Jung-Hyun Ahn
email: jung-hyun.ahn@neoma-bs.fr

Pierre Six
email: pierre.six@neoma-bs.fr

A study of first generation commodity indices: Indices based on financial diversification $

Keywords: commodity indices, investment, diversification, commodity futures, JEL: G11, G12, G13 commodity indices, investment, diversification, commodity futures, JEL: G11, G12, G13

This study compares first generation, i.e. long-only passive, commodity indices based on financial diversification criteria, such as equal weight, equal risk contribution and global minimum variance, to the main first generation commodity indices used as vehicles for investment, namely the S&P GSCI (GSCI) and the BCOM. The GSCI and the BCOM are mainly computed according to the world individual commodity production. We find that commodity indices based on financial diversification offer better Sharpe ratios, lower volatilities, and lower correlation with bonds and equity. Finally, we provide evidence for a low-volatility anomaly of the commodity market.

Introduction

Standard and Poor's Dow Jones (SP GS) has recently launched two commodity indices based on financial diversification: an Equal Weight Index in September 2010 (GSEW) and an Equal Risk Contribution Index in April 2013 (GSRW). The S&P GSCI (GSCI) and, to a lesser extent, the BCOM 1 are the two main indices used for investment and benchmark [START_REF] Sanders | Measuring index investment in commodity futures markets[END_REF][START_REF] Bianchi | Combining momentum with reversal in commodity futures[END_REF]. The compositions of these two indices are mainly computed based on world production weights of individual commodities. The goal of our study is to compare the indices based on financial diversification to those based on the world commodity production. Our article thus belongs to the growing literature on commodity indices [START_REF] Miffre | Long-short commodity investing: A review of the literature[END_REF][START_REF] Daskalaki | Diversification benefits of commodities: A stochastic dominance efficiency approach[END_REF][START_REF] Fernandez-Perez | The skewness of commodity futures returns[END_REF]. We restrict our study to first generation commodity indices, i.e., long passive investment in commodities. 2 The introduction of the GSEW and GSRW reflects the increasing demand for investment in commodity indices in the 2000s [START_REF] Basak | A model of financialization of commodities[END_REF], which has triggered an increase in the size of the commodity futures markets [START_REF] Bhardwaj | Facts and fantasies about commodity futures ten years later[END_REF]. This socalled financialization period of the commodity market [START_REF] Tang | Index investment and the financialization of commodities[END_REF] has been accompanied by renewed interest in the study of investment strategies in commodity markets (e.g., [START_REF] Miffre | Momentum strategies in commodity futures markets[END_REF][START_REF] Fuertes | Tactical allocation in commodity futures markets: Combining momentum and term structure signals[END_REF][START_REF] You | A markowitz optimization of commodity futures portfolios[END_REF][START_REF] You | A markowitz optimization of commodity futures portfolios[END_REF], recently, Fan et al., 2018;[START_REF] Fernandez-Perez | The skewness of commodity futures returns[END_REF].

While these studies have mainly focused on strategies based on signals and/ or longshort positions, the so-called second and third generations of commodity indices respectively, less attention has been paid to the first generation commodity indices, i.e., long passive investments, in spite of their relevance to the commodity market. On the one hand, the majority of institutional investors have long-only mandates and were primary contributors to commodity financialization [START_REF] Sanders | Measuring index investment in commodity futures markets[END_REF]; 3 the bulk of the investment is carried out through the GSCI index [START_REF] Basu | The performance of simple dynamic commodity strategies[END_REF]. On the other hand, the construction of first generation indices is still underdeveloped in the commodities market 1 Formerly the Dow Jones-UBS Commodity Index (DJ-UNSCI).

2 Most studies of commodity indices have investigated second generation i.e., active strategies, and/ or third generation, active strategies with possible short positions (e.g., [START_REF] Bianchi | Combining momentum with reversal in commodity futures[END_REF][START_REF] Cotter | Predictability and diversification benefits of investing in commodity and currency futures[END_REF][START_REF] Fuertes | Commodity strategies based on momentum, term structure and idiosyncratic volatility[END_REF][START_REF] Daskalaki | Diversification benefits of commodities: A stochastic dominance efficiency approach[END_REF]. However, long investment in commodity indices, such as the GSCI and the BCOM, constitute an important part of commodity investment, as implied by [START_REF] Sanders | Measuring index investment in commodity futures markets[END_REF]. Studies that focus on diversification inside the commodity universe are scarce. A recent study of this topic was conducted by [START_REF] Bernardi | Maximum diversification strategies along commodity risk factors[END_REF]; however, the strategies in [START_REF] Bernardi | Maximum diversification strategies along commodity risk factors[END_REF] are built on concepts such as minimum torsion and entropy. Because of their complexity, we rule them out as potential candidates for first generation commodity indices.

3 [START_REF] Adams | Financialization in commodity markets: A passing trend or the new normal?[END_REF]: "Conservative estimates show that from 2000 to 2010 the number of commodity index traders, i.e. long-only investors such as pension funds and insurance companies, more than quadrupled."

compared to the equity market; major commodity indices providers do not publish indices based on global minimum variance, 4 whereas equity indices suppliers such as FTSE Russell edit indices based on such objectives for various equity markets. In addition, recent studies outline the importance of minimum variance as an objective for investors [START_REF] Bednarek | Understanding the outperformance of the minimum variance portfolio[END_REF].

Against this backdrop, we investigate whether it adds value to invest in these new first generation indices, computed according to financial diversification objectives, as explained in [START_REF] Demey | Risk-based indexation[END_REF]. For this purpose, we construct the three following commodity indices: 5 Equal Weight (EW), Equal Risk Contribution (ERC) 6 and Global Minimum Variance (GMV). The building of our own indices based on financial diversification enables robustness checks for the following reasons. The GSEW (GSRW) is backdated to only mid-January (mid-February) 1995, which is a relatively short period of time for the monthly analyses that are typical in such studies [START_REF] De Miguel | Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy?[END_REF]. 7 These two indices are not directly comparable to the GSCI and BCOM: the number of commodities in the GSEW ( 14) is small compared to the numbers in the GSCI (24) and the BCOM ( 22), 8 and the risk contributions of the GSRW are the risk contributions of the sector subindices of the GSCI, not the risk contribution of individual commodities. For additional robustness checks, we include the light energy version of the GSCI: the GSLE.

We compare the Sharpe ratios, volatilities and correlations with the S&P 500 and the U.S. Treasury 10-year bond (hereafter T10YR) in our set of first generation indices. We conduct our study for the periods for which all the indices are available, beginning in March 1995. We find that the indices based on financial diversification offer better Sharpe ratios, 4 Global Minimum Variance indices do exist in the form of Eexchange Traded Funds (ETF), as demonstrated by the introduction in 2016 of such an ETF by Direxion, an American mutual funds and Exchange Traded Funds provider.

5 [START_REF] Demey | Risk-based indexation[END_REF] provide an additional diversification strategy: the most-diversified portfolio. However, this strategy requires the use of the Sharpe ratio. It is thus a directional strategy that qualifies for the second generation set of commodity indices. In addition, it is well known that the Sharpe ratio is hard to pin down, as opposed to the estimation of the variance-covariance matrix, i.e., volatilities and correlations. 6 The composition of Equal Weight is a long-standing issue in investment, as explained by De Miguel et al.

( 2009). The Equal Risk Contribution is a much more recent financial diversification objective, as explained by [START_REF] Maillard | On the properties of equally-weighted risk contributions portfolios[END_REF]. The risk contribution of an individual asset in a portfolio is its contribution to the volatility of the portfolio; the mathematical details are given in section 2 of this article.

7 We obtain EW, ERC and GMV data from August 1970 onward because the GSCI data began in January 1970; we need 150 returns from that point forward to compute the variance-covariance matrix. We end our study in August 2017. 8 The GSCI and BCOM indices are not identical. The GSCI and BCOM comprise 24 and 22 individual commodities, respectively; 19 of these commodities are common to both indices. We do not see any reason to favour one particular universe of the two commodity indices. Therefore, we consider the union of the two commodity universes, and our commodity universe comprises 27 individual commodities at the end of our study period.

lower volatilities and lower correlations with equity and bonds. Our results are not affected by the various kinds of robustness checks: analyses with the periods covering the full range of the GSCI and the BCOM, since January 1970 and 1991, respectively; and sub-sample analysis breaking our studying period down into financial expansionary and recessionary cycles [START_REF] Bessler | Do commodities add value in multi-asset portfolios? an out-of-sample analysis for different investment strategies[END_REF]. The remainder of the article is organized as follows. Section 2 gives a technical account how our commodity indices are built. Section 3 summarizes our results for the period starting in March 1995 (the backdating of the GSRW). Section 4 presents the robustness checks. Section 5 offers some concluding remarks.

The construction of the ERC, EW and GMV indices

Let us denote by Ω nt the time-t variance-covariance matrix of commodity futures returns, where n stands for the number of commodities in the portfolio. We consider the individual commodities present in the GSCI and BCOM and build our indices backward by including these futures contracts when they are available in the futures market. The number of individual commodity futures contracts, n, ranges from 10 at the beginning of our study in August 1970 to 27 in August 2017. 9 The prices of the futures contracts were downloaded from the Bloomberg database. We estimate the variance-covariance matrix using historical moments with 150 past daily commodity futures returns. 10Let us denote by π

(GM V ) nt and π (ERC) nt
the time-t n-dimensional portfolios of the GMV and ERC indices invested in the commodity futures markets. According to [START_REF] Demey | Risk-based indexation[END_REF], these portfolios are solutions to the following constrained optimization programs:

   π (GM V ) nt = argmin πnt π nt Ω nt π nt s.t. π (GM V ) nt 1 n = 1, π (GM V ) nt ≥ n 0 n ; (1)    π (ERC) nt = argmin πnt i,j (RC it -RC jt ) 2 s.t. π (ERC) nt 1 n = 1, π (ERC) nt ≥ n 0 n , (2)
where 1 n (0 n ) is the n-dimensional vector of ones (zeros) and ≥ n is the n-dimensional component-wise inequality sign. RC it denotes the marginal contribution to the risk of the portfolio of the commodity futures contract i. It is proportional to the marginal contribution of the futures contract i to the volatility of the portfolio: RC it ≡ π int ∂σ (π nt ) /∂π int , where π int is the weight of asset i in the portfolio π nt . The Equally Weighted portfolio is given simply by π

(EW ) nt = (1/n)1 n .
The ERC, EW and GMV indices then start on July 1970 and are normalized at that date to the value of 100. With obvious notations, the variation of an index I between two trading dates is computed as follows:

I (J) t+1 = I (J) t + I (J) t n i=1 π (J) int F i(t+1) -F it F it , J ∈ {ERC, EW, GM V } , (3)
where F it is the time t futures price of the commodity associated with the weight i. We focus on the Sharpe ratio of indices, their volatilities and their correlation with bonds and equity. As such, we do not need to collateralize our indices. The futures price, F it , is defined as the nearest futures contracts; when a futures contract expires, the second nearest futures contract is used. We ensure that our strategies are attainable by an investor in the commodity futures markets. The composition of an index is changed only when a futures contract expires.

The empirical comparison of first generation commodity indices

This section compares all the first generation commodity indices: the BCOM, ERC, EW, GMV, GSCI, GSLE, GSEW, GSRW and GSLE. The period starts in March 1995 (the backdating of the GSRW) and includes 270 monthly returns until August 2017. We start with some summary statistics in Table 1. Table 1 demonstrates that the indices computed after the risks of the commodity futures contracts, namely, the GMV, ERC, EW, GSEW and GSRW, have higher Sharpe ratios than do the indices based on commodity world production, the BCOM, GSCI and even the GSLE. Table 1 shows that the Sharpe ratios of the commodity indices are lower than the Sharpe ratio of the S&P 500. This result is unusual in the literature, as shown by [START_REF] Bhardwaj | Facts and fantasies about commodity futures ten years later[END_REF], and is related to our short data set.11 Table 1 also shows that the correlation between commodity indices and the S&P 500 (T10YR) is low and positive (negative). This low correlation with other asset classes is typical in the literature [START_REF] Bessler | Do commodities add value in multi-asset portfolios? an out-of-sample analysis for different investment strategies[END_REF].

[INSERT TABLE 1 AROUND HERE]

Table 2 displays the difference in Sharpe ratios and tests its statistical significance according to the test of [START_REF] Memmel | Performance hypothesis testing with the sharpe ratio[END_REF], as in [START_REF] Basu | The performance of simple dynamic commodity strategies[END_REF]. Table 2 presents some interesting features: the differences in Sharpe ratios between the GMV, ERC, EW, GSEW and GSRW, on the one side, and the BCOM, GSCI and GSLE, on the other side, are all positive. Further, they are almost all statistically significant when the differences are computed with respect to the BCOM and GSLE. 12

[INSERT TABLE 2 AROUND HERE] [INSERT TABLE 3 AROUND HERE]
Table 3 compares the ratios of volatilities of the commodity indices and analyses their statistical significance using the Pitman-Morgan test. 13 First, Table 3 shows the (very) low volatility of the GMV. By construction, the volatility of the GMV is lower than that of other commodity indices. However, the theory does not indicate the extent to which this volatility is lower or statistically significant. All the ratios are highly statistically significant, and the ratio of the volatility of the GMV and the volatilities of the BCOM, GSCI and GSLE is, at most, 56.3%. Moreover, the volatility of the GMV is always less than 80% that of other indices. Second, Table 3 shows that the indices based on equal risk weighting perform second best in terms of volatility, after the GMV. The indices based on equal weighting perform third best. 14 Finally, the indices based on world commodity production perform very poorly in terms of volatility. The evidence supports a low volatility anomaly for the commodity markets: indices with low volatility tend to offer better Sharpe ratios than do indices with high volatility.

[INSERT TABLE 4 AROUND HERE] [INSERT TABLE 5 AROUND HERE]

Table 4 (5) shows the differences in the correlations of the commodity indices with the S&P 500 (T10YR). 15 The statistical significance of these differences relies on the insight of [START_REF] Meng | Comparing correlated correlation coefficients[END_REF]. Table 4 presents a striking result: the correlation of the GMV index with the S&P 500 is lower than that of any other commodity index. In addition, the difference is very significant, except for the correlation with the GSCI. 16 As far as the correlations of the commodity indices with the T10YR are concerned, the ERC and EW indices offer the best perspective: both have lower correlations with the T10YR than they have with other indices, with statistical significance outside the GSLE and GSCI indices. It is worth noting that their SP GS counterparts, namely the GSEW and GSRW, do not offer such results.

12 The exception is the difference computed between the GMV and GSLE. 13 Del Brioa et al. (2018), for example show the importance of studying commodity volatility. 14 The EW built on more commodities performs better in terms of volatility than the GSEW based on fewer commodities. 15 The reader can refer to [START_REF] Narayana | How do bond, equity and commodity cycles interact?[END_REF] for economic analysis between the commodity, bond and equity markets 16 Interestingly, the GSCI is the second-best index in terms of its correlation with the S&P 500. The GSLE also displays a low, but not statistically significant, correlation with the S&P 500.

Robustness checks

In this section, we extend our study to cover a longer backdated data set of the GSCI and GSLE. The five indices available for this period are then the ERC, EW, GMV, GSCI and GSLE. The study period starts in August 1970, with a dataset of 565 monthly returns. The results provided by this robustness check are in line with those of section 3, with the following minor differences. The Sharpe ratios of the ERC, EW and GMV are of the same order of magnitude as the S&P 500 [START_REF] Bhardwaj | Facts and fantasies about commodity futures ten years later[END_REF], whereas those of the GSCI and GSLE are approximately half those of the Sharpe ratios. The positive statistical differences in the Sharpe ratios between the indices based on financial diversification and those based on the production of commodities are greatly improved. The correlations of the commodity indices with the S&P 500 are lower than are the correlations in the short period.

We also extend our analysis of correlation with equity to other stock markets such as Canada (S&P TSX), France (CAC 40), Germany (DAX 30), Japan (NIKKEI 225) and United-Kingdom (FTSE100). 17 We convert all indices in U.S. dollars and obtain similar results: the correlations of the GMV with the equity markets are lower than the correlations of other commodity indices with the same equity markets. The difference in correlations is also statistically significant -the significance is reduced for the NIKKEI 225, though.

We perform additional robustness checks. First, we test our results for the period covered by the backdating of the BCOM to January 1991. The results offered by this period are in line with those of section 3 and those offered by the period covered by the GSCI explained in the preceding paragraph. We also break our sample down into a financial expansionary (recessionary) subperiod [START_REF] Belousova | On the diversification benefits of commodities from the perspective of euro investors[END_REF][START_REF] Bessler | Do commodities add value in multi-asset portfolios? an out-of-sample analysis for different investment strategies[END_REF]. 18 In sum, the results of our paper remain unchanged in expansionary and recessionary financial cycles. Finally, while a complete study of transaction costs is outside the scope of this short article, our indices based on financial risks should not produce abnormal transaction costs for the following reasons. The transaction costs in the commodity futures markets are low [START_REF] Bianchi | Combining momentum with reversal in commodity futures[END_REF]. Our results are valid for our own indices: the EW, ERC and GMV. The construction of these indices minimizes the transaction costs since a change in portfolio is only made when a futures contract expires and is then necessary.

Conclusion

In 2010 and 2013, Standard and Poor's Dow Jones launched two first generation commodity indices based on equal weight and equal risk contribution [START_REF] Demey | Risk-based indexation[END_REF]. 17 We thank an anonymous referee for this suggestion. 18 We follow the same methodology used in [START_REF] Bessler | Do commodities add value in multi-asset portfolios? an out-of-sample analysis for different investment strategies[END_REF] to distinguish an expansionary period from a recessionary period: we combine monetary and stock market signals based on the 24-month moving averages of the S&P 500 and the federal funds target rate.

This article compares these indices based on financial diversification with the two indices mainly used for investment: the GSCI and BCOM. The GSCI and BCOM are based on the world production of commodities. We introduce another type of first generation index based on global minimum variance. Interestingly, such indices do not exist for the commodity markets, while they are largely edited in the equity market. Our study demonstrates that first generation commodity indices based on financial risk diversification offer better Sharpe ratios, lower volatilities and lower correlations with equity and bonds than do the two main indices used for investment based on commodity production. Our study thus encourages economic agents to invest in indices based on financial risk diversification and index providers to launch an index based on global minimum variance for the commodity market.

Table 1 :

 1 Summary statistics of monthly return for indices

		GMV	ERC	EW	BCOM GSCI GSEW GSRW GSLE T10YR S&P500
	Mean (%)	3.104	4.476	5.379	0.654	5.878	6.447	5.128	3.753	2.535	8.573
	Vol. (%)	8.771 10.988 13.930 15.584 21.727 15.786 11.324 15.722	8.682	14.692
	Sharpe ratio	0.354	0.407	0.386	0.042	0.271	0.408	0.453	0.239	0.292	0.584
	Corr. (S&P500) 0.190	0.371	0.382	0.318	0.253	0.355	0.311	0.320	-0.228	1.000
	Corr. (T10YR) -0.067 -0.157 -0.159 -0.099 -0.134 -0.120	-0.094 -0.135	1.000	-0.228
	Obs.	270	270	270	270	270	270	270	270	270	270
	Note: This table provides the summary statistics of monthly returns from March 1995 to
	August 2017.

Table 2 :

 2 Comparison of Sharpe ratiosThis table provides the difference of the row and the column indices' Sharpe ratio, with Z-value below. * * * , * * , and * indicate significance at the 1%, 5%, and 10% levels when the difference in the Sharpe ratios is positive. For example, the value of 0.097 with *** in the third row and fourth column means that the Sharpe ratio of EW is greater than that of BCOM and that this is significant at the 1% level. The sample period is March 1995 to August 2017.

		GMV	ERC	EW	BCOM	GSCI	GSEW	GSRW	GSLE	T10YR	S&P500
	GMV	-	-0.015	-0.008	0.089 *	0.025	-0.014	-0.027	0.033	0.017	-0.061
		-	(-0.377) (-0.155) (1.528)	(0.382) (-0.237) (-0.598) (0.588)	(0.196) (-0.789)
	ERC	0.015	-	0.006	0.103 * * *	0.039	0.001	-0.012	0.048 *	0.032	-0.047
		(0.377)	-	(0.365)	(3.355)	(0.877)	(0.023) (-0.451) (1.576)	(0.344) (-0.683)
	EW	0.008	-0.006	-	0.097 * * *	0.033	-0.006	-0.019	0.041 * *	0.026	-0.053
		(0.155) (-0.365)	-	(4.429)	(0.885) (-0.253) (-0.607) (1.810)	(0.275) (-0.783)
	BCOM	-0.089	-0.103	-0.097	-	-0.064	-0.102	-0.116	-0.056	-0.071	-0.150
		(-1.528) (-3.355) (-4.429)	-	(-2.265) (-4.907) (-3.489) (-2.827) (-0.789) (-2.101)
	GSCI	-0.025	-0.039	-0.033	0.064 * *	-	-0.038	-0.052	0.008	-0.007	-0.086
		(-0.382) (-0.877) (-0.885) (2.265)	-	(-1.241) (-1.181) (0.353) (-0.079) (-1.152)
	GSEW	0.014	-0.001	0.006	0.102 * * *	0.038	-	-0.013	0.047 * * *	0.031	-0.048
		(0.237) (-0.023) (0.253)	(4.907)	(1.241)	-	(-0.419) (2.517)	(0.342) (-0.685)
	GSRW	0.027	0.012	0.019	0.116 * * *	0.052	0.013	-	0.060 * *	0.044	-0.034
		(0.598)	(0.451)	(0.607)	(3.489)	(1.181)	(0.419)	-	(2.106)	(0.492) (-0.479)
	GSLE	-0.033	-0.048	-0.041	0.056 * * *	-0.008	-0.047	-0.060	-	-0.016	-0.094
		(-0.588) (-1.576) (-1.810) (2.827) (-0.353) (-2.517) (-2.106)	-	(-0.170) (-1.324)
	T10YR	-0.017	-0.032	-0.026	0.071	0.007	-0.031	-0.044	0.016	-	-0.079
		(-0.196) (-0.344) (-0.275) (0.789)	(0.079) (-0.342) (-0.492) (0.170)	-	(-0.824)
	S&P500	0.061	0.047	0.053	0.150 * *	0.086	0.048	0.034	0.094 *	0.079	-
		(0.789)	(0.683)	(0.783)	(2.101)	(1.152)	(0.685)	(0.479)	(1.324)	(0.824)	-
	Note:

Table 3 :

 3 Comparison of volatilities indicate significance at the 1%, 5%, and 10% levels. For example, the ratio 0.798 with *** in the first row, second column means that the ratio of volatilities between GMV and ERC is different from 1 and that this is significant at the 1% level; in other words, the volatility of GMV is significantly lower than that of ERC. The sample period is March 1995 to August 2017.

		GMV	ERC	EW	BCOM	GSCI	GSEW	GSRW	GSLE	T10YR S&P500
	GMV	-	0.798 * * * 0.630 * * * 0.563 * * * 0.404 * * * 0.556 * * * 0.775 * * * 0.558 * * *	1.010	0.597 * * *
		-	(6.225) (10.169) (11.879) (18.896) (12.170) (6.166) (12.378) (0.167)	(8.987)
	ERC	1.253 * * *	-	0.789 * * * 0.705 * * * 0.506 * * * 0.696 * * *	0.970	0.699 * * * 1.266 * * * 0.748 * * *
		(6.225)	-	(13.748) (12.064) (17.709) (12.292) (1.120) (12.555) (3.940)	(5.193)
	EW	1.588 * * * 1.268 * * *	-	0.894 * * * 0.641 * * * 0.882 * * * 1.230 * * * 0.886 * * * 1.604 * * *	0.948
		(10.169) (13.748)	-	(5.311) (13.025) (5.642)	(6.935)	(5.456)	(8.134)	(0.944)
	BCOM 1.777 * * * 1.418 * * * 1.119 * * *	-	0.717 * * *	0.987	1.376 * * *	0.991	1.795 * * *	1.061
		(11.879) (12.064) (5.311)	-	(12.329) (0.637) (10.257) (0.455) (10.182) (1.019)
	GSCI	2.477 * * * 1.977 * * * 1.560 * * * 1.394 * * *	-	1.376 * * * 1.919 * * * 1.382 * * * 2.502 * * * 1.479 * * *
		(18.896) (17.709) (13.025) (12.329)	-	(10.847) (17.122) (14.233) (17.369) (6.791)
	GSEW 1.800 * * * 1.437 * * * 1.133 * * *	1.013	0.727 * * *	-	1.394 * * *	1.004	1.818 * * *	1.074
		(12.170) (12.292) (5.642)	(0.637) (10.847)	-	(11.143) (0.222) (10.456) (1.259)
	GSRW 1.291 * * *	1.031	0.813 * * * 0.727 * * * 0.521 * * * 0.717 * * *	-	0.720 * * * 1.304 * * * 0.771 * * *
		(6.166)	(1.120)	(6.935) (10.257) (17.122) (11.143)	-	(12.109) (4.419)	(4.536)
	GSLE	1.792 * * * 1.431 * * * 1.129 * * *	1.009	0.724 * * *	0.996	1.388 * * *	-	1.811 * * *	1.070
		(12.378) (12.555) (5.456)	(0.455) (14.233) (0.222) (12.109)	-	(10.397) (1.172)
	T10YR	0.990	0.790 * * * 0.623 * * * 0.557 * * * 0.400 * * * 0.550 * * * 0.767 * * * 0.552 * * *	-	0.591 * * *
		(0.167)	(3.940)	(8.134) (10.182) (17.369) (10.456) (4.419) (10.397)	-	(9.257)
	S&P500 1.675 * * * 1.337 * * *	1.055	0.943	0.676 * * *	0.931	1.297 * * *	0.934	1.692 * * *	-
		(8.987)	(5.193)	(0.944)	(1.019)	(6.791)	(1.259)	(4.536)	(1.172)	(9.257)	-

Note: This table provides the ratio of row index to column index volatility, with t-value below. * * * , * * , and *

Table 4 :

 4 Comparison of coefficients of correlation with S&P 500 This table provides the difference, between the row and column indices, of the mean of the coefficient of correlation with the S&P 500, with Z-value below. * * * , * * , and * indicate significance at the 1%, 5%, and 10% levels in a 1-tailed test. For example, the figure -0.181 with *** in the first row, second column means that the coefficient of correlation of GMV with the S&P 500 is lower than that of ERC with the S&P 500 and that this is significant at the 1% level. The sample period is March 1995 to August 2017.

		GMV	ERC	EW	BCOM	GSCI	GSEW	GSRW	GSLE
	GMV	-	-0.181 * * * -0.192 * * * -0.128 * *	-0.064	-0.165 * * * -0.122 * * * -0.131 * * *
		-	(-4.916)	(-3.878)	(-2.294) (-1.013) (-2.981)	(-2.797)	(-2.424)
	ERC	0.181	-	-0.011	0.053	0.118	0.016	0.059	0.050
		(4.916)	-	(-0.692)	(1.852)	(2.787)	(0.541)	(2.290)	(1.783)
	EW	0.192	0.011	-	0.064	0.129	0.027	0.071	0.061
		(3.878)	(0.692)	-	(3.190)	(3.684)	(1.283)	(2.428)	(2.904)
	BCOM 0.128	-0.053 * * -0.064 * * *	-	0.065	-0.037 * *	0.006	-0.003
		(2.294) (-1.852)	(-3.190)	-	(2.389)	(-1.934)	(0.206)	(-0.148)
	GSCI	0.064 -0.118 * * * -0.129 * * * -0.065 * * *	-	-0.102 * * *	-0.058 *	-0.067 * * *
		(1.013) (-2.787)	(-3.684)	(-2.389)	-	(-3.458)	(-1.392)	(-2.981)
	GSEW	0.165	-0.016	-0.027 *	0.037	0.102	-	0.044	0.035
		(2.981) (-0.541)	(-1.283)	(1.934)	(3.458)	-	(1.479)	(1.984)
	GSRW	0.122	-0.059 * * -0.071 * * *	-0.006	0.058	-0.044 *	-	-0.009
		(2.797) (-2.290)	(-2.428)	(-0.206)	(1.392)	(-1.479)	-	(-0.341)
	GSLE	0.131	-0.050 * * -0.061 * * *	0.003	0.067	-0.035 * *	0.009	-
		(2.424) (-1.783)	(-2.904)	(0.148)	(2.981)	(-1.984)	(0.341)	-
	Note:

Table 5 :

 5 Comparison of coefficients of correlation with 10-year U.S. T-bond This table provides the mean of the difference between the row and columns indices of their coefficients of correlation with the 10-year U.S. T-bond, with the Z-value below. * , and * indicate significance at the 1%, 5%, and 10% levels in a 1-tailed test. For example, the value of -0.090 with *** in the second row and first column means that the coefficient of correlation of ERC with the 10-year U.S. T-bond is lower than that of GMV with the 10-year U.S. T-bond and that this is significant at the 1% level. The sample period is March 1995 to August 2017.

		GMV	ERC	EW	BCOM	GSCI	GSEW	GSRW	GSLE
	GMV	-	0.090	0.092	0.032	0.067	0.053	0.027	0.067
		-	(2.358) (1.767)	(0.555)	(1.034)	(0.909)	(0.599)	(1.205)
	ERC	-0.090 * * *	-	0.001	-0.058 * *	-0.024	-0.038	-0.063 * *	-0.023
		(-2.358)	-	(0.075) (-1.917) (-0.536) (-1.217) (-2.307) (-0.768)
	EW	-0.092 * *	-0.001	-	-0.059 * * *	-0.025	-0.039 * * -0.065 * *	-0.024
		(-1.767) (-0.075)	-	(-2.781) (-0.682) (-1.734) (-2.091) (-1.079)
	BCOM	-0.032	0.058	0.059	-	0.034	0.021	-0.005	0.035
		(-0.555)	(1.917) (2.781)	-	(1.228)	(1.006)	(-0.159)	(1.806)
	GSCI	-0.067	0.024	0.025	-0.034	-	-0.014	-0.040	0.001
		(-1.034)	(0.536) (0.682) (-1.228)	-	(-0.453) (-0.913)	(0.034)
	GSEW	-0.053	0.038	0.039	-0.021	0.014	-	-0.026	0.015
		(-0.909)	(1.217) (1.734) (-1.006)	(0.453)	-	(-0.822)	(0.802)
	GSRW	-0.027	0.063	0.065	0.005	0.040	0.026	-	0.040
		(-0.599)	(2.307) (2.091)	(0.159)	(0.913)	(0.822)	-	(1.433)
	GSLE	-0.067	0.023	0.024	-0.035 * *	-0.001	-0.015	-0.040 *	-
		(-1.205)	(0.768) (1.079) (-1.806) (-0.034) (-0.802) (-1.433)	-
	Note:

* * * , *

The list of futures contracts as of 2017 is given in the appendix.

Equity indices, such as the FTSE Global Equal Risk Contribution Index and the FTSE Global Minimum Variance Index, rely on an estimation window of approximately 500 days. We believe that 150 returns is a good compromise between a reliable estimation and a past horizon too long to reflect the dynamics of the variance-covariance matrix. Given the important numbers of securities in their index, FTSE also relies on a PCA analysis to compute the variance-covariance matrix. Our number of commodities, 27, is sufficiently limited for us to not have to resort to this methodology.

Please refer to section 4 on robustness checks for the statistics over a longer period, which are consistent with the results in the existing literature.


A Chalot

L Michel

E Ferrier

Claude Bernard Lyon

EXPERIMENTAL STUDY OF EXTERNAL BONDED CFRP-CONCRETE INTERFACE UNDER LOW CYCLE FATIGUE LOADING

Keywords: Fatigue, Carbon fibre, Adhesion, Polymer (textile) fibre

The main problem with carbon fibre reinforced polymers (CFRPs) in reinforcement is the bonding of the reinforcement to the concrete because the failure generally occurs by debonding. In the case of seismic loading, a high level of stress with cyclic loading may appear in the adhesive zone that's why this study focuses on the low-cycle fatigue of a concrete/CFRP interface under a high level of shear stress. It focuses on the influence of the fatigue loading amplitude on the service life (number of cycles prior to failure). This paper presents the results of an experimental campaign conducted on double shear test, an analysis of the failure mechanism, and the development of an analytical model to predict the lifetime of a CFRP/concrete interface. We studied two different CFRPs, one based on a unidirectional textile, composite A, and the other reinforcement based on a bidirectional textile (with a 70/30 ratio), composite B. The results of this study will allow the development of an empirical model based on Wöhler's fatigue law describing

Introduction

The need to protect buildings and people against seismic risk has become a major preoccupation of legislators. To reduce such risks, infrastructure standards are becoming increasingly strict. A large proportion of buildings and transport infrastructure built during the 1970s and 1980s do not adhere to the latest regulations.

To respond to the requirements of the new standards and ensure the stability of a structure, it is necessary to provide solutions to reinforce or repair existing structures.

The reinforcement and repair of structures using an externally bonded composite, and more particularly, those using an epoxy matrix and carbon fibre reinforced polymer (CFRP), have shown significant potential for many years. This technology is already being widely used around the world. CFRPs have extremely good mechanical properties with a strength of up to 2,000 MPa and a modulus of elasticity on the order of 230 GPa.

They can act as complementary reinforcements in bent elements, as transverse reinforcements for the shear strength, or as a confinement for a compressed element to increase the strength. The good mechanical performance of CFRPs generally leads to a failure in CFRP-concrete prior to a tensile FRP failure.

The use of CFRP reinforcements has been well documented in the literature. For the flexural strengthening of the beams [START_REF] Zhao | Ductility Research of Reinforced Concrete Beams Strengthened with CFRP[END_REF][START_REF] Spadea | Structural Behavior of Composite RC Beams with Externally Bonded CFRP[END_REF][START_REF] Yail | Ductility and Cracking Behavior of Prestressed Concrete Beams Strengthened with Prestressed CFRP Sheets[END_REF][START_REF]Evaluation of Bending Strength of RC Beams Strengthened with FRP Sheets[END_REF][START_REF] Ascione | A numerical evaluation of the interlaminar stress state in externally FRP plated RC beams[END_REF][START_REF] Bocciarelli | Failure of RC beams strengthened in bending with unconventionally arranged CFRP laminates[END_REF], or to increase the shear strength [START_REF] Dong | Structural behaviour of RC beams with external flexural and flexural-shear strengthening by FRP sheets[END_REF][START_REF] Bocciarelli | Shear failure of RC elements strengthened with steel profiles and CFRP wraps[END_REF], it is also possible to reinforce the slabs [START_REF] Shen | Bending and vibration characteristics of damaged RC slabs strengthened with externally bonded CFRP sheets[END_REF]. CFRPs are also used for confinement [START_REF] Bocciarelli | Failure of RC beams strengthened in bending with unconventionally arranged CFRP laminates[END_REF][START_REF] Dong | Structural behaviour of RC beams with external flexural and flexural-shear strengthening by FRP sheets[END_REF]. It is also possible to combine several of these effects on the same structure [START_REF] Promis | Performance indices to assess the efficiency of external FRP retrofitting of reinforced concrete short columns for seismic strengthening[END_REF].

The main problem with a CFRP reinforcement is anchoring the reinforcements to the existing structure. In most of the tests, the failure of the reinforcement is caused by debonding of the CFRP strips. Numerous experimental, numerical and theorical studies [START_REF] Bizindavyi | Transfer Lengths and Bond Strengths for Composites Bonded to Concrete[END_REF][START_REF] Chen | Anchorage Strength Models for FRP and Steel Plates Bonded to Concrete[END_REF][START_REF] Yao | Experimental study on FRP-to-concrete bonded joints[END_REF][START_REF] Lu | Bond-slip models for FRP sheets/plates bonded to concrete[END_REF][START_REF] Nakaba | Bond Behavior between Fiber-Reinforced Polymer Laminates and Concrete[END_REF][START_REF] Savoia | Non linear bond-slip law for frp-concrete interface. Fibre-Reinforced Polymer Reinforcement for Concrete Structures[END_REF][START_REF] Wu | Fracturing behaviors of FRP-strengthened concrete structures[END_REF][START_REF] Buyukozturk | Progress on understanding debonding problems in reinforced concrete and steel members strengthened using FRP composites[END_REF][START_REF] Lu | Bond-slip models for FRP sheets/plates bonded to concrete[END_REF][START_REF] Carloni | Direct determination of cohesive stress transfer during debonding of FRP from concrete[END_REF][START_REF] Mohamad | Analysis of Scaling and Instability in FRP-Concrete Shear Debonding for Beam-Strengthening Applications[END_REF][START_REF] Subramaniam | Width effect in the interface fracture during shear debonding of FRP sheets from concrete[END_REF][START_REF] Pellegrino | Experimental Study on Bond Behavior between Concrete and FRP Reinforcement[END_REF][START_REF] Mohamad | Experimental Investigation and Fracture Analysis of Debonding between Concrete and FRP Sheets[END_REF] have therefore been conducted on the behaviour of a CFRP/concrete interface.

Such studies have indicated the existence of an effective bond zone where the shear stress is concentrated. The effective bond length depends on the Young's modulus and the thickness of the reinforcement. Most static tests end in concrete failure. The ultimate load of the joint is therefore directly related to the tensile-shear strength of the concrete substrate and the mechanical properties of the reinforcement.

Most studies on fatigue behaviour have focused on a large number of low-intensity loading cycles. However, in the case of a seismic event, the structures are subjected to a low number of high intensity cycles. Thus, this paper focuses on a fatigue collapse during low cycle fatigue (LCF). J. F. Chen and J. G. Teng (2001) [START_REF] Chen | Anchorage Strength Models for FRP and Steel Plates Bonded to Concrete[END_REF] studied the effects of the effective bond length on the ultimate strength of bonded joints. Eq. [START_REF] Zhao | Ductility Research of Reinforced Concrete Beams Strengthened with CFRP[END_REF] indicates that the effective length depends on the tensile strength of the concrete as well as the modulus of elasticity (Ef) and the thickness of the reinforcement (tf). They also showed that an anchorage length that is longer than the effective length (Le) does not increase the bond strength.

(1) J.F. [START_REF] Dong | Structural behaviour of RC beams externally strengthened with FRP sheets under fatigue and monotonic loading[END_REF] [START_REF] Dong | Structural behaviour of RC beams externally strengthened with FRP sheets under fatigue and monotonic loading[END_REF] studied the fatigue and post-fatigue behaviour of a series of shear-reinforced beams. Their study shows that, in addition to increasing the strength and stiffness of the specimens, the reinforcement increases the fatigue life (1,000,000 cycles) of the RC by increasing the spacing of cracks and reducing their opening. E. Demers (1997) [START_REF] Demers | Fatigue strength degradation of E-glass FRP composites and carbon FRP composites[END_REF] studied the tensile fatigue behaviour of several glass fibre and carbon composite reinforcements. CFRP has a higher fatigue resistance than GFRP. Eq.

(2) corresponds to the lifetime of an unidirectional CFRP, whereas Eq. ( 3) corresponds to that of 0/90° bidirectional CFRP. Note that the equations quoted are equations designed with 95% more resistant specimens than the model applied. These results indicate that bidirectional CFRP has a higher fatigue life than unidirectional CFRP for an equivalent stress rate, and thus has more fatigue resistance than unidirectional CFRP.

0.8227 -0.0519 ( ) (2)
0.8133 -0.0415 ( )

J. Zhu et al. (2015) [START_REF] Zhu | Analysis of interfacial bonding characteristics of CFRP-concrete under fatigue loading[END_REF] studied the fatigue behaviour of CFRP/concrete interfaces on modified beam specimens. The experimental campaign, which included three static tests and five fatigue tests, allowed them to develop a model (Eq. ( 4)) in which the fatigue life of the interface depends mainly on the loading amplitude (T) and, on a smaller scale, the concrete compressive strength (fc').

Experimental program 2.1 Materials and specimens

There are several methods used to test the CFRP/concrete interface, namely single shear tests [START_REF] Ferrier | Fatigue of CFRPs externally bonded to concrete[END_REF][START_REF] Carloni | Experimental determination of FRP-concrete cohesive interface properties under fatigue loading[END_REF][START_REF] Carloni | Investigation of sub-critical fatigue crack growth in FRP/concrete cohesive interface using digital image analysis[END_REF], double shear tests [START_REF] Lu | Bond-slip models for FRP sheets/plates bonded to concrete[END_REF][START_REF] Wu | Fracturing behaviors of FRP-strengthened concrete structures[END_REF][START_REF] Buyukozturk | Progress on understanding debonding problems in reinforced concrete and steel members strengthened using FRP composites[END_REF], and modified beam tests [START_REF] Zhu | Analysis of interfacial bonding characteristics of CFRP-concrete under fatigue loading[END_REF]. To study the fatigue behaviour of the CFRP/concrete interface, double shear tests (DS) were conducted. Fig. 1 presents the three CFRP/concrete interface characterisation test configurations.

The plan of the specimens and all dimensions are detailed in Fig. Once the concrete has cured (28 days), the reinforced areas are sanded. This step is carried out using an angle grinder with a diamond disc. The surface is sanded down to the aggregates, the aim is to increase the bond between CFRP and concrete. During all these steps the faces of the two blocks remain parallel with each other because of the threaded rod. The tolerance on the alignment of the gluing faces is 0.25 mm under a 20-cm ruler. Composite reinforcements of 440 mm x 50 mm and dimensions are laminated on both sides using wet lay-up method. The reinforcing sheets and substrate are impregnated with epoxy resins, and the sheets are then placed on the concrete. Once in place, the strips are squeezed to remove any air bubbles, and a final layer of resin is applied to the CFRP strips. The geometry of the specimens is detailed in Fig. 2, and the preparation of the support and the application of the reinforcements are conducted in accordance with the technical advice provided by the products [START_REF] Carloni | Direct determination of cohesive stress transfer during debonding of FRP from concrete[END_REF][START_REF] Mohamad | Analysis of Scaling and Instability in FRP-Concrete Shear Debonding for Beam-Strengthening Applications[END_REF] and the temporary recommendations of the AFGC [START_REF] Afgc | Renforcement parasismiques de structures en béton armé par matériaux composites -Recommandations provisoires[END_REF]. The properties of the reinforcement are summarised in Table 1.

Loading

The loading applied is a cyclic type under tension with Pmax and Pmin exceeding zero.

The double shear tests do not allow the interfaces to be stressed during compression owing to the risk of buckling. Traditionally, the fatigue life of an element is controlled based on the rate of stress. The loading rate corresponds to the ratio between the maximum applied force and the static failure force of the specimen, which is noted as %Pu (Eq. ( 5)). Some models described in the literature have linked the life span to the amplitude of the loading [START_REF] Zhu | Analysis of interfacial bonding characteristics of CFRP-concrete under fatigue loading[END_REF]. In the case of a tensile fatigue test, the amplitude of the load, T, is also noted (Eq. ( 6)). Here, Pmax corresponds to the maximum load of the loading cycle, and Pmin corresponds to the minimum load (see Fig. 4). In general, the number of cycles is expressed as a logarithmic, namely, log(N) or le(N).

% (5) % -() (6)
In this experimental program, the loading rate ranges from 60% to 95%, and the loading amplitude ranges from 18 to 90%. The results of the static tests are detailed in Table 2.

These results allow the ultimate load values (Pu) of the fatigue tests to be set.

Table 3 shows the loading parameters and properties of the concrete, as well as the number of cycles. As a reminder, the properties of the reinforcements are as indicated in Table 1.

The tests were conducted on a Zwick tensile compression press with a capacity of 65 kN. The tensile forces were applied using hinges located at the clamping jaws to suppress any parasitic bending. Monotonic tests were are driven at a speed of 1 mm/min until the specimen failed. The fatigue tests were controlled while moving at a speed of 60 mm/min. The loading frequency varied from 0.2 to 0.33 Hz depending on the intensity of the loading. This frequency is much lower than 20 Hz, which prevented an overheating during the test [START_REF] Demers | Fatigue strength degradation of E-glass FRP composites and carbon FRP composites[END_REF]. Fig. 4 shows the shape of the cyclic loading imposed on the specimen. Tensile/tensile fatigue tests were applied, and thus we consistently have 0 < Pmin < Pmax < Pu. These tests ended when the specimen failed.

Sensor setup

At the equipment level, the displacement between the two blocks (noted ΔL1) was measured using two linear variable differential transformers (LVDTs). An LVDT has an amplitude of ±5 mm and an accuracy of 10 -3 mm. Two strain gauges (SG1 and SG2)

were placed on the tensile part of the composite. The arrangement of all sensors is shown in Fig. 5. SG1 and SG2 were used to measure the strain of the tensile CFRP during the test. SG2 is located was located on the other side of the specimen, in the middle of the tensile part. The elongation of the tensile part is noted as ΔL2, as indicated in Eq. [START_REF] Dong | Structural behaviour of RC beams with external flexural and flexural-shear strengthening by FRP sheets[END_REF]. For some specimens, several gauges were placed on one side of the reinforcement to follow the distribution of the local shear stress and the evolution of the debonding during the cycle. The strain gauges measured 10 mm and have a resistance of 120 Ω. The displacement between the blocks is due to the elongation of the tensile composite and the sliding of the glued composite. Knowing that we have the distance between the blocks (ΔL1) and the elongation of the composite (ΔL2), we can compute the average sliding value. The sliding is noted as ΔL (8). In the following equation, L2

is equal to 40 mm, which is the length of the tensile part (see Fig. 2).

∆L + , -+ , + 2 * + (7) ∆L ∆L --∆L + 2 (8)
On some of the specimens, five strain gauges (SG3 to SG7) were placed along the composite to follow the evolution of the distribution of the deformation on one of the four interfaces. The following relationship ( 9) is used to calculate the average shear stress between two gauges,τ where dε corresponds to the strain variation between 2 gauges and dx to the spacing between them.

Strain distribution

Fig. 8 shows the evolution of the strain distribution along the bonding during the fatigue tests. Each curve shows the distribution for a full cycle. We can see that, during the first loading cycles, the strain was at maximum in the first part of the bonding joint (SG3).

The strain in this zone increased slightly during the test. This increase was not caused by a modification of the bonding but was probably due to the damaging of the CFRP itself. At SG4 and SG5, we can see that the deformation increased until ε3, namely, the deformation of the front part of the bonding joint. This phenomenon was visible during all tests. We can also see that at SG6 and SG7, the most distant locations from the tensile end of the bond, the deformation was close to zero during the first cycles, which indicates that the force had already completely transited. The values then increased up to the failure of the specimen. In the case of A-F1-1 and B-F1-5, the most loaded specimens, the strain is was on all gauges at the end of the tests. In the case of A-F3-1, ε6 moved to ε3 when ε7 did not exceed 1000 μm/m. To understand this phenomenon at work, it is necessary to study the local shear stress distribution.

Local shear stress distribution

Fig. 9 shows the local shear stress distribution throughout the bonding. The local shear stress is calculated using Eq. ( 9). As described in the previous section, we note that the behaviours of the different tests were similar to those described in the previous section.

During the first cycles, the shear stress was high in the part of the adhesive closest to the tensioned end. The values of τ1 and τ2 were close to 10 MPa. The values of τ4 and τ5

were low or zero, except for specimen A-F1-1 shown in Fig. 9 (a), where a shear stress of 1.5 MPa was observed from the first cycle at τ4. The stress was concentrated at the bonding front. The rest was almost unsolicited. The length of the transfer zone of composite A was 100 mm, which corresponds to the calculation by Le [START_REF] Chen | Anchorage Strength Models for FRP and Steel Plates Bonded to Concrete[END_REF]. For composite B, this zone measured 80 mm, which is less than Le's calculation.

As we progress, we observe a change in the distribution. Indeed, from the tenth cycle, the stress profile shifted to the right. There is a decrease in stress at the front part of the FRP band, and this drop is associated with an increase in stress in the central part. The phenomenon continued for the duration of the test until a brutal failure of the specimen occurred. The length of the transfer zone was approximately constant throughout the duration of the tests.

The decrease in stress in the front part of the adhesive reflects a loss of adhesion caused by the strip debonding. When the bonded composite surface became too small, it caused a sudden increase in the local shear stress, leading to a brutal failure of the specimen.

Evolution of sliding and stiffness

Fig. 10 shows the evolution of the sliding (ΔL) and stiffness (k = Pmax/ΔL) as a function of the log(N). The pattern observed for the two types of reinforcements is identical, although the different fatigue loading intensity groups (F1, F2, and F3) do not show the same behaviours. The higher the load rate, the bigger the initial and final slips that occurred. The initial stiffness of the tests with low fatigue loading was greater, whereas the stiffness at the time of failure was extremely close to this value. The decrease in stiffness was linear when plotted to log(N).

Debonding process

As described in the previous section, it was observed that the length of the transfer zone remained constant during the test. The increase in slip and the decrease in stiffness were therefore caused by the propagation of the debonding. The increase in the slip observed from the first cycle was caused by an increase in the length of the tensile CFRP. Using

Eqs. ( 7) and ( 8), it is possible to return to the debonding length. Note that this is the average debonding of the 4 gluing joist. Fig. 11 shows the evolution of the average debonding during the tests. The debonding was initiated from the first cycle of the test, and then progressed in a linear manner (in the function of log(N)). At the end of the test, this process accelerated to failure. The final detachment length during the composite B tests was smaller (80 mm) despite a smaller transfer zone than that of composite A.

It could be interesting to precisely follow the debonding of a bond by using a digital image correlation (DIC). This will provide the true debonding length of each bounding zone and not the average value of the four.

Analysis and discussion

Fatigue life model (Wöhler curve) Ncycle (%Pu)

In this study, we considered the impact of the fatigue loading on the service life of CFRP/concrete interfaces. A Wöhler curve is a widely used technique to predict the fatigue life of a material. The study was conducted on two different reinforcements to study the effects of the type of textile on the fatigue resistance of the bonding.

The relationships between the number of cycles up to failure of the two types of reinforcements and the loading rate are presented in Fig. 12 and Fig. 13. The relationship between these two variables is linear. An order 1 linear regression makes it possible to obtain the fatigue model of each of the two reinforcements. Eq. [START_REF] Tao | Axial Loading Behavior of CFRP Strengthened Concrete-Filled Steel Tubular Stub Columns , Axial Loading Behavior of CFRP Strengthened Concrete-Filled Steel Tubular Stub Columns[END_REF] provides the trend of composite A, with a correlation coefficient (R²) of 0.65 and a standard deviation (RMSE) of 0.739. Eq. ( 11) is the equation for composite B, with an R² of 0.76 and an RMSE of 0.783. The large dispersion obtained during the fatigue tests does not allow a good correlation. We can see that, for a high loading rate (%Pu = 95%), the bi-directional reinforcement (B) is 4% more enduring than reinforcement A.

However, this tendency is reversed when one takes a lower loading rate. For %Pu = 60%, reinforcement A has 5% more endurance. More testing would be appropriate to confirm these observations.

The two models are still very close. It is therefore possible to combine the two groups to calculate a common model. The model written in Eq. ( 11) has an R² of 0.7 and an RMSE of 0.760.

% 2 -0.051 log( ) + 0.925

% 2 -0.065 log( ) + 0.971

% 2 -0.058 log( ) + 0.949

Testing and comparison of the proposed model

A fatigue test database from the literature was edited to test our fatigue model, which was calibrated as described in the previous section. The model was also compared to the model developed by [START_REF] Zhu | Analysis of interfacial bonding characteristics of CFRP-concrete under fatigue loading[END_REF], which is indicated in Eq. ( 4). The test parameters and results of the two models are detailed in Table 4.

In Fig. 15, we can see that the proposed model provides more reliable results than the Zhu model [START_REF] Zhu | Analysis of interfacial bonding characteristics of CFRP-concrete under fatigue loading[END_REF], with a determination coefficient of 0.72 against 0.41 for the Zhu model. These results confirm the validity of the proposed model despite the fatigue tests being conducted under different experimental protocols.

Predictive preliminary model

In addition to the analytical model, a predictive preliminary model was also calibrated.

The distribution of the results follows a normal rule. The different series of tests show the same deviations. By studying the distribution of the tests, we can thus obtain a trend curve that delimits the lower 5% from the upper 95%. The predictive preliminary model is given through Eq. ( 13).

% 2 0.861 -0.058 log( )

Conclusion

The objective of this study was to determine the fatigue behaviour of CFRP/concrete interfaces. The following conclusions can be highlighted:

The fatigue tests show significant dispersion. To counter this experimental bias, it is therefore necessary to conduct many tests and control this dispersion. ( 12) and the designed model in Eq. ( 13). The proposed model was compared to an existing model based on a database provided in the literature.

[MPa] [kN] [%] [kN] [%] [kN] [%] [cycle] [cycle]

Composite A

2 .

 2 The concrete blocks have dimensions of 140 mm x 140 mm x 260 mm and are installed in a 140 mm x 140 mm x 560 mm mould with a 40-mm thick separating element. The connection between the two blocks is ensured by two threaded rods connected by a bolt. This assembly ensures a perfect alignment of the side faces of the blocks because they are both formworked. For all batches, samples are cast to test their compressive and tensile strengths (by splitting). The average concrete strength is 39.6 MPa with a standard deviation of 0.9 MPa. Concrete characterisation tests are conducted in accordance with the European standard [35]. The composition of the mixture is 350, 190, 850, and 1020 kg/m 3 of cement, water, sand, and gravel, respectively.

Fig. 3

 3 Fig. 3 shows an image of the formwork (left) with a Styrofoam separating element

  . Chen and J. G. Teng (2001) [10] distinguished six failure modes for single or double shear tests: 1. Concrete delamination 2. Tensile failure of the CFRP 3. Adhesive failure 4. Internal delamination of the CFRP 5. Adhesive-concrete interface failure 6. Adhesive-reinforced interface failure None of the tested specimens failed in the stretched part of the reinforcement (failure mode 2). In the monotonic tests, failure mode 1 was predominant with a concrete delamination failure (Fig.7 (a)), the cracks is located inside the concrete block and there is no damage in the adhesive or in the composite. A failure at the glue/concrete interface (failure mode 5) could also be observed on some specimens, although this phenomenon remained marginal. Fatigue loading completely modified the failure mode of the specimens. Indeed, failure mode 5 was predominant with a failure at the interface between the glue and concrete. As shown in Fig.7 (b) and (c), a local concrete pull-out could also be observed at the front and rear ends of the CFRP strip. On a certain specimen of reinforcement A, a failure quickly occurred at the fibre/resin interface. As indicated in Fig.7(d), the fatigue loading intensity did not appear to affect the type of failure mode during the experimental campaign. Both reinforcements also showed the same failure mode.

  Using 29 fatigue tests, this experimental campaign provides a faithful picture of the behaviour, although additional tests will have to be conducted to increase the relevance of the analytical model. The tests performed showed that the fatigue life (number of cycles prior to failure) of the specimens is a function of the loading rate which depends only of the static strength of the bond. The other parameters do not seem to have any influence on the fatigue behaviour of the interface Cyclic fatigue loading changes the failure mode. We move from a concrete pull-out, namely, a type 1 failure mode (concrete delamination), to a type 5 (adhesive/concrete interface failure) and type 6 (adhesive/CFRP interface failure) mixed failure. It is therefore the resin and not the concrete which is the weak point. Cyclic fatigue loading produces debonding, which spreads along the adhesive joint until the bonded surface is too small to take up the load. The final failure remains significant. CFRP A (unidirectional fabric) and CFRP B (bidirectional carbon fabric) show similar fatigue behaviours. The data collected made it possible to calibrate the model in Eq.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Diagram of the different test configurations: (a) single shear test, (b) double shear test, and (c) modified beam test.

Fig. 3 :

 3 Fig. 3: Specimen manufacturing.

Fig. 4 :

 4 Fig. 4: Fatigue loading profile.

Fig. 5 : 25 Fig. 6 : 1 Fig. 7 : 4 Fig. 8 :Fig. 9 :Fig. 10 :Fig. 11 :

 5256174891011 Fig. 5: Strain gauges and LVDT detail.

Fig. 12 :

 12 Fig. 12: Loading rate of Log(N) curve: Composite A.

Fig. 13 :Fig. 14 :Fig. 15 :

 131415 Fig. 13: Loading rate of Log(N) curve: Composite B.

Fig. 16 :

 16 Fig. 16: Wöhler curve: Designed model.

Table 2 :

 2 Static test results.

		Specimen	fcu	Pu
	CFRP
		ID	[MPa] [kN]
		A-M-1	38	36.8
	COMPOSITE	A-M-2	38	38.3
	A	A-M-3	38	39.8
		Average	38	38.3
		B-M-1	38	28.2
	COMPOSITE	B-M-2	39	29.2
	B	B-M-3	38	29.6
		Average	38	29.0

Table 3 :

 3 Fatigue loading descriptions and fatigue test results.

	A-F1-1	40	2.4	6%	31.3 82% 38.3 75% 387	3
	A-F1-2	40	2.3	6%	31.4 82% 38.3 76%	56	2
	A-F1-3	40	2.3	6%	31.3 82% 38.3 76%	24	1
	A-F1-4	40	6.3	16% 35.8 93% 38.3 77%	80	2
	A-F1-5	40	2.8	7%	35.1 92% 38.3 84%	6	1
	A-F2-1	40	5.5	14% 28.1 73% 38.3 59% 16458	4
	A-F2-2	40	5.4	14% 28.1 73% 38.3 59% 534	3
	A-F2-3	40	2.7	7%	29.1 76% 38.3 69% 120	2
	A-F2-4	40	2.8	7%	29.1 76% 38.3 69% 251	2
	A-F2-5	40	6.2	16% 32.2 84% 38.3 68% 159	2
	A-F3-1	40	8.5	22% 24.9 65% 38.3 43% 99491	5
	A-F3-2	40	9.9	26% 28.8 75% 38.3 49% 3609	4
	A-F3-3	40	9.8	26% 28.9 75% 38.3 50% 14018	4
	A-F3-4	40	9.8	26% 28.7 75% 38.3 49% 1699	3
	A-F3-5	40	9.9	26% 28.5 74% 38.3 49% 5075	4
	B-F1-1	38	1.4	5%	27.2 94% 29.0 89%	62	2
	B-F1-2	38	1.9	7%	27.0 93% 29.0 86%	24	1
	B-F1-3	38	4.9	17% 25.9 89% 29.0 72%	3	0
	B-F1-4	39	1.4	5%	24.4 84% 29.0 79% 1574	3
	B-F1-5	39	2.0	7%	24.5 85% 29.0 78% 451	3
	B-F2-1	38	2.7	9%	24.3 84% 29.0 75%	40	2
	B-F2-2	38	4.8	17% 24.3 84% 29.0 67%	21	1
	Composite B
	B-F2-3	38	3.3	11% 22.9 79% 29.0 68% 291	2
	B-F2-4	42	4,8	17% 21,4 74% 29,0 57% 7053	4
	B-F2-5	39	4.6	16% 21.6 75% 29.0 59% 554	3
	B-F3-1	39	3.2	11% 20.1 69% 29.0 59% 9353	4
	B-F3-2	42	4.6	16% 20.1 69% 29.0 53% 6941	4
	B-F3-3	39	5.9	20% 20.1 70% 29.0 49% 7761	4
	B-F3-4	42	6.0	21% 18.8 65% 29.0 44% 52877	5

A

Composite A

B

Composite B

Fabric properties

Fibre

Carbon Carbon