. f0cc Model Error (%) Ecc EXP. Ecc Model Error (%) Eh EXP. Eh Model Error (%) Source

36 50 48 4 12,050 11,985 1 8471 8675 �2 [22]37 71 82 �15 21,500 26,049 �21 9059 13,631 �5038 75 82 �9 22,320 26,049 �17 9529 13,631 �4339 83 116 �40 24,690 33,491 �36 7765 14,075 �8140 86 116 �35 23,150 33,491 �45 8824 14,075 �5941 78 76 3 13,400 12,763 5 9647 7429 2342 84 76 9 15,770 12,763 19 8824 7429 1643 92 86 6 16,600 13,295 20 8235 6169 2544 96 86 10 17,310 13,295 23 7765 6169 2145 74 87 �17 11,310 12,700 �12 7294 7745 �646 78 87 �12 8214 12,700 �55 8118 7745 547 107 99 7 14,060 14,054 1 7882 6469 1848 106 99 6 13,330 14,054 �5 8118 6469 2049 102 99 3 10,950 14,054 �28 7529 6469 1450 107 104 3 25,259 19,701 22 9019 7769 14 [36]51 129 121 6 27,472 24,960 9 8877 5966 3352 158 136 14 35,111 22,165 37 8454 6197 2753 46 48 �4 2712 3656 �35 1982 2024 �254 53 56 �5 11,890 12,447 �5 16,687 12,239 2755 50 47 6 9188 10,665 �16 8929 12,083 �3556 63 63 0 10,052 12,455 �24 9295 8522 857 85 77 9 17,725 13,595 23 10,038 7820 2258 83 77 6 12,999 13,595 �5 9112 7820 1459 51 51 0 13,690 12,276 10 11,320 9115 19 [34]60 54 51 5 13,010 12,276 6 9847 9115 761 70 64 9 19,610 13,185 33 9847 7076 2862 72 64 11 18,030 13,185 27 9202 7076 2363 85 88 �3 22,400 19,886 11 8306 8524 �364 97 88 9 24,810 19,886 20 9571 8524 1165 58 53 9 14,570 17,110 �17 19,030 15,716 1766 77 69 10 22,010 19,924 9 16,560 12,680 2367 57 57 0 11,990 12,111 �1 10,630 9551 10 [35]68 57 57 0 11,740 12,111 �3 10,630 9551 1069 82 68 17 21,890 13,646 38 11,210 7935 2970 81 68 15 21,210 13,646 36 11,380 7935 3071 55 52 4 20,670 16,971 18 * – – [37]72 66 61 7 25,460 18,115 29 * – –73 39 38 4 12,710 10,697 16 13,310 8220 38 [38]74 66 57 12 14,820 12,526 15 13,230 5351 6075 66 67 �2 11,248 17,591 �56 * – – [29]76 90 85 5 15,702 18,653 �19 * – –77 108 104 3 18,091 20,322 �12 * – –78 78 80 �3 11,251 16,748 �49 12,114 13,653 �1379 113 104 8 18,135 18,868 �4 13,570 10,928 1980 137 121 12 21,306 19,638 8 12,522 9360 2581 62 63 �2 7451 16,479 �121 10,094 12,152 �2082 89 75 16 14,559 14,357 1 12,183 7587 3883 113 101 10 18,376 20,141 �10 11,786 8654 2784 76 89 �17 7072 17,104 �142 11,198 14,276 �2785 106 115 �8 12,612 19,477 �54 11,906 11,379 486 138 133 3 17,439 19,435 �11 11,968 9691 1987 55 56 �2 8667 12,307 �42 * – – [33]88 67 63 7 17,440 12,896 26 * – –89 52 63 �21 9573 12,896 �35 * – –90 42 43 �2 8394 11,510 �37 * – –91 40 43 �7 12,300 11,510 6 * – –92 49 51 �6 10,300 12,372 �20 * – –93 51 51 �1 10,740 12,372 �15 * – –94 63 59 7 16,880 12,517 26 * – –95 67 59 12 16,880 12,517 26 * – –96 65 59 10 16,810 12,517 25 * – –97 61 50 18 15,660 18,799 �20 15,710 11,512 27 [30]98 100 68 33 28,070 14,917 47 13,570 5292 6199 93 65 30 14,750 8591 42 4388 1963 55

100 53 44 16 19,300 14,772 23 16,460 13,595 17 [31]101 66 65 2 24,630 17,908 27 15,820 10,580 33102 83 80 4 30,220 20,006 34 16,460 8688 47103 56 51 9 10,470 12,323 �18 9658 8975 7104 68 69 �1 16,270 14,571 10 11,960 6414 46105 96 97 �1 22,580 15,744 30 9485 3983 58

* No data available.

O. Youssf et al. / Engineering Structures 79 (2014) 70–85 75

Table 5Statistical parameters for the errors in the proposed model.

Confined concrete property Ultimate strength (f0cc) Ultimate axial strain (Ecc) Ultimate hoop strain (Eh)

M (%) SD (%) M (%) SD (%) M (%) SD (%)

Data used to develop the model 2.3 8.5 7.4 20 14.5 23.3Data used to validate the model 2.7 11.6 �3.2 33.6 13.3 27.8All data 2.6 10.7 0.3 29 13.7 26.3

M: Mean. SD: Standard deviation.

Fig. 7. Experimental results versus model predictions for both calibration data (left) and validation data (right) of: (a) f0cc, (b) Ecc and (c) Eh.

76 O. Youssf et al. / Engineering Structures 79 (2014) 70–85

Fig. 8. Variation of effective rupture strain with respect to FRP rupture strain fromcoupon test at constant E1/f0c.

O. Youssf et al. / Engineering Structures 79 (2014) 70–85 77

and development of the expression for x (see Table 1). A total of 70additional specimens were then used to test the proposed model(see Table 2). In the collected database, f0c ranged from 26 to62.5 MPa, E1 ranged from 181 to 4364 MPa, the specimens’ diame-ters ranged from 76 to 200 mm, and the specimens’ slendernessratios ranged from 1.6 to 4.0. The FRP confinements in this studywere formed using carbon, glass, and aramid fibres. Filamentwound tubes, and FRP wraps were considered in this study.Researchers [40,41] reported that there was no difference in behav-iour between concrete encased in these two different types of FRPtubes.

Fig. 9. Effect of E1/f0c on the % er

3. Finite element modelling

3.1. Model geometry

The concrete core of a test cylinder was modelled using 8-nodeconstant stress solid hexahedron elements, which have six degreesof freedom at each node. Single point volume integration was car-ried out by Gaussian quadrature. Hourglass control with an hour-glass coefficient of 0.03, as recommended by LS-DYNA support[42], was provided in order to avoid the zero energy modes. FRPjackets were modelled using 4-node shell elements with sixdegrees of freedom at each node. This element includes membrane,bending and shear deformation capabilities. The section attributefor this element was thickness alone. The shell thickness of a givenspecimen was selected to be equal to the FRP thickness that wasreported in the database (Tables 1 and 2). The Belytschko–Tsay[43] element formulation was used for the shell elements, whichis the default theory for shell elements in LS-DYNA [44] due toits computational efficiency. It is based on a combined co-rota-tional and velocity strain formulation. The co-rotational portionof the formulation avoids the complexities of non-linear mechanicsby embedding a co-ordinate system in the element [25].

Fig. 4 shows the geometry of a 150 � 300 mm specimen with itsboundary conditions. Each specimen was axially loaded with a dis-placement control using a rate of 0.01 mm/s. The displacement wasuniformly applied to all nodes at the top surface of the cylinder,using the BOUNDARY_PRESCRIBED_ MOTION card, by linking themtogether in a node set to simulate the rigid loading plate of thecompression testing-machine. For this node set, nodal constraints

ror of: (a) f0cc, (b) Ecc and Eh.

Fig. 10. Effect of E1 on the % error at f0c = 30–40 MPa (left) and at f0c = 40–50 MPa (right) of: (a) f0cc, (b) Ecc and (c) Eh.

78 O. Youssf et al. / Engineering Structures 79 (2014) 70–85

were applied for translation in the global X and Y directions,whereas the nodes were free to translate in the Z direction, seeFig. 4(b). Another similar node set was generated at the bottomsurface of the concrete cylinder and was constrained in the globalX, Y, and Z directions. The node-set constraint actions were definedusing the BOUNDARY_SPC_SET card.

In order to choose the optimum mesh size for the FE model, amesh sensitivity analysis was carried out on a 150 � 300 mm cyl-inder using element aspect ratios ranging between 1.0 and 3.0, thenumber of solid elements ranging between 96 and 2100 and the

number of shell elements ranging between 64 and 400. Increasingthe number of elements was carried out using constant aspectratios which resulted in smaller element sizes. The FRP shell ele-ment mesh was modelled like the concrete outer surface elementmesh. The best fit to the experimental results was achieved usingan aspect ratio of 1.5, 1024 solid elements and 256 shell elements.The shell lengths in the circumferential and longitudinal directionswere about 469 mm and 296 mm, respectively. All other specimensizes were then modelled using the same procedures with anelement aspect ratio of 1.5.

Fig. 11. Effect of f0c on the % error at E1 = 181–1000 MPa (left) and at E1 = 1000–4000 MPa (right) of: (a) f0cc, (b) Ecc and (c) Eh.

O. Youssf et al. / Engineering Structures 79 (2014) 70–85 79

3.2. Material modelling

The K–C model, *MAT_CONCRETE_DAMAGE_RELIII, was usedfor simulating the concrete material [45]. This model has the capa-bility of auto-generating the required model parameters basedsolely on the unconfined compressive strength of concrete. It wasused in conjunction with an equation of state, EOS_TABULATED_COMPACTION, which gives the current pressure, P, as a functionof current and previous volumetric strain. In this tabulatedcompaction model, pressure P = C (Ev), where Ev is the volumetric

strain. The function C (Ev) was entered as a series of C, Ev pairs inthe keyword input file using the automatically generated values.Once the pressure is known, the stress tensor can be calculatedas being a point of a moveable surface that can be a yield surfaceor a failure surface. One of the features of this model is the abilityto control the fracture energy of the system which controls thetension softening of the concrete. The fracture energy can be intro-duced using two different approaches. In the first approach, theuser may select the input fracture energy. Alternatively, the K–Cmodel can automatically generate the fracture energy for a given

Fig. 12. Effect of slenderness ratio on the % error of: (a) f0cc, (b) Ecc and Eh.

80 O. Youssf et al. / Engineering Structures 79 (2014) 70–85

f 0c using the recommendations of CEB. The second approach wasused throughout this study. More details and features of this mate-rial model can be found in [23,24].

The FRP jacket was modelled using the MAT_ORTHOTRO-PIC_ELASTIC_002 material model [44]. This is used to define ortho-tropic materials such as unidirectional layers in composite shellstructures. In addition, it allows the user to control the FRP shellproperties in three dimensions and control the FRP-shell fibredirection. This material model adopts laminated shell theory forthe purpose of correcting the assumption of a uniform constantshear strain throughout the thickness of the composite shell; thus,avoiding very stiff results [46]. The MAT_ADD_ EROSION card wasused to control the failure strain of the FRP shell. The failure strainwas selected based on either the coupon test results or the manu-facturer’s data, as reported by each individual study in thedatabase.

A perfect bond was assumed between the FRP jacket and theconcrete core by sharing the same nodes at the contact surfaceas recommended by previous FE studies (e.g. Elsanadedy et al.[25] and Mohammed [47]). In addition, the AUTOMATIC_SUR-FACE_TO_SURFACE card was used to model the interface betweenthe FRP shell elements and the concrete solid elements. The shellelements were modelled as a slave segment set and the solid ele-ments were modelled as master segment set.

4. Estimation of dilation parameter

The FE model described above was used to model the behaviourof 35 standard confined concrete specimens. These specimens had

f 0c ranging from 26 to 55.2 MPa and E1 ranging from 181 to4364 MPa. For each specimen, the dilation parameter was changedand the resulting axial stress–strain was used to calculate thetoughness. The toughness from the experimental results was thencompared to that from the FE analysis. The value of the dilationparameter corresponding to the least root mean square errors inthe toughness was selected as the best fit value for this specimen.Calibration of the dilation parameter resulted in the relationshippresented in Eq. (3) and Fig. 5.

Dilation parameter ðxÞ ¼ �0:195 lnE1

f 0c

!" #þ 0:6115 ð3Þ

As shown in the figure, the shear dilation parameter decreased asE1/f 0c increased. This means for a given f0c, by increasing the number

of FRP layers increasing E1 ¼2Ef tf

D

� �the dilation of the concrete

decreased. In addition, the value of the dilation parameterapproached zero at E1/f 0c = 23.0. Beyond that increasing E1/f 0c ledto contraction rather than expansion in the specimen volume.

Fig. 6(a) shows an example of the volumetric behaviour for lowand high carbon FRP confinement of 53.5 MPa concrete. As shownin the figure, each specimen’s volume contracted until it reachedits f0c. Beyond that, for the low confinement (1-layer), the inade-quate lateral restriction led the contraction to change to expansion.But, for the relatively high confinement (3-layers), the high lateralrestriction led to ongoing contraction. Selecting the appropriatedilation parameter x is able to control the volumetric behaviour(expansion or contraction) of the FE models to closely match theexperimental results. Fig. 6(b) shows the dilation rate, defined as

Fig. 13. Model verification versus Youssf et al. [29] experimental data: (a) f0c = 41.6 MPa, (b) f0c = 53.5 MPa and (c) f0c = 62.5 MPa.

O. Youssf et al. / Engineering Structures 79 (2014) 70–85 81

the rate of change of hoop strains with respect to axial strains[29,48,49], of the same specimens. The relatively high dilation rateof the low confinement specimen confirms its relatively highexpansion compared to the higher confinement specimen thathas lower dilation rate leading to lower expansion (or contraction)as shown in Fig. 6(a).

5. Results and discussion

Results obtained from the FE modelling, using the proposeddilation parameter model, were compared with those of the exper-imental results for the database specimens. Tables 3 and 4 showthe experimental results and the proposed FE model predictions,using the proposed dilation parameter formula (Eq. (3)), for f0cc,Ecc, and Eh. In this table, the error % was calculated as:

Error ð%Þ ¼ EXP: result� FE resultEXP: result

� 100 ð4Þ

Table 5 summarizes the means and the standard deviations oferrors in the predicted f0cc, Ecc, and Eh for both data used to developthe model and data used to validate the model, as well as, the wholedatabase. The proposed model has mean errors of 2.6%, 0.3%, and13.7% and standard deviations of 10.7%, 29.0%, and 26.3% in the pre-dicted values of f0cc, Ecc, and Eh, respectively, for the whole database.Thus, the proposed model was able to predict f0cc more accuratelycompared to predictions of Ecc, and Eh.

Fig. 7 shows the experimental results versus the predicted onesfor f0cc, Ecc, and Eh for both calibration data (left side) and validation

data (right side). The trend line is close to the neutral line inFig. 7(a) which indicates good predictions of f0cc. However, mostof the data for Ecc, and Eh are below the neutral line which revealsthat the model underestimates Ecc, and Eh. The relatively high scat-ter in the Ecc and Eh predictions, Fig. 7(b and c), is attributed to theissue of selecting the appropriate value of the effective rupturestrain for the FRP shell. The value of the effective rupture strainis hard to predict because of its high variability in the availableexperimental database. Fig. 8 shows the high scatter of Eh/Ef at con-stant E1/f0c. The value ranges between 0.34 and 1.31. Someresearchers have recommended using a value of 0.50–0.85 of theultimate strain obtained from coupon tests [50–54]. Most empiri-cal models use average constant values with respect to the mate-rial properties of the FRP jacket. It is worth noting that theeffective rupture strain value that was used in the present FE anal-ysis was the full value of material properties that were mentionedin each individual study (Eh/Ef = 1).

6.1. Model errors

Fig. 9 shows the error variations of the proposed model predic-tions for f0cc, Ecc, and Eh with respect to E1/f0c. As shown in the figure,there are no strong correlations between E1/f0c and the error varia-tion (low R2 values). The scattering in f0cc errors was less than thatof Ecc, and Eh. Considering that errors of 20% are generally acceptedin concrete models, the proposed model was able to predict 94.2%,51.4%, and 37.0% of the presented data in that range (Fig. 9) for f0cc,

Fig. 14. Model verification versus Xiao and Wu [22] experimental data: (a) f0c = 33.6 MPa, (b) f0c = 43.7 MPa and (c) and f0c = 55.2 MPa.

Fig. 15. Model verification versus Lam and Teng [34] experimental data: (a) f0c = 35.0 MPa and (b) f0c = 38.5 MPa.

82 O. Youssf et al. / Engineering Structures 79 (2014) 70–85

Ecc, and Eh, respectively. This demonstrates again that the model isable to predict the ultimate strength well, and predict the ultimateaxial and hoop strains moderately well.

The factors affecting errors in the model predictions are thoserelated to the experimental work as well the numerical modelling.During the experimental work, strain measurements are discrete at

few locations while FRP rupture may occur far from these loca-tions. Hence, these strain measurements are less than the actualultimate strains. Many of the tested cylinders were wrapped inFRP sheets where the quality of the wrapping and workman shipplay crucial role. For the numerical model, the FRP and concretecharacteristics significantly affect the model performance. One of

Fig. 16. Model verification versus Lam et al. [35] experimental data: (a) f0c = 38.9 MPa and (b) f0c = 41.1 MPa.

Fig. 17. Model verification versus Teng et al. [37] experimental data(f0c = 39.6 MPa).

O. Youssf et al. / Engineering Structures 79 (2014) 70–85 83

the main inaccuracies in modelling FRP is the assumed linear rela-tionship. Many experimental tests on FRP coupons showed thatFRP is not perfectly linear especially close to failure and some non-linear behaviour was observed. However, since this insignificantnonlinearity does not affect the global (macro) performance of con-crete encased in FRP, there is consensus among researchers toignore such nonlinearity. However, such nonlinearity may affectthe local (micro) behaviour of concrete encased in FRP. Fig. 10shows the effect of E1 E1 ¼

2Ef tf

D

� �on the model errors at a given

f0c. In addition, Fig. 11 shows the effect of f0c on the model errorsat a given E1. As shown in the figures, there are no strong correla-tions between either E1 or f0c and the error variation (as indicatedby low R2 values). In Fig 10, the model errors in f0cc and Ecc predic-tions decrease as E1 increases for the f0c ranging from 30 to 40 MPa,Fig. 10(a and b). From Fig. 11, it can also be observed that increasesin f0c have no significant effect on the f0cc predictions (most errorsare lower than 20%) at both low and high confinement moduli,Fig. 11(a). In addition, increasing f0c does not have significant effecton the errors in predicting Ecc, and Eh at low confinement modulus,Fig. 11(b and c) left. However, it decreases the model errors for pre-dictions of Ecc, and Eh at high confinement modulus, Fig. 11(b and c)right.

6.2. Effect of concrete slenderness ratio

The effect of the slenderness ratio (H/D) of the concrete speci-men on the model prediction errors is shown in Fig. 12. The avail-able range of H/D was from 1.6 to 4.0 (Table 1). This analysis wascarried out by comparing model results at similar E1/f0c. As shownin the figure, the errors do not have a strong correlation with dif-ferent values of slenderness ratios (low R2 values). Nevertheless,Fig. 12(a) shows no change in the error range of f0cc predictionsup to H/D = 2.85. Beyond that, at H/D = 4.0, the model had rela-tively higher errors compared to other H/D ratios. Thus, there isno significant effect of changing cylinder slenderness ratio on themodel predictions.

6.3. Stress–strain predictions

The proposed model predictions, in terms of axial and hoopstress–strain behaviours, were compared to the database experi-mental results. Figs. 13–17 show some of the predicted stress–strain curves with respect to the corresponding experimental oneswhich were provided by Youssf et al. [29], Xiao and Wu [22], Lamand Teng [34], Lam et al. [35], and Teng et al. [37]. In these figures,the positive axial strain values represent compressive strains and

the negative hoop strain values represent tensile strains. As shownin the figures, the proposed model closely predicted the stress–strain path for both axial and hoop strains, particularly with spec-imens in a normal concrete strength range of 30–40 MPa, (seeFigs. 14a, 15a, b, 16a, and 17). Using f0c ranging from 40 to50 MPa resulted in underestimation of the plastic behaviour ofthe axial stress–strain curve as shown in Figs. 13(a) and 14(b).The level of underestimation depended on the confinement ratio.Increasing the value of the confinement ratio decreases the abilityof the model to predict the plastic behaviour slope in the axialstress–strain curve. However, the hoop stress–strain curves werepredicted well. In the case of using high strength concrete (f0c > 50 -MPa), the difficulty of predicting the axial stress–strain plasticbehaviour slope increased, see Figs. 13(b and c) and 14(c).

6.4. Axial-hoop strain response

The experimental versus FE model predictions for the relation-ship between the axial strains and hoop strains of some confinedspecimens are shown in Fig. 18. The dilation rate (R) is also shownfor each specimen in the figure. As shown in the figure, at a givenf0c, increasing the value of the confinement modulus decreased theR value of the confined specimens for both experimental and FEresults. In addition, it decreased the error % in the FE model predic-tions for the R values. For example, by increasing the confinementmodulus by 50% the model error % decreased by 23.1%, 72.1%,

Fig. 18. Axial strain versus hoop strain for confined specimens. (a) f0c = 33.6 MPa, (b) f0c = 43.7 MPa, (c) f0c = 53.5 MPa and (d) f0c = 62.5 MPa.

84 O. Youssf et al. / Engineering Structures 79 (2014) 70–85

59.0%, and 19.2% for concrete with f0c of 33.6 MPa, 43.7 MPa,53.5 MPa, and 62.5 MPa, respectively.

Increasing the f0c value showed no similar trend in the modelpredictions of R values. Increasing f0c by 22.4% increased the modelerror by 65.1%, Fig. 18(b and c). However, increasing f0c by 59.2%decreased the model error by 51.0%, Fig. 18(a and c). It wasobserved that the FE model gives the closest predictions for theaxial–hoop strain response when f0c = 43.7 MPa, Fig. 18(b).

7. Summary and conclusions

This study presents a simple to use empirical model to predictthe FRP-circular-confined-concrete dilation parameter which iscrucial for finite element modelling of concrete encased in FRP.The proposed model can be used with plasticity models to predictthe volumetric behaviour of concrete encased in FRP tubes. Estima-tion of the dilation parameter was first developed through calibra-tion of finite element while results of 35 cylinders to those ofexperimental results. During this process, the difference in tough-ness between the numerical and the experimental results wereminimized to determine the optimum values for the dilationparameters. Once developed, the model was validated against theexperimental results of another 70 cylinders. The main conclusionsof this study are summarized in the following points:

� The proposed model provides a simple FRP-confined concretedesign tool for engineers in practical applications using onlythe main material properties of the concrete and FRP sheets.The proposed model closely predicted the stress–strain pathfor both axial and hoop constitutive relationships, particularlyfor concrete in the conventional strength range between 30

and 40 MPa. Beyond unconfined concrete strength of 40 MPa,the proposed model underestimated the post-elastic behaviourof the axial stress–strain relationships. The level of underesti-mation depended on the confinement ratio. Increasing the valueof confinement ratio decreases the ability of the model to pre-dict the plastic slope in the axial stress–strain curve. However,the hoop stress–strain curves were predicted quite well. Theproposed model is able to predict the volumetric behaviour oflow and high FRP confined concrete.� The proposed model provides conservative predictions of the

ultimate strength. However, it underestimated the ultimateaxial and hoop strains.� The proposed model predicted the ultimate strength, ultimate

axial strain, and ultimate hoop strain with absolute mean errorsand standard deviations of: 2.6% and 10.7% for the ultimatestrength, 0.3% and 29.0% for the ultimate axial strain, and13.7% and 26.3% for the ultimate hoop strain, respectively.� There was no significant effect on the model predictions due to

increasing E1/f0c.� The proposed model is not significantly affected by changing

the cylinders slenderness ratio.

While the proposed model presents significant improvementsover existing models in predicting the confined concrete strengthf0cc, ultimate confined concrete strain Ecc, and FRP ultimate strainEh, the model still needs improvements to reduce the standarddeviation in predicting the ultimate strains of FRP and concrete.A significant contribution to the high values of the standard devi-ation is the high variability and inaccuracy in the experimentalwork data. Improving the quality and quantity of the availableexperimental work would help in developing more accurate

O. Youssf et al. / Engineering Structures 79 (2014) 70–85 85

prediction of concrete dilation parameter. Using more advancedsystems such as image correlation system will certainly improvethe quality of the measured strains. Furthermore, this manuscriptfocused on specimens loaded under pure axial loads, other speci-mens subjected to combined axial and flexural loading, e.g.[55,56], need to be considered for future studies.

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