Modified Johnston Failure Criterion from Rock Mechanics to Predict the Ultimate Strength of Fiber Reinforced Polymer (frp) Confined Columns

The failure criteria from rock mechanics, Hoek-Brown and Johnston failure criteria, may be extended and modified to assess the ultimate compressive strength of axially loaded circular fiber reinforced polymer (FRP)-confined concrete columns. In addition to the previously modified Hoek-Brown criterion, in this study, the Johnston failure criterion is extended to scope of FRP-confined concrete, verified with the experimental data and compared with the significant relationships from the current literature. Wide-range compressive strengths from 7 to 108 MPa and high confinement ratios up to 2.0 are used to verify the ultimate strengths in short columns. The results are in good agreement with experimental data for all confinement levels and concrete strengths.


Introduction
Fiber reinforced polymer (FRP) composites are increasingly being applied for the seismic retrofitting and strengthening of reinforced concrete structures.Currently, FRPs are primarily used for two types of applications.One is a thin layer of FRP jacket applied for seismic rehabilitation of damaged and undamaged reinforced concrete structures, and another is the application of FRP tubes in rebuilding and new construction.FRP composites are suitable for use in coastal and marine structures as well as civil infrastructure facilities due to their properties such as high strength-to-weight ratio, high-tensile strength and modulus, corrosion resistance and durability.FRP confinement provides superior seismic performance with enhancing lateral confinement level, energy absorption capacity and ductility [1][2][3][4][5].Nowadays, new types of cheaper and eco-friendly materials (e.g., recycled pet bottles, etc.) are investigated [6] in addition to common FRP materials (carbon, glass or aramid fiber reinforced plastics; CFRP, GFRP, AFRP, respectively).
Most empirical confinement models address the Mohr-Coulomb failure criterion for actively (hydrostatic pressure) or passively (steel, FRP) confined concrete [3,[7][8][9][10][11][12][13][14].This study focuses on two failure criteria from rock mechanics, the Hoek-Brown's [15] and especially the Johnston's failure criteria [16], for FRP-confined concrete.The Mohr-Coulomb criterion is valid only in the compression region [17], whereas Hoek-Brown's and Johnston's failure criteria from rock mechanics exist both in compressive and tensile regions to complete it (Figure 1).Previously, in order to extend the scope to concrete, the Hoek-Brown's and Johnston's criteria were successfully verified to predict the ultimate strength in high strength (60-132 MPa) concrete specimens under active confining pressure [17].More recently, the Hoek-Brown criterion was applied to reinforced concrete and FRP-confined circular columns by the author [18].This study addresses the applicability of other failure criterion, the Johnston failure criterion, from rock mechanics.The criterion is validated for FRP-confined circular short columns through the wide-range experimental data (7 to 108 MPa) from the current literature.The comparisons confirm the applicability of the Johnston failure criterion to FRP composites as well.

Overview to Confinement Models with FRP
Under triaxial compressive stresses, the columns are subjected to major compressive stresses σ 1 uniformly applied along the axial axis of the column and lateral confining pressure σ 3 (Figure 1).This pressure may be provided by passive means such as steel (hoops, ties, spirals, jackets, etc.) and FRP (sheets, tubes, etc.) confinement around the concrete core or actively through hydrostatic pressure.On the contrary of steel, FRP behaves elastically until failure.The inward radial pressure increases with the lateral expansion of the concrete, so that the assumption of constant confining pressure is no longer valid.The models [7,8,[46][47][48] developed for steel confinement may unsafely overestimate the strength of FRP-confined columns.
In FRP confined columns, the fibers are generally aligned primarily along the hoop direction to provide the confinement of concrete while the fibers in the axial direction provide the flexural strength and stiffness.While the concrete is stressed triaxially, the FRP jacket is loaded uniaxially and, at the FRP-concrete interface, the confining radial pressure σ 3 (Figure 2) develops by: where t, D, σ frp denote the thickness of FRP, the diameter of concrete core and the hoop tensile strength of FRP, respectively.The Mohr-Coulomb failure criterion frequently used for confined concrete is essentially based on triaxial soil data: here σ 1 signifies the major principal stress at failure (ultimate strength), σ 3 is the minor principal stress (confining pressure), c is the cohesive strength of soil, and  is the internal-friction angle.Unconfined strength (f l = σ 3 = 0) is defined as: The positive effect of confinement on concrete cylinders was first observed and modelled by Richart et al. [7] by defining the internal-friction angle  as 37°.Then Goodman [49] suggested  in the range of 36°-45° for most of concrete strengths, Rochette and Labossière formulated  and c [30].
The confinement effectiveness coefficient k is defined in terms of  : and Equation ( 2) may be expressed to assess the ultimate strength f cc of confined concrete: and often in the following normalized form: where f cc /f co is described as the strengthening ratio.Richart et al. [7] who suggested k value as 4.1 corresponding to  = 37° and then many authors have widely used these forms [8][9][10][11][12][13][14].
Saatcioglu and Razvi [47] found that the coefficient k decreases with increasing confining pressure by approaching a constant value in high lateral stresses.While Candappa et al. [50] proposed k = 5.3 for low confinement levels, Ansari and Li [10] found k = 2.6 for high confinement levels.Dahl [51] has shown that the traditional value of k = 4.1 overestimates the ultimate strength for the confinement ratios exceeding 0.5.For FRP-confined concrete, while some authors suggested a constant value for k [6,14,22,[52][53][54], according to other authors, k should be a function decreasing with the confinement ratio [3,[11][12][13]20,55].
As a great number of relationships according to Mohr-Coulomb criterion have been proposed so far, in this study, only the variation of k is displayed for FRP-wrapped or FRP tube encased specimens.For the averaged database of this study, two significantly different cases are observed especially for FRP-wrapped specimens (Figure 3).In the first case, a specific trend with a dashed line is under consideration and while k value is high in low confinement levels it declines toward a constant value of 1.8 in high confinement levels.For the second case, there are data scattering in the confinement ratios lower than about 0.7.It may be suggested that k has either a conservative value of about 2 or lower and variable values within the dotted curve.The dotted line converges towards the first trend at medium confinement levels.Spoelstra and Monti [13] previously defined lower effectiveness for f l /f co < 0.07 and expressed that the confinement effectiveness is never greater than 3 and that it reaches maximum at about f l /f co < 0.3.According to Li [56], both insufficient coefficient and higher concrete strength may lead to lower confinement effectiveness.Recently Teng et al. [57] defined f l /f co to be the product of the confinement stiffness ratio and strain ratio.In the literature, k coefficient was expressed in terms of normalized lateral jacket rigidity [24], confinement pressure nonlinearly [28], confinement pressure and cylinder concrete strength nonlinearly [3,12,13], and more recently normalized axial rigidity [20].

Hoek-Brown Failure Criterion
Hoek and Brown [15] introduced a failure criterion for rocks (σ c ≥ 20 MPa): where σ c is the uniaxial-unconfined-compressive strength of intact rock specimens, m and s are the material constants.The value of m that governs the curvature of failure envelope depends on the type of rock, e.g., the representative value of m is 15 for sandstone and quartzite.The other constant s describes the discontinuities in rock and ranges from 0 (heavily jointed rocks) to 1 (intact rocks).
For confined concrete, it can be shown that Equation ( 7) may take the form: Equation ( 8) can further be expressed as [17]: where

B sf 
Concrete containing no discontinuity can be treated as intact rock material (s = 1).Using f l and f cc values resulting from triaxial tests on confined concrete, Y and X values are determined.Then, the material constant m may be assigned from the linear regression analysis between Y and X values and thus the failure envelope is described.In the pure tension case f cc = 0 → f l = −f t where f t is the uniaxial-direct-tensile strength (Figure 1).
In this failure criterion adapted from rock mechanics, the first step is to precisely predict m constant.The predicted m values regarding the active and FRP confinement are displayed according to the strength ranges in Table 1.For FRP confined concrete, while m is 4.8 to 3.3 in normal strength range (20 to 40 MPa), it has a very low value (m = 0.1) in the high-strength concrete especially over 80 MPa [18].Meanwhile, it should be mentioned for cylinder strengths over 108 MPa the accuracy of prediction may be decreased due to very low and constant m coefficient.As for actively confined concrete, the highest m value (m = 13) is under consideration [17] and approaches to the lower range of rocks [15].The variations [18] between confinement effectiveness (strengthening ratio) and confinement ratio are demonstrated for differently confined concrete (FRP, steel, FRP + steel) and rock specimens in Figure 4.

Johnston's Failure Criterion
Johnston's failure criterion [16] is valid for all intact soils and rocks under triaxial compression ranging from 0.008 MPa (lightly overconsolidated clays) to 600 MPa (extremely hard rocks).It is defined as: where M and B correspond to the material coefficients.B defines the nonlinearity of the strength criterion and M describes the slope of the failure envelope at f l = 0.They are reasonably related to the compressive strength σ c .Johnston expressed B coefficient as a parabolic curve: , (0.008 MPa  σ c  600 MPa) (11) in which the unit of σ c is kPa.It can be realized that B converges to 1 (σ c → 1) for normally consolidate clays, to 0.5 for rocks.Johnston identified M coefficient with parabolic curves in four groups regarding geomaterial formations.
If the Johnston criterion is extended to confined concrete, Equation ( 10) may be adapted for confined concrete as follows: B coefficient minimizing the IAE ratio was previously determined to be 0.5 for high strength concrete (f co = 60-132 MPa) under active pressure (Table 2) and M/B ratio was defined in terms of splitting tensile and compressive strength [17] and this ratio seems to be a function of the uniaxial compressive strength f co and the type of rock.In this study, the Johnston criterion is modified and extended to FRP-confined concrete.Herein, Equation ( 13) derived from the classical relationship Equation ( 11) is employed for B coefficient by covering all the strength ranges: For M coefficient, the following correlations were developed with min.IAE ratios and deviations: 2 0 0035 0 056 2 83 Thus, by knowing B and M coefficients, the failure envelope can be established easily.For the strength levels lower than about f co = 25 MPa, M values decreases from 3.8 down to about 2.6 (f co = 7.3 MPa).From 25 MPa to upper strength levels, M values gradually decrease, e.g., M coefficient is 0.75 for f co = 108 MPa.

Evaluations and Comparisons of Modified Johnston Failure Criterion
In this section, the prediction capability of modified Johnston criterion will be verified through the averaged database (n = 135) and with the current models in Table 3. Data were classified according to strength ranges as groups and all the models were individually investigated according to these ranges to be independent from the definition range of the model.The number of data for each strength range is 24, 21, 41, 37, 12, respectively.
The Integral Absolute Error (IAE) previously defined [17,18] and average deviations (  ) were used in comparisons.When comparing different models, the smallest value of the IAE can be judged as the most reliable one.IAE ratio 10% may be regarded as the limit for a acceptable prediction.
High IAE and  ratios of the models developed for steel confinement [7,[46][47][48] usually indicate an over-estimation for FRP confined concrete.In FRP models, the IAE ratio usually increases in high-strength concrete especially for f co  70 MPa [3,4,[11][12][13]32,50,54,57,60] or in poor strength levels lower than 20 MPa [4,12,22,55,60].Karbhari and Gao [3] and Saafi et al.'s models [11] have good assessment capability beyond the strength range as well.Within the models based on Mohr-Coulomb criterion, the most reliable results for constant k coefficient are provided with k = 2 [32,53,54] for the range 7-52 MPa.Rousakis et al.'s model [20] was individually defined for carbon and glass jackets, and in this study the predictions were executed only for carbon sheets with different elastic modulus and very good accuracy was achieved especially for f co  30 MPa.Modified Johnston criterion, similar to previously modified Hoek-Brown criterion, yields the best prediction with the smallest IAE ratios (4.7%) and deviations (−4.8%, +4.1%) (Tables 1-3) for all the strength ranges (f co = 7 to 108 MPa) from low to high confinement ratios.

Figure 1 .
Figure 1.Mohr-Coulomb, Johnston and Hoek-Brown's failure envelope to estimate the ultimate strength under triaxial compression.

Figure 3 .
Figure 3.The variation of confinement efficiency in FRP-wrapped and FRP-encased specimens.

Figure 4 .
Figure 4. Failure envelopes of intact rock specimens and concrete confined by different confining materials.Reprinted with permission from [18].Copyright 2009.ACI.

a
Analysis was carried out by taking co co f f   , co f   in-place unconfined compressive strength of concrete [the ratio of unconfined strength in-place; co f  in the column to standard cylinder strength f co is generally taken as 0.85]; b  f = 4t f /d, E f = 10 MPa (for units compliance); α = −0.336,β = 0.0223 for FRP sheets ; α = −0.23,β = 0.0195 for FRP tube;Y = f cc /f co − 1, X =  f E f /f co , Y = AX, carbon: A = 0.0151 (E f = 234 GPa), 0.0093 (E f =377GPa), 0.0021 (E f = 640 GPa), glass: A = 0.0187 (E f = 80.1 GPa);c In this study, predictions was carried out for only carbon sheets 234, 377 and 640 GPa.

Figure 8 .
Figure 8. Predicted versus experimental ultimate compressive strengths according to modified Johnston criterion.

Table 2 .
The variation of predicted B and M coefficients.

Table 3 .
Prediction of ultimate compressive strength in FRP-confined concrete.