Isabelle Roussos

Quentin Carboué
email: quentin.carboue.qc@gmail.com

Catherine Rébufa

Rayhane Hamrouni

Sevastianos Roussos

Isabelle Bombarda

Statistical approach to evaluate effect of temperature and moisture content on the production of antioxidant naphtho-gamma-pyrones and hydroxycinnamic acids by Aspergillus tubingensis in solid-state fermentation

Keywords: Solid-state fermentation, Aspergillus tubingensis, Naphtho-gamma-pyrones, Hydroxycinnamic acids, Principal component analysis, Vine shoots

Statistical approach to evaluate effect of temperature and moisture content on the production of antioxidant naphtho-gamma-pyrones and hydroxycinnamic acids by Aspergillus tubingensis in solid-state fermentation

Quentin Carboué, Catherine Rebufa, Rayhane Hamrouni, Sevastianos

Introduction

Among the culture methods available, solid-state fermentation (SSF) is a technic that is thousands of years old, first used by mankind for food production but now owning a diversity of applications [START_REF] Soccol | Recent developments and innovations in solid state fermentation[END_REF]. It is defined as a microbial culture involving a solid medium in the absence (or near absence) of free water; however, substrate must possess enough moisture to support growth and metabolism of micro-organism [START_REF] Pandey | Solid-state fermentation[END_REF]. Another important parameter to consider is the temperature inside the bioreactor bed. For that matter, heat removal is one of the major constraints in SSF, especially in large-scale processes where metabolic heat is proportionally generated [START_REF] Figueroa-Montero | Improvement of heat removal in solid-state fermentation tray bioreactors by forced air convection[END_REF]. By the way, heat removal through water evaporation is a possible way to manage the energy inside the bioreactor but requires water replenishment through aeration with vapor saturated air [START_REF] Singhania | Recent advances in solid-state fermentation[END_REF].

The influence of moisture content and temperature is essential in the growth and secondary metabolite production of fungi [START_REF] Nielsen | Mould growth on building materials under low water activities. influence of humidity and temperature on fungal growth and secondary metabolism[END_REF][START_REF] Parra | Modelling the effect of temperature and water activity on growth of Aspergillus niger strains and applications for food spoilage moulds[END_REF]. However, although some tendencies have been described regarding the effects of moisture content and temperature on secondary metabolism, each fungal strain has its own behavior when facing environmental conditions changes [START_REF] Dhungana | Effects of temperature, water activity, and fungal isolate on ochratoxin A accumulation in oat grain inoculated with Penicillium verrucosum and development of a methodology to screen oat cultivars for ochratoxin A accumulation[END_REF][START_REF] Pardo | Prediction of fungal growth and ochratoxin A production by Aspergillus ochraceus on irradiated barley grain as influenced by temperature and water activity[END_REF]. It is, therefore, important to empirically study those effects when dealing with a strain of industrial interest to optimize the production.

Another essential parameter is the medium composition which has a direct influence on numerous anabolic pathways and, therefore, on the production yields. Besides the 1 3 selection of a proper medium having economic repercussions on the overall process, the valorization of local byproducts as culture medium should be considered to the extent possible [START_REF] Yazid | Solidstate fermentation as a novel paradigm for organic waste valorization: a review[END_REF][START_REF] Hamrouni | From flasks to single used bioreactor: scale-up of solid state fermentation process for metabolites and conidia production by Trichoderma asperellum[END_REF]. In the case of southern France, the vine shoots are common by-products and to date, various studies investigated them as a solid medium for the SSF, for the production of feruloyl esterases using Aspergillus terreus and for the production of cellulolytic enzymes by Aspergillus uvarum [START_REF] Pérez-Rodríguez | Vine trimming shoots as substrate for ferulic acid esterases production[END_REF][START_REF] Pérez-Rodríguez | Feruloyl esterase production by Aspergillus terreus CECT 2808 and subsequent application to enzymatic hydrolysis[END_REF][START_REF] Salgado | Enhancing the bioconversion of winery and olive mill waste mixtures into lignocellulolytic enzymes and animal feed by Aspergillus uvarum using a packed-bed bioreactor[END_REF].

The black aspergilli (Aspergillus section Nigri) are an important group of species in food mycology, medical mycology and biotechnology [START_REF] Samson | Diagnostic tools to identify black aspergilli[END_REF]. In biotechnology, they are used in many processes for the production of biomass, organic acids, enzymes, and secondary metabolites [START_REF] Frisvad | Safety of the fungal workhorses of industrial biotechnology: update on the mycotoxin and secondary metabolite potential of Aspergillus niger, Aspergillus oryzae, and Trichoderma reesei[END_REF][START_REF] Kang | Production of cellulases and hemicellulases by Aspergillus niger KK2 from lignocellulosic biomass[END_REF][START_REF] Roukas | Citric and gluconic acid production from fig by Aspergillus niger using solid-state fermentation[END_REF]. Aspergillus tubingensis is commonly described as a xerophilic fungus with the optimum growth temperature ranging from 10 to 44 °C [START_REF] García-Cela | Ecophysiological characterization of Aspergillus carbonarius, Aspergillus tubingensis and Aspergillus niger isolated from grapes in Spanish vineyards[END_REF]. For that reason, A. tubingensis is a post-harvest mold frequently encountered in grapes where it might cause mycotoxin problems [START_REF] Perrone | Biodiversity of Aspergillus species in some important agricultural products[END_REF]. A. tubingensis strain G131 isolated from grape from a vineyard in Southern France can produce naphtho-gamma-pyrones (NγPs) [START_REF] Bouras | Occurrence of naphtho-gamma-pyrones-and ochratoxin A-producing fungi in French grapes and characterization of new naphtho-gamma-pyrone polyketide (Aurasperone G) isolated from Aspergillus niger C-433 1[END_REF]. These secondary metabolites exhibit a broad spectrum of biological activities and particularly radical scavenging activities which depend on the number of hydroxyl groups on the NγP skeleton [START_REF] Choque | Fungal naphtho-γ-pyrones-secondary metabolites of industrial interest[END_REF][START_REF] Xu | Bioactive natural products derived from mangrove-associated microbes[END_REF][START_REF] Carboué | Naphtho-gammapyrones produced by Aspergillus tubingensis G131: new source of natural nontoxic antioxidants[END_REF]. Also, this fungus is non-mycotoxigenic as genome sequencing revealed that the synthetic genes required for the production of ochratoxin A and fumonisin are absent [START_REF] Choque | Whole-genome sequencing of Aspergillus tubingensis G131 and overview of its secondary metabolism potential[END_REF]. A study carried out in a liquid medium showed that the production could be improved by decreasing the water activity or by adding an oxidative stress [START_REF] Lopez De Leon | Influence of the culture conditions on the production of NGPs by Aspergillus tubingensis[END_REF].

Overall compounds of interest found in a fermentation extract can be classified into two categories: the compounds directly produced by the fungus-like the secondary metabolites-and the compounds resulting from the fungal enzymatic degradation of the medium-composed by byproducts of vegetal origin. Indeed, plants produce during the phenylpropanoid biosynthesis pathway phenolic hydroxycinnamic acids which have a defensive role as they ensure the transversal junction between polysaccharides-hemicellulose and pectin-and phenolic compounds like lignin [START_REF] Dilokpimol | Diversity of fungal feruloyl esterases: updated phylogenetic classification, properties, and industrial applications[END_REF]. The action of specific fungal enzymes that release them is particularly valuable, e.g., hydroxycinnamic acids-p-coumaric acid or trans-ferulic acids-exhibit antioxidant properties [START_REF] Max | Ferulic acid and p-coumaric acid solubilization by alkaline hydrolysis of the solid residue obtained after acid prehydrolysis of vine shoot prunings: effect of the hydroxide and pH[END_REF].

An antioxidant prevents the oxidation of oxidizable matter such as foodstuffs and reduces oxidative stress in living cells, the latter being the physiological consequence of factors such as sunburns, airborne pollutants, or disease [START_REF] Molyneux | The use of the stable free radical diphenylpicryl-hydrazyl (DPPH) for estimating antioxidant activity[END_REF]. Antioxidant applications, therefore, include various industrial sectors such as health, food, and cosmetics [START_REF] Sindhi | Potential applications of antioxidants-a review[END_REF]. Moreover, as the demand of products of natural origin increases, microbial non-toxic antioxidant compounds are particularly sought-after for their ability to substitute their synthetic counterparts.

The present study aims to investigate the effect of moisture content and temperature on the NγP and hydroxycinnamic acid production by A. tubingensis G131 cultivated under SSF on a medium composed of vine shoots and wheat bran as sole nutrient source. Kinetics of production of various metabolites-including secondary metabolites like NγPs and degradation products-and conidia have been carried out under various levels of moisture contents and temperatures. A statistical approach was used through a principal component analysis (PCA) to visualize rapidly and simultaneously the collected data from all kinetics of production to compare them easily, to highlight eventual production patterns following the various experimental conditions and to interpret the correlations between the different identified compounds to optimize the antioxidant potential of the crude extract. Furthermore, this study contributes to a better knowledge of the growth of A. tubingensis in SSF using byproducts as a culture medium.

Materials and methods

Fungal strain

A. tubingensis G131 was provided by École Nationale Supérieure Agronomique de Toulouse (France). The strain was conserved at 4 °C in a 5-ml bottle on potato dextrose agar (PDA). The inoculum stock was prepared by propagating the fungus in Erlenmeyer flasks containing PDA. The cultures were incubated at 25 °C for 7 days. The inoculum suspension was prepared by adding 0.01% (v/v) Tween 80 and scraping with a magnetic stirrer to recover the conidia. The number of conidia was counted using a Malassez cell before the inoculation of the solid medium.

Solid-state fermentation (SSF)

SSF was carried out in duplicate in Erlenmeyer flasks of 250 mL on 5 g (dry matter) occupying nearly 40% of the volume. Particle size ranged from 0.5 to 1 cm. The previously autoclaved solid medium was composed of 50% vine shoots-acting as a solid support, bringing physical properties like an adapted porosity to the medium-and 50% wheat bran (w/w)-acting mainly as substrate, bringing nutrients to the fungus-and was inoculated with 2 × 10 7 conidia g -1 (dry matter) and put in an incubator at various temperatures. The vine shoots were provided by the Laboratoire Européen d'Extraction (La Laupie, France) after they were previously exhausted of their polyphenol content through ethanol extraction to produce value-added compounds [START_REF] Carboué | Experimental design and solid state fermentation: a holistic approach to 1 3 improve cultural medium for the production of fungal secondary metabolites[END_REF].

3

Effect of temperature and moisture content

The temperature effect was investigated at a fixed moisture value of 66% following four incubation temperatures: 25, 30, 35, and 40 °C. Moisture effect was studied at a fixed temperature of 25 °C following three initial moisture values: 50, 60, and 66%. In both cases, production kinetics were established for the following variables: NγPs including asperpyrone E, dianhydroaurasperone C, fonsecin, fonsecin B and ustilaginoidin A; some other fungal alkaloids including nigragilin and aspernigrin A; some degradation products from the solid medium tied to the fungal enzymatic activity such as β-dglucose, p-coumaric acid and trans-ethyl ferulate; a membrane lipid, ergosterol, and the conidia. The cultures were carried out during 14 days and samples have been collected every 2 days.

Ergosterol extraction

The extraction protocol has been adapted from [START_REF] Saxena | Relationship of mould count, ergosterol and ochratoxin A production[END_REF] [START_REF] Saxena | Relationship of mould count, ergosterol and ochratoxin A production[END_REF]. Briefly, 1 g of fermented material was introduced in 10 mL of methanol (> 99.5%, Sigma-Aldrich, Saint-Louis, USA) solution containing 0.2 M of potassium hydroxide and put in a water bath at 55 °C for 20 min. The methanol extracts (5 mL) were collected and stirred with water (5 mL) and hexane (10 mL) (Sigma-Aldrich, Saint-Louis, USA). After decantation, the hexane phase was removed. The hexane extraction step was carried out two times. The obtained hexane phase was then evaporated under reduced pressure at 30 °C. The dry extract was solubilized in methanol (2 mL), filtered with a 0.2-μm filter (GE Healthcare Systems, Chicago, USA) for subsequent quantification by high-performance liquid chromatography (HPLC) analysis.

Other compound extraction

The extraction of the investigated compounds (NγPs, degradation products, and other fungal alkaloids) was performed on 1 g of moist fermented material with analytical ethanol (10 mL) (> 99.5%, Sigma-Aldrich, Saint-Louis, USA) following a self-developed protocol. The mixture was stirred during 1 h. The crude ethanol extract was then filtered with a 0.2-μm filter before HPLC analysis.

HPLC analysis

The HPLC system (Agilent Technologies, Santa Clara, USA) was equipped with a diode array detector (DAD), an RP18 analytical column (150 × 5 mm, 4.6 μm particle size, Zorbax Eclipse XDB, Agilent Technologies, Santa Clara, USA), fitted to an RP18 guard column (10 × 4 mm, Agilent Technologies, Santa Clara, USA) and kept at 30 °C during the analysis. The metabolites were monitored at 280 nm and the subsequent peaks were observed using ChemStation B.02.01 (Agilent Technologies, Santa Clara, USA).

Ergosterol quantification

The methanol extracts (30 μL) were injected using an autosampler. The isocratic elution was carried out with analytical methanol at a flow rate of 1 mL min -1 for 15 min. Ergosterol concentration was determined using a commercial ergosterol standard (Sigma-Aldrich, St. Louis, USA). The results were expressed in mg per g of dry matter based on the calibration curve obtained from the analysis of standard solutions ranging from 10 to 500 mg L -1 .

Other compound quantification

The crude ethanol extracts (40 μL) were injected using an auto-sampler. The mobile phase consisted of acidified water (0.2% glacial acetic acid) (A) and analytical acetonitrile (B) (Sigma-Aldrich, Saint-Louis, USA). The linear gradient started with 30-100% of B for 45 min and then 100% B for 5 min at a flow rate of 1 mL min -1 . The secondary metabolites investigated being mostly NγPs, the concentrations were determined using the commercial standard rubrofusarin A (> 98% purity, Bioviotica Naturstoffe GmbH, Dransfeld, Germany). The results were expressed in mg equivalent rubrofusarin per gram of dry matter based on the calibration curve obtained from the analysis of standard solutions ranging from 2 to 200 mg L -1 .

Isolation and radical scavenging assays of the compounds

Isolation and radical scavenging capacity assays of the compounds were studied in a previous work [START_REF] Carboué | Naphtho-gammapyrones produced by Aspergillus tubingensis G131: new source of natural nontoxic antioxidants[END_REF]. Briefly, the radical scavenging capacity assay was carried out as follows: a solution of radical monocation ABTS •+ reacted with 10 µmol L -1 of the assayed compound for 5 min, at 25 °C, in 96-well plates. The subsequent decolorization of the monocation in the presence of a radical scavenging compound was measured at 734 nm using a microplate reader Infinite 200 (Tecan©, Mennedorf, Switzerland). The results were expressed in Trolox equivalent antioxidant capacity using a calibration curve of 3,4-dihydro-6-hydroxy-2,5,7,8-tetramethyl-2H-1-benzopyran-2-carboxylic acid (Trolox) ranging from 15.625 to 1000 μmol L -1 . Among the various metabolites that own antioxidant properties, fonsecin and ustilaginoidin A exhibit the highest potentials of 13.32 ± 0.08 and 14.59 ± 1.75, respectively (not significantly different from the radical scavenging activity of ascorbic acid: 13.23 ± 0.05), when measured with an ABTS assay. This antioxidant potential then decreases as follows: trans-ethyl 1 3 ferulate (8.32 ± 0.85), p-coumaric acid (6.86 ± 0.43), fonsecin B (5.12 ± 0.08), dianhydroaurasperone C (4.57 ± 0.36) and asperpyrone E (3.29 ± 0.11) [START_REF] Carboué | Naphtho-gammapyrones produced by Aspergillus tubingensis G131: new source of natural nontoxic antioxidants[END_REF].

Conidium counting

1 g of each sample was collected, put in distilled water with Tween 80 (Sigma-Aldrich, St. Louis, USA) 0.01% (v/v), and stirred for 5 min. After appropriate dilution, the conidia were counted using a Malassez cell.

Water content

The results were expressed in dry weight. To determine the dry weight of a sample, 1 g of moist fermented material was put in an oven at 105 °C for 24 h and was reweighted to measure the percentage of lost water.

Statistical analysis

Standard deviation was calculated from the duplicate samples and kinetic production curves were made using R software (4.0.0., R Core Team).

PCA was used for reducing the dimensionality of datasets, increasing interpretability but at the same time minimizing information loss. It allowed an interpretable overview of the main information in a multidimensional array [START_REF] Jolliffe | Principal component analysis: a review and recent developments[END_REF]. PCA was performed using Unscrambler X software version 10.3 (CAMO, Oslo, Norway).

13 explicative variables were considered: β-d-glucose, p-coumaric acid, trans-ethyl ferulate, nigragilin, aspernigrin A, asperpyrone E, dianhydroaurasperone C, fonsecin, fonsecin B, ustilaginoidin A, ergosterol concentrations and conidia. All were centered, reduced, and divided with the standard deviation to avoid giving importance to high values. The concentration values have not been determined for all moisture contents and temperatures: two matrices were generated. One using the data of kinetics realized under constant moisture content (66%) at four different temperatures (25, 30, 35 and 40 °C) and another with kinetic data with variable moisture contents (50, 60 and 66%) at constant temperature (25 °C). The number of principal components (PCs) was determined through a cross-validation. Result interpretation was done from the score plots that allowed explaining the position of samples in the two-dimensional plane generated by the PCs, thanks to certain variables, identified on the plot of correlation loadings. For the analysis related to the temperature effect, 74 samples were used with 962 values for metabolite concentrations and conidium production. Concerning the moisture content effect, 70 samples and 910 concentration values were used.

Results and discussion

The effects of temperature and moisture content were tested on the growth and metabolite productions within a range of 25 and 40 °C and 50 and 66%, respectively. Those values were chosen because these ranges were typically used for A. tubingensis optimization and included optimal values for mycotoxins and enzyme production [START_REF] Imran | Beta-glucosidase production optimization from newly isolated Aspergillus tubingensis IMMIS2 using taguchi statistical design[END_REF][START_REF] Lahouar | Influence of temperature, water activity and incubation time on fungal growth and production of ochratoxin A and zearalenone by toxigenic Aspergillus tubingensis and Fusarium incarnatum isolates in sorghum seeds[END_REF][START_REF] Trivedi | Optimization of inulinase production by a newly isolated Aspergillus tubingensis CR16 using low cost substrates[END_REF].

All the tested moisture contents correspond to modalities in which the solid medium is saturated with water, with the presence of a thin aqueous film. Moreover, these moisture contents are constant during the culture process. First of all, we can observe the evolution of the number of conidia and the biomass from the ergosterol production (Fig. 1). Indeed, SSF final product is a fermented matrix for which the separation between the fungus and the culture medium is very challenging, especially when this medium is made of agricultural by-products that are consumed during the fungal growth. Ergosterol is a sterol from the fungal membrane, so it can be used as an indirect biomass indicator [START_REF] Djajakirana | Ergosterol and microbial biomass relationship in soil[END_REF]. However, it is difficult to convert ergosterol content to a precise biomass value because the ergosterol production depends on the physiological state of the fungus itself depending on many parameters [START_REF] Gessner | Ergosterol-to-biomass conversion factors for aquatic hyphomycetes[END_REF]. In this study, two tendencies can be observed: higher temperatures-35 and 40 °C-seem to favor the fungal growth as the fungus produces more ergosterol and earlier than when the fungus is cultivated under lower temperatures. This production tends to equalize itself from day 7. This observation is in accordance with the study by Parra and Magan [START_REF] Parra | Modelling the effect of temperature and water activity on growth of Aspergillus niger strains and applications for food spoilage moulds[END_REF] for A. niger which showed that for a fixed water activity value, the growth rate was higher for temperatures of 35, 30, 25, and 20 °C, respectively [START_REF] Parra | Modelling the effect of temperature and water activity on growth of Aspergillus niger strains and applications for food spoilage moulds[END_REF]. The moisture content does not lead to major differences in the ergosterol production. Conidia seem to be produced with delay as temperature increases whereas moisture content seems to have a small effect on it.

During the stationary phase, when biomass production decreases, the secondary metabolism starts through endogenous and environmental regulations. Ecologically speaking, this environmental regulation corresponds to a change toward more stressful medium conditions (an oxidative stress, a decline of the carbon source, a temperature rise, etc.) against which the fungus will develop mechanisms of protection through active compound production [START_REF] Keller | Fungal secondary metabolism: regulation, function and drug discovery[END_REF].

The fonsecin and ustilaginoidin A are proportionally produced as moisture content increases and temperature decreases (Figs. 2,3). This tendency seems to be amplified for the fonsecin production as temperatures of 30, 35 and 40 °C almost completely shut down the production. Although the increasing moisture contents favor fonsecin B, diahydroaurasperone C and asperpyrone E, temperature 1 3 exhibits different effect on the production as compared with the previous NγPs. Indeed, fonsecin B production is maximized for temperatures of 30 and 35 °C, dianhydroaurasperone C, for 35 °C and asperpyrone E for 30 °C.

As for the hydroxycinnamic acids, p-coumaric acid and trans-ethyl ferulate are proportionally released into the medium as temperature decreases and as moisture content increases.

Concerning the NγP yields, a study dealing with the production of NγPs using the same strain of A. tubingensis G131 cultivated in Erlenmeyer flask reported a production of 20 mg g -1 of dry biomass cultivated in liquid conditions using Czapek yeast broth with an osmotic stress-the water activity was set at 0.95-during 9 days in static conditions [START_REF] Lopez De Leon | Influence of the culture conditions on the production of NGPs by Aspergillus tubingensis[END_REF]. Comparing these results with the one obtained in our study-a total NγP production of 0.62 mg g -1 of dry matter obtained after 7 days of growth at 25 °C and with a moisture content of 66%-is complicated because unlike in SSF although the quantifications of the NγPs were carried out using the same HPLC technic, the downstream quantification of fungal biomass is easier in liquid systems as it is direct. Using the dry biomass per dry solid medium ratio (0.02 g g -1 ) obtained by [START_REF] Steudler | Biomass estimation during macro-scale solid-state fermentation of basidiomycetes using established and novel approaches[END_REF] through the biomass determination of ergosterol measurement for a strain of Trametes hirsuta grown in SSF on corn silage after 7 days, our NγP production yield would be of 31 mg g -1 of dry biomass [START_REF] Steudler | Biomass estimation during macro-scale solid-state fermentation of basidiomycetes using established and novel approaches[END_REF]. This is, however, speculative as there are many different parameters between the two studies, for example, T. hirsuta is a Basidiomycota whereas A. tubingensis is an Ascomycota. To carry out a proper comparison between the solid and liquid modalities of culture, efficient biomass estimation has to be developed for the solid system. This determination could be carried out by determining the right relation between ergosterol and A. tubingensis G131 biomass production.

As a complement of the observation of the kinetic production curves, another possibility to analyze the data is to use a PCA approach to observe the metabolite behavior under different experimental conditions (kinetics of production related to the temperature and the moisture content variations) and compare them.

In the case of the PCA of the data related to the temperature variations, the cluster of samples in (PC1, PC2) score plot suggests a temporal evolution on PC1 axis with an explained variance of 46% (Fig. 4a). This PC1 axis gathers in its negative part the first 2 days of fermentation of the temperature conditions at 25, 30 and 35 °C and the whole kinetic carried out at 40 °C, suggesting a strong negative impact of a temperature of 40 °C on the metabolism when compared to the others. The PC2 with an explained variance of 23% shows nonetheless a similar behavior of the data for later times of the kinetics (after day 2) related to 35 and 40 °C which points are clustered in the positive part of this axis. Points corresponding to higher times of fermentation at 25 °C follow a distinct pattern as they are located in the negative part of PC2. The points associated with the same time values but with a temperature of 30 °C exhibit an intermediary behavior as they are placed in the center of PC2. Those various patterns associated with the different conditions are explained by differential production quantities related to each compound.

The interpretation of the correlation loadings allows highlighting the correlations existing between the response variables. Two groups appear: at 25 °C, days 5-13 are characteristic of a group of molecules comprising fonsecin, ustilaginoidin A in addition to the hydroxycinnamic acids such as the p-coumaric acid and the trans-ethyl ferulate (Fig. 4b). By the way, this result suggests that the fungal p-coumaroyl and feruloyl esterases have a temperature optimum of 25 °C. Unlikely, feruloyl esterases of an A. niger strain growing in solid state have shown a temperature optimum for the enzyme activity of 40 °C [START_REF] Hegde | Isolation and partial characterization of alkaline feruloyl esterases from Aspergillus niger CFR 1105 grown on wheat bran[END_REF]. As suggested by the correlation loadings, a second group of compounds which variables are negatively correlated with the previous ones emerges at [START_REF] Trivedi | Optimization of inulinase production by a newly isolated Aspergillus tubingensis CR16 using low cost substrates[END_REF] °C and after 5 days of fungal growth including aspernigrin A, nigragilin, dianhydroaurasperone C, sugars and to a lesser extent fonsecin B.

A difference of 10 °C is thus responsible for radical change in the metabolite production.

Concerning data related to the moisture content variation, again the discrimination of samples in (PC1, PC2) score plot suggests a temporal evolution on PC1 axis with this time an explained variance of 63% (Fig. 5a).

The PC1 axis gathers in its negative part the points associated with the first days of culture (days 1-4) related to moisture contents of 66 and 60% and almost all the points of the kinetic (except day 15) related to the moisture content of 50%. The kinetic data obtained for a moisture level of 50 and 60%, for a duration superior to 3 days are, however, more clustered along the PC1 than the data obtained for a moisture level of 66%, indicating that an increase of moisture content is characteristic of a higher metabolic maturity. On the other hand, points associated with the early days of fermentation (until the day 3) for the data related to moisture contents of 50 and 60% are associated in the negative part of PC2 (explained variance of 13%) with the whole kinetic production point with a moisture content of 66%. Positive part of the PC2 gathers points higher to day 3 of the kinetics assigned to 50 and 60% of moisture contents.

The observation of correlation loadings shows that most of the NγPs as well as the hydroxycinnamic acids are produced in higher quantities with moisture content levels of 60 and 66% for a culture duration ranging between 7 and 15 days. Fonsecin is particularly favored by a moisture content of 66% (Fig. 5b). Moreover, there is a negative correlation for conidia and aspernigrin productions with the other compounds, suggesting that, for a culture time exceeding 7 days, the production of conidia and aspernigin are favored at a lower moisture content of 50%.

In a strategy of maximization of the antioxidant properties of the crude extract, fonsecin and ustilaginoidin A are the most desirable compounds of the crude extract as they are the most antioxidant compounds [START_REF] Carboué | Naphtho-gammapyrones produced by Aspergillus tubingensis G131: new source of natural nontoxic antioxidants[END_REF]. Therefore, to obtain a crude extract with maximum antioxidant properties and because there is a net quantitative difference in the production between fonsecin and the other compounds, fonsecin production must be favored. This pattern of production that maximizes fonsecin production also includes ustilaginoidin A and the hydroxycinnamic acids p-coumaric acid and transethyl ferulate.

The NγP production process is sensitive to a variation of temperature: this one should be fixed at 25 °C. Indeed, a temperature rise to a value of 35 °C increases proportionally the production of less hydroxylated NγPs that are, therefore, less antioxidant. Finally, a temperature of 40 °C disrupts the production. The impact of moisture content on the secondary metabolite production is also significant, in particular, fonsecin production-and to a lesser extent, ustilaginoidin A and hydroxycinnamic acids. Temperature and moisture content have to be carefully monitored around 25 °C and 66%, respectively. This affirmation is equally more important if the process is up-scaled. Indeed, the cultures have been performed in flasks at laboratory scale and it is notorious in SSF; an increase of production scale leads to the formation of important thermic and water gradients in the medium bed that can negatively affect the production [START_REF] Da Silveira | Modeling the microbial growth and temperature profile in a fixed-bed bioreactor[END_REF]. However, encouraging results have been obtained at pilot scale for NγP production [START_REF] Carboué | Solid state fermentation pilot-scaled plug flow bioreactor, using partial least square regression to predict the residence time in a semicontinuous process[END_REF].

Also, although the focus is made on the most antioxidant compounds in the crude extract, the complexity of the latter may increase its overall radical scavenging potential through synergistic effects due to other bioactive compounds presence. Further studies have thus to be carried out to determine an eventual synergy. This point supports the fact that the use of the crude extract may be more convenient; also because it would not require implementing an expensive purification process and is notably allowed by the non-mycotoxigenic nature of the fungus. It is also interesting to mention that the culture medium is composed of 50% of vine shoots and 50% of wheat bran, making the process based on the valorization of by-products. Concerning vine shoots, it is a local by-product widely produced during the vine pruning in wine-producing French regions. Precisely, the vine shoots used here are previously exhausted of their polyphenol compoundsthe recovered polyphenol are used as value-added molecules-thus making the NγP production through SSF, a second way of valorization for this by-product. To complete the circle in a biorefinery approach, it is possible to consider a final way of valorization through composting of the residual fermented matrix to produce fertilizing material for the vine culture.

Conclusion

The PCA approach on the kinetics of fungal compound production under various temperatures and moisture contents rapidly allowed highlighting the patterns of the compound production during the time. Unlike the conidia production which was relatively stable with the various levels of moisture contents and temperatures, the NγP production was sensitive to the variation of these parameters. Therefore, although NγP production was associated with conidiogenesis on a temporal aspect, it was not correlated to it during the variation of the studied factors. The approach used in this study, growth kinetics under controlled levels of temperature and moisture content, allowed the insight of the metabolic process that was developed during the time. Fonsecin and to a lesser extent ustilaginoidin A and hydroxycinnamic acids are the compounds with higher radical scavenging properties and they follow a similar production pattern depending on temperature and moisture contents changes. In the case of fonsecin, a rise of temperature of 5 °C over the optimum of 25 °C drastically diminished the production. This approach also allowed us to determine the optimal temperature and moisture content conditions to maximize the production of antioxidant compounds. Potential applications-for example, included in cosmetic formulations or as a food additive-of the obtained crude extract enriched in non-toxic antioxidant compounds have yet to be investigated.

Fig. 1

 1 Fig. 1 Fungal growth curves of ergosterol and conidia under various temperatures (■ 25 °C ◆ 30 °C + 35 °C × 40 °C) and moisture contents (+ 50% ◆ 60% • 66%). Error bars represent the variability associated with the measurements of samples (color figure online)

Fig. 2

 2 Fig. 2 Fungal growth kinetic curves of metabolites under various temperatures (square 25 °C diamond 30 °C plus symbol 35 °C multiplication symbol 40 °C). Error bars represent the variability associated with the measurements of samples (color figure online) ◂

Fig. 3

 3 Fig. 3 Fungal growth kinetics curves of metabolites under the variation of moisture content (plus symbol 50% diamond 60% circle 66%). Error bars represent the variability associated with the measurements of samples (color figure online)

Fig. 4

 4 Fig. 4 PCA of the metabolic production related to the temperature variations (square 25 °C diamond 30 °C plus symbol 35 °C multiplication symbol 40 °C) in the (PC1, PC2) plane: score plot (a) and correlation loadings (b) (color figure online)

Fig. 5

 5 Fig. 5 PCA of the metabolic production related to the moisture content variations (plus 50% diamond 60% circle 66%) in the (PC1, PC2) plane: score plot (a) and correlation loadings (b) (color figure online)

Acknowledgements Support from Vinovalie (CIFRE ANRT number 2015/0027) is gratefully acknowledged. Funding Not applicable.

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Keywords: Structural robustness, progressive collapse, disproportionate collapse, local failure, yield design approach, nonlinear analysis

A progressive collapse modelling strategy coupling the yield design theory with non-linear analysis

Mohammad EL HAJJ DIAB1 , André ORCESI2* , Cédric DESPREZ3 and Jérémy BLEYER4

ABSTRACT: Several examples of structures have been severely damaged, or even reached the total collapse, after the propagation of some local failure, resulting from an accidental or exceptional event. These catastrophic events highlight the importance of not limiting safety assessment in structural design under identified conditions, but also to investigate how much the structural integrity can be preserved so that progressive collapse is avoided under an exceptional event unidentified at the design stage.

The main objective of this work is to propose a progressive collapse modelling approach in order to characterize structural robustness. Considering the occurrence of some exceptional event, the yield design theory is used to identify the part of the structure concerned by a failure mechanism. As the yield design theory is based on the infinitesimal strain assumption, an iterative coupling between the yield design approach and a non-linear analysis of the directly affected part is proposed to (i) better analyze the structural response of the part concerned by the failure mechanism, (ii) consider the development of materials and geometrical nonlinearities, and (iii) model progressive collapse within the structure. The successive iterations of the yield design approach including the deformed geometrical configuration allow to check the potential ability of the structure to develop a second line of defence. This coupling strategy is applied to a steel framed structure for illustration of the proposed concepts.

INTRODUCTION

The world has witnessed many incidents of progressive collapses on various types of structure (bridges, airports, towers, etc.), under several types of triggering events (natural hazard, blast, impact, construction errors, etc.). The associated consequences, whether human, economic or environmental, have often put robustness issues on the table (Faber et al., 2006[START_REF] Baker | On the assessment of robustness[END_REF], COST Action TU0601, 2011[START_REF] Kagho-Gouadjio | Quantification of structural robustness: application to the study of a prestressed concrete beam[END_REF][START_REF] Hoffmann | Robust Impact Design of Steel and Composite Buildings[END_REF], Demonceau et al. 2018) and pushed the civil engineering community to further analyze structural behaviour under abnormal actions.

After the Second World War, Baker et al. [START_REF] Baker | The design of framed buildings against high explosive bombs[END_REF] studied the behaviour of structures against bomb explosions and debris impacts in London during the war and identified progressive collapse in several cases of structural failures. The partial and progressive collapse of the Ronan Point Tower, in London on 16 May 1968 is one of the most famous historical failures, where an internal gas explosion on the 18th floor dislodged one of the exterior walls, which propagated and led to the collapse of the entire corner of the building [START_REF] Pearson | Ronan Point Apartment Tower Collapse and its Effect on Building Codes[END_REF]. This collapse was a decisive moment when structural robustness began to gain importance. As the consequences were considered unacceptable to the initial damage, some regulations for design against disproportionate collapse were introduced in the British design code "The Building (fifth Amendment) Regulations 1970". In 2001 the collapse of the World Trade Centre (WTC) Twin Towers, due to the impact of large commercial aircraft in New York City (NIST, 2006), led to a significant interest in the structural robustness issue. The research effort intensified in parallel of the development of structural design codes that introduce novel instructions and provisions (Demonceau 2008, Droogné et al. 2018[START_REF] Adam | Research and practice on progressive collapse and robustness of building structures in the 21st century[END_REF]. Many other collapses also highlight the issue of structural robustness, such as the partial collapse of Terminal 2E at Charles-de-Gaulle Airport in France in 2004 (El Kamari et al., 2015), the collapse of the I-35W Bridge in the USA in 2007 [START_REF] Ntsb | Collapse of I-35W Highway Bridge Minneapolis[END_REF], the collapse of the Genoa Bridge in Italy in August 2018, and of Nanfang'ao Bridge in Taiwan in October 2019. For each of these examples, a local failure led to an extended collapse.

During its service life, a structure might not be able to resist some of the actions not considered at the design stage, or for example in case of advanced degradation of structural components.

An event might lead to a local failure with a partial collapse of some structural elements that can propagate and cause the failure of other components in the structure [START_REF] Nafday | Consequence-based structural design approach for black swan events[END_REF].

The spread of failure propagation not only depends on the magnitude of the local failure, but also on the capacity of the structure to prevent or to mitigate this propagation.

Progressive collapse analysis represents an essential tool in the structural robustness assessment.

The features needed for progressive collapse modelling are related to the identification of the main aspects in the approach of the structural robustness quantification. These features are described as below:

 to perform an overall assessment of structural robustness, a large number of scenarios have to be considered. Therefore, a simplified structural modelling method is required to model the structural response under the applied scenario with a reasonable computation time;

 to assess the degree of failure propagation, the initial and the final states of failure have to be identified. Thus, the structural modelling method should follow the propagation of failure, starting from the initial failure until the final stage of failure, including the successive loss of structural elements.

 With the view to achieve a simulation of progressive collapse close to reality, the essential aspects of progressive collapse analysis have to be taken into account, which relate to the geometrical and materials non-linearities as well as the dynamic effects due to the loss of structural elements.

There are different levels of discretization for structural modelling of progressive collapse (local, global, and semi-global), where the choice depends on the dimensions of the structure and the level of precision required [START_REF] Ulm | Un modèle d'endommagement plastique: application aux bétons de structure[END_REF]. The local approach accurately simulates the structural response, using proper material constitutive modelling but at a significant computational cost, especially when dealing with non-linearities or numerous scenarios. Its application to exceptional situations can be found in [START_REF] Bao | Macromodel-Based Simulation of Progressive Collapse: RC Frame Structures[END_REF], [START_REF] Sadek | Testing and Analysis of Steel and Concrete Beam-Column Assemblies under a Column Removal Scenario[END_REF][START_REF] Gao | Capacity of semi-rigid composite joints in accommodating column loss[END_REF]. In a global approach, the structure is modelled with beam/shell elements, and a global constitutive law expressed in terms of force resultants. Significant computation time savings can be obtained at the expanse of less detailed representation of local material behaviour, especially in case of heterogeneous sections (reinforced concrete, damage, etc.). The global approach has been used to simulate an entire building in three-dimensional (3D) with nonlinear dynamic analysis [START_REF] Fu | 3-D nonlinear dynamic progressive collapse analysis of multi-storey steel composite frame buildings -Parametric study[END_REF][START_REF] Li | An improved tie force method for progressive collapse resistance design of reinforced concrete frame structures[END_REF][START_REF] Song | Experimental and analytical progressive collapse assessment of a steel frame building[END_REF]). An intermediate scale of discretization is presented by the semi-global approach, where the element section can be discretized using multi-layer or multi-fiber elements, each layer or fiber having its own constitutive law [START_REF] Mazars | Simplified strategies based on damage mechanics for concrete under dynamic loading[END_REF]. It has been widely used in the progressive collapse analysis, whether under threat-dependent scenarios [START_REF] Sun | Progressive collapse analysis of steel structures under fire conditions[END_REF][START_REF] Lu | Progressive Collapse Analysis of a Typical Super-Tall Reinforced Concrete Frame-Core Tube Building Exposed to Extreme Fires[END_REF], or threatindependent scenarios [START_REF] Bao | Robustness Assessment of RC Frame Buildings under Column Loss Scenarios[END_REF][START_REF] Kazemi-Moghaddam | Progressive collapse evaluation of Murrah Federal Building following sudden loss of column G20[END_REF][START_REF] Li | Probability-based progressive collapseresistant assessment for reinforced concrete frame structures[END_REF].

Furthermore, [START_REF] Tagel-Din | Applied element method for simulation of nonlinear materials: theory and application for RC structures[END_REF] developed the Applied Element Method to model the entire collapse process which consists in an assembly of small rigid elements, connected together by sets of normal and shear springs. This method can simulate the structure progressive collapse, starting from crack initiation and propagation, element separation, formation of debris and collision between falling debris and other structural components in reasonable time. It has been used in recent years to simulate the progressive collapse of structures as [START_REF] Sasani | Response of a reinforced concrete infilled-frame structure to removal of two adjacent columns[END_REF] and [START_REF] Marginean | Simulation of the dynamic response of steel moment frames following sudden column loss. Experimental calibration of the numerical model and application[END_REF]. Moreover, in the context of applying the Alternative load paths method, an analytical approach was developed at the University of Liège that allows predicting the response of building frames submitted to a column loss, and in particular, the associated catenary actions (Demonceau, 2008;[START_REF] Jaspart | Robustness of steel and composite building structures[END_REF]Demonceau et al., 2018).

The non-linear dynamic analysis allows to simulate the structural response by taking into account non-linearities and dynamic effects. Therefore, it is the most accurate method to simulate the progressive collapse (DCLG and CPNI, 2011;[START_REF] Stylianidis | Progressive collapse response of steel and composite buildings[END_REF]. However, the non-linear dynamic analysis is often very time-consuming and vulnerable to non-convergence issues. Hence, a non-linear static analysis is commonly used for the progressive collapse simulation. Furthermore, many studies and international codes propose to consider the dynamic effects by amplifying the gravity loading with a dynamic amplification factor (EN 1990(EN , 2003;;UFC 4-023-03, 2009;[START_REF] Marchand | Facts for Steel Buildings-Blast and Progressive Collapse[END_REF]. Besides, [START_REF] Izzuddin | Progressive collapse of multi-storey buildings due to sudden column loss -Part I: Simplified assessment framework[END_REF] propose an energy-based approach to define the pseudo-static response of building structures subject to sudden column loss.

In this work, one aims to provide a framework for progressive collapse modelling. The key question addressed here is how to propose a simplified modelling of progressive collapse by considering the dynamic effects due to some element loss, the geometrical and the material nonlinearities. For structural robustness assessment purposes, there is a high uncertainty level associated with the type and intensity of exceptional actions, so a large number of scenarios should be investigated and the structural modelling should effectively describe failure propagation.

To address these questions, the proposed method takes advantage of coupling the yield design and the non-linear finite element modelling strategies.

On the one hand, the yield design approach is used to detect the successive failure mechanisms that could occur in the structure. This direct method enables to determine the ultimate load of a structure, thus avoiding a step-by-step non-linear analysis of the structural response along the full loading path until failure (De Buhan, 2007a). On the other hand, the non-linear analysis is restricted to the sub-structure directly affected to compute structural evolution in non-linear domain. This coupling strategy is an iterative procedure to check if an alternative load path can be found in the area close to the directly affected part of a structure, and if not, how the failure propagation can be discretized until one reaches a final stage.

The paper is organized as follows. The key points of the yield design theory are reminded in Section 2, with the presentation of the static and the kinematic approaches. Then, the coupling between yield design and non-linear analyses is described in Section 3 and the numerical modelling of both approaches used for progressive collapse analysis is introduced in Section 4.

Finally, this coupling strategy is applied in Section 5 to a steel framed case study for illustration.

YIELD DESIGN APPROACH BACKGROUND

The theory of yield design (or limit analysis in the context of perfectly plastic materials) is based upon the analysis of the conditions ensuring the compatibility between the equilibrium of the structure and the local verification of a strength criterion at any point of the structure (Salençon, 2013). This method consists of the application of two approaches, static and kinematic, that enable to approximate and bracket the ultimate load of the structure by lower and upper bounds, respectively. The main benefit of this method is its ability to investigate the ultimate state of the structure with avoiding complex non-linear computations with potential convergence issues, and significantly saving computational costs. Besides, this method provides valuable information such as the failure mechanism and the most critical areas of the structure. Principles of the yield design theory can be explicitly found in the strut-and-tie modelling (NF EN 1992-1-1, 2005). Strut-and-tie models consist of struts representing compressive stress fields, of ties representing the reinforcement, and of the connecting nodes.

The forces in the elements of a strut-and-tie model should be determined by maintaining the equilibrium with the applied loads in the ultimate limit state. In some way, this approach can be related to the lower bound static approach of yield design [START_REF] Vincent | Yield design based numerical analysis of three-dimensional reinforced concrete structures[END_REF].

The yield design approach is based upon the infinitesimal strain theory, meaning that the geometrical deformation is neglected. The important concepts are now recalled for a 3D continuum 𝛺. The applied loading mode is defined through 𝑄 (𝑄 𝑖 , 𝑖 = 1, … , 𝑛) depending on n parameters. The loading can be body forces (gravity for example) or surface forces on the edge of the structure. The constituent material strength properties are characterized, at each point 𝑥 of the system 𝛺, by a convex strength domain 𝐺(𝑥). Therefore, an admissible stress tensor 𝜎(𝑥) has to comply with the following strength condition:

∀𝑥 ∈ 𝛺, 𝜎(𝑥) ∈ 𝐺(𝑥) (1)
The yield design approach enables to identify the domain of potentially safe loads 𝐾 based on the compatibility between the equilibrium of the structure and the local resistance at any point of the structure. The domain of potential safe loads 𝐾 is defined as the set of loads 𝑄, for which there exists a stress tensor 𝜎(𝑥) statically admissible with the loading mode 𝑄 and for which the strength condition is satisfied at each point of the system i.e.:

𝑄 ∈ 𝐾 ⟺ { 1. ∃ 𝜎(𝑥) statically admissible in the loading mode with load 𝑄 (equilibrium condition)

2. 𝜎(𝑥) ∈ 𝐺(𝑥), ∀ 𝑥 ∈ 𝛺 (resistance condition) (2)
Restricting the above definition to a finite set of statically admissible stress fields, a lower bound estimate of the potentially safe loads domain can be obtained.

Let us also mention that yield design (or limit analysis) theory offers an alternative characterization of the domain of potentially safe loads through the so-called kinematic approach. Relying on the use of virtual collapse mechanisms, equilibrium equations are written in a weak fashion using the virtual work principle. The structure stability is ensured if and only if the work of external forces is less than the maximum resisting work, see more details in (de Buhan, 2007b;Salençon, 2013). Applying the kinematic approach on a finite set of potential collapse mechanisms, obtained through a finite-element discretization for instance, yields an upper bound estimate to the structure collapse load.

The yield design approach enables to approximate and bracket the ultimate load 𝑄 𝑢 of the structure by a lower bound 𝑄 𝑢,𝑠 with the static approach and upper bound 𝑄 𝑢,𝑘 with the kinematic approach ( 𝑄 𝑢,𝑠 < 𝑄 𝑢 < 𝑄 𝑢,𝑘 ), and it also identifies the corresponding failure mechanism. One can then identify the capacity of the structure and estimate if the structure can resist a load 𝑄 under an exceptional situation. The structure is assumed to be able to resist an applied load if 𝑄 < 𝑄 𝑢,𝑠 , which indicates that the structure can support the applied load without any failure. Conversely, the structure is unable to resist an applied load if 𝑄 > 𝑄 𝑢,𝑘 , meaning that the structure fails under the mechanism identified by the kinematic approach of yield design. By comparing the applied load 𝑄 with 𝑄 𝑢,𝑠 , which is the lower bound of the ultimate load estimation, one avoids the overestimation of the structural capacity.

The corresponding failure mechanism identifies the directly affected part, and the critical points of the structure in which the structural element cannot resist the applied forces. Besides, the failure mechanism also allows to estimate if there is either a loss of stability of the directly affected part, or the possibility to develop an alternative functioning. When the failure mechanism indicates the mechanical instability of some elements under the applied load, there is no interest to simulate the geometrical displacements of the affected part after the failure.

Conversely, if the failure mechanism reveals a potential alternative equilibrium state, the geometrical displacements might lead to a change in the redistribution of the forces in the elements, and the structure might be able to function in a different manner. Furthermore, this alternative functioning can increase the structural capacity to withstand the applied load.

Therefore, there is an interest to simulate the geometrical changes in this case, and to study the evolution of the structural capacity with the geometrical displacements in order to investigate the change in ability of the structure to resist the applied load. As the yield design approach is based on infinitesimal strain assumption, considering that the materials are elastic perfectly plastic, the main challenge to use this method for robustness analysis is to consider the geometrical non-linearities and to iteratively simulate the propagation of the failure. Using a non-linear analysis on the affected part identified by the yield design is proposed thereafter to offset the limits of the yield design calculation.

COUPLING BETWEEN THE YIELD DESIGN APPROACH AND A NON-LINEAR ANALYSIS

The principle of coupling a yield design approach with another type of analysis was used by [START_REF] Pham | Analyse par le calcul à la rupture de la stabilité au feu des panneaux en béton armé de grandes dimensions[END_REF] and [START_REF] Bleyer | Méthodes numériques pour le calcul à la rupture des structures de génie civil[END_REF] to model the geometrical and material changes. The behaviour of a reinforced concrete panel under the fire action was considered and a thermo-elastic analysis was used to identify the geometrical displacements. The materials degradation was then integrated in a yield design calculation in order to identify the stability factor of the deformed structure.

This study proposes an embedded yield design approach within a non-linear analysis. In order to follow the propagation of failure and identify the successive structural states, an iterative procedure of yield design calculation is proposed (El Hajj Diab et al., 2019, El Hajj Diab, 2019).

At each step 𝑖 of the iterative process, the yield design calculation checks the ability of the structure to resist the applied load 𝑄, and identifies the failure mechanism if 𝑄 > 𝑄 𝑢,𝑠 𝑖 . The proposed structural modelling strategy is based on the coupling between the yield design approach and the non-linear analysis in an iterative procedure. This procedure is illustrated in Figure 1, with the following steps (each step is identified in Figure 1):

1) The yield design approach is applied to identify the ultimate load 𝑄 𝑢,𝑠 and the failure mechanism.

2) 𝑄 𝑢,𝑠 and applied loads 𝑄 are compared.

3) If 𝑄 < 𝑄 𝑢,𝑠 , the current structural configuration can support the applied load, and the failure stops at this stage.

4) If 𝑄 ≥ 𝑄 𝑢,𝑠 , a failure mode potentially occurs under the applied loads.

5) The failure mechanism identified by the kinematic approach allows to identify the affected part and to estimate if there is either a loss of stability or the possibility to develop an alternative functioning.

6) The possibility of developing an alternative functioning by the tensile membrane action is investigated by checking whether the joints on both sides of the directly affected part (Figure 2a) are connected to the indirectly affected part or not.

7) If there is no continuity of the directly affected part beams with the indirectly affected parts on both sides, the failure mechanism indicates a mechanical instability of some elements, and the affected part is removed for the next iteration of the yield design calculation. If the entire structure is affected, then the overall structure collapses.

8) If the failure mechanism reveals that the directly affected part may develop an alternative functioning after large displacement, a non-linear analysis is applied to this part, in order to calculate the geometric displacements. The effect of the indirectly affected part on the substructure of the directly affected part is taken into account by considering horizontal springs at the extremities 𝑁 𝑖 of the substructure, as shown in Figure 2b. The rotation and the vertical displacement are restrained at 𝑁 𝑖 with allowing the horizontal displacement, and the springs are attached to fixed supports. In order to characterize these springs, the indirectly affected part is assumed to be perfectly elastic as assumed in Demonceau ( 2008), and the springs are assumed as fully elastic. The stiffness 𝐾 𝑖 of each spring is calculated by applying a unitary force (1𝑘𝑁) at the 𝑖 𝑡ℎ corresponding extremity (Figure 2c), where 𝐾 𝑖 is described as follow:

𝐾 𝑖 = 1 𝑢 𝑁 𝑖 (3)
where 𝑢 𝑖 is the horizontal displacement of the joint 𝑁 𝑖 . 9) A non-linear analysis is applied on the substructure identified in the previous step with a view to identify the geometrical change under the applied loads 𝑄.

10) The geometric changes are integrated in the next iteration of the yield design calculation by updating the structure geometry.

11) The iterative procedure continues until the end of collapse, for which the ultimate load on remaining elements is larger than the applied load, or until total collapse of the structure. In order to take into account the dynamic effects resulting from a geometrical modification after the loss structural component, though preventing the use of a full dynamic analysis, the structural response can be estimated from a non-linear static response under amplified gravity loading using a dynamic amplification factor. This aspect is mentioned in many codes and standards, such as EN 1990as EN (2003)), GSA (2003), andUFC 4-023-03 (2009). The recommended value of the dynamic amplification factor is often 2.0, and the load amplification should only be applied over the bays adjacent to the removed elements. Otherwise, [START_REF] Marchand | Facts for Steel Buildings-Blast and Progressive Collapse[END_REF] mentioned that the factor of 2.0 is very conservative, and that a value between 1.3 and 1.5 is more realistic for a non-linear analysis when members achieve significant plastic rotation and displacements.

The idea of the proposed approach is to discretize the failure propagation associated with progressive collapse and to exploit the benefits of yield design approach in the progressive collapse simulation. The principal advantage is to identify the ultimate load and the failure mechanism in a direct manner, then avoiding some convergence issues, and significantly saving computational costs. The role of the iteration of the yield design calculation is to identify whether or not there is a failure under the applied load. Furthermore, the identification of the failure mechanism allows to identify the directly affected part and then to identify whether or not there is a need to apply a detailed analysis with a non-linear analysis. Hence, the importance of the proposed method is to avoid the detailed analysis in the cases where there is no need, and to apply the detailed analysis on a localised part of the structure.

PROGRESSIVE COLLAPSE ANALYSIS FOR FRAMED STRUCTURES

Numerical modelling of yield design approach

The model proposed by Bleyer and de Buhan ( 2013) is used herein to approximate the ultimate load by some lower and upper bounds for frame structures. The static approach requires the maximization of a variable parameter with solving a linear optimization problem under nonlinear convex constraints. Conversely, the kinematic approach requires the minimization of a variable parameter with solving a non-linear convex optimization problem under linear constraints. The model considers that the element is infinitely resistant with respect to shear and torsion forces. The yield criterion 𝐺 𝑢 is identified in the 3D space involving axial force 𝑁 and the bending moments 𝑀 𝑦 and 𝑀 𝑧 based on an analysis of the cross-section material properties.

To check the local resistance of structural components in a global structure, [START_REF] Bleyer | Yield surface approximation for lower and upper bound yield design of 3D composite frame structures[END_REF] proposed an approximation procedure to describe the yield surface with a few parameters only.

In addition to strength limits based on the cross-section material properties, critical buckling loads are also verified according to Euler load formulation (NF EN 1992-1-1, 2005), which is expressed as:

𝑁 𝐸𝑢𝑙𝑒𝑟 = 𝜋 2 𝐸𝐼 𝑚𝑖𝑛 𝑙 𝑓 2 (4)
where 𝐸𝐼 𝑚𝑖𝑛 is the minimum bending stiffness, and 𝑙 𝑓 is the effective length of the beam element. The boundary conditions of the element are considered as fixed for the numerical mockup of the following case study. Hence, 𝑙 𝑓 = 0.5𝑙, where 𝑙 is the element length. It is worth to note that the Euler formula represents the capacity of a perfectly elastic column to resist flexural buckling under compression, as a function of its elastic stiffness (EI) and interaction with yielding [START_REF] Zhao | Members Subjected to Bending and Compression[END_REF]. This equation is considered in the following as a first approximation and further detailed interaction relationship between axial compression and bending should be used in future studies for beam-column elements.

The ultimate load computation, from either a static or a kinematic approach and after a finiteelement discretization, finally consists in solving a second-order cone programming (SOCP)

problem, for which very efficient interior-point solvers are available, such as Mosek [START_REF] Mosek | The MOSEK optimization software[END_REF] for instance. More detailed information can be found in the article of Bleyer and de Buhan (2013).

Numerical modelling of non-linear analysis

The MATLAB toolbox FEDEASLab [START_REF] Filippou | FEDEASLab getting started guide and simulation examples[END_REF]) is used to apply the non-linear static analysis, where the multilayer approach is applied in the structural modelling.

The choice of this approach is based on (i) the possibility to consider the geometrical and material non-linearities with providing local information about the state of section and materials, and (ii) the computation cost of this method which is lower than with a local approach regarding the need for multi-scenario analysis. FEDEASLab enables to study the response of a 2D beam element with distributed inelasticity under non-linear geometry [START_REF] Spacone | Fibre beam-column model for non-linear analysis of r/c frames: part I[END_REF]. The nonlinear analysis is performed by using the imposed displacement method, where the displacement increment enables to control structural behaviour close to the collapse and prevent the frame from drifting apart suddenly in the ultimate state [START_REF] Nutal | Robustness of Steel and Concrete Structures[END_REF]. The corotational formulation [START_REF] Le | Efficient formulation for dynamics of corotational 2D beams[END_REF][START_REF] Le | Nonlinear dynamics of lexible structures using corotational beam elements[END_REF][START_REF] Le | Corotational formulation for nonlinear dynamics of beams with arbitrary thin-walled open cross-sections[END_REF] is chosen to take into account the geometrical non-linearities using the FEDEASLab toolbox that integrates the effect of the large displacements and rotations). It should be mentioned that the shear effects are neglected herein when modelling the directly affected part, which is accepted herein for medium to large span to depth ratios of the member. For the examples considered in the following, the constitutive law of materials in the non-linear analysis is assumed as a bilinear elastic, perfectly plastic relationship. Such assumption is considered in the non-linear analysis with a view to be compatible with the yield design calculation.

APPLICATION TO A STEEL-FRAMED STRUCTURE

The structure considered for illustration is a 2D representative five-storey steel-framed building with ten bays. The geometric layout of the numerical mockup is presented in The combination of actions 𝐷𝐿 and 𝐿𝐿 refers to an ultimate state as recommended in the Eurocodes NF EN 1990[START_REF] Nutal | Robustness of Steel and Concrete Structures[END_REF] for an accidental situation, for which one can mitigate the reserve on the applied loads by using smaller partial factors than those with a persistent situation, as follows:

 persistent situation: 𝑊 𝑝 = 1.35𝐷𝐿 + 1.5𝐿𝐿

 accidental situation: 𝑊 𝑎 = 𝐷𝐿 + 0.5𝐿𝐿

This design configuration respects the Eurocodes serviceability and ultimate limit states.

Moreover, the load capacity of the intact structure is verified under the combination of loads in the persistent situation 𝑊 𝑝 = 52.5 𝑘𝑁 𝑚 ⁄ . In the structural robustness assessment, the combination of loads used under the local failure scenarios corresponds to the accidental situation 𝑊 𝑎 = 31.6 𝑘𝑁 𝑚 ⁄ . Further, the dynamic amplification factor used in this example is 1.5. The load amplification is applied only on the directly affected part, which normally contains all the beams, columns and beam-to-column joints located just above lost column(s).

The structural response under the applied local failure scenarios can be classified into four categories, as described below:

 C1 without consequences: the local failure does not lead to any failure mechanism,  C2 without collapse of the directly affected part: a failure mechanism is initiated according to the yield design theory. For a frame structure, the directly damaged part contains beams, columns and beam-to-column joints located just above lost column(s). However, the structure succeeds to find a second line of defence (i.e. The affected or the collapsed parts are quantified by the length of associated beams, where this value represents the consequences and the part of a structure that becomes out of service. The ultimate loads identified by the yield design calculation are noted as 𝑄 𝑢,𝑠 𝐶𝑖,𝑗 and 𝑄 𝑢,𝑘 𝐶𝑖,𝑗 for static and kinematic approaches, respectively, where 𝑖 represents the category of the local failure scenario 𝐶𝑖, and 𝑗 is the iteration number of yield design calculation.

Illustration of local failure scenario C1

The first scenario of local failure is the notional loss of column # 6. Figure 6 The non-linear static analysis on the whole structure indicates that the ultimate load is 𝑄 𝑢,𝑁𝐿 𝐶1 = 38.8 𝑘𝑁 𝑚 ⁄ (Figure 7), where 𝑄 𝑢,𝑠 𝐶1,1 < 𝑄 𝑢,𝑁𝐿 𝐶1 < 𝑄 𝑢,𝑘 𝐶1,1 . The computation cost with the nonlinear static analysis is 𝑡 𝐶1 𝑁𝐿 = 1,230.0𝑠, while it can decrease by using the proposed coupling strategy to 𝑡 𝐶1 𝐶𝑆 = 14𝑠 with one iteration of yield design calculation.

Illustration of local failure scenario C2

The second scenario is the notional loss of column # 26, as shown in Figure 8 The substructure corresponding to the initial damaged part is presented in Figure 10. The stiffness of the spring at each extremity of the substructure is identified according to Equation

(3). As the indirectly affected part is symmetric for both sides of the directly affected part, the stiffness of springs are similar for both sides of each story, as shown in Table 1. The non-linear analysis of the substructure corresponding to the initial damaged part (Figure 10) indicates that it achieves to support the applied loads after a certain deflection, where Figure 11 presents the deflection 𝛿 𝑁 6 at Joint 𝑁 6 and the load (𝑄) diagram, which shows the increment of the substructure capacity due to the development of the tensile membrane action. The substructure achieves to find an alternative equilibrium state under 𝑊 𝑎 = 31.6 𝐾𝑁 𝑚 ⁄ with 𝛿 𝑁 6 = 0.63𝑚. The tensile membrane action is revealed by the normal force and bending moment diagrams at the Figure 12 shows the effect of large deflections at the joint 𝑁 1 , where after the bending resistance limit is reached, there is an increase of tensile stress in beams and decrease of bending moment, which helps the structure to reach an alternative equilibrium configuration. The non-linear analysis identifies if the substructure can develop an alternating functioning state, and it identifies the geometrical changes that correspond to the new equilibrium state.

However, the development of a second line of defence in the substructure does not mean that the indirectly affected part of the structure can resist the horizontal forces developed at the joints.

Here, the second iteration of the yield design calculation allows to check if the structure can resist these forces. geometrical deformation of the substructure identified in the non-linear analysis is integrated in the model for the next iteration of yield calculation, as shown in Figure 13. In order to evaluate the assumption to calculate the deformation of the damaged structure by isolating the affected part with considering horizontal springs at the extremities of the substructure, a non-linear static analysis is also performed on the whole structure to identify the margin of error. Figure 14 shows that the maximum vertical deflection of the joint on the top of the removed column 𝛿 26 = 0.67𝑚. As 𝛿 26 (0.67𝑚) > 𝛿 𝑁 6 (0.63𝑚), the deflection identified in the analysis of the whole structure is larger than that of the substructure with a relative error about 6%. It is worth to note that the stiffnesses of the horizontal springs at the extremities of the substructure are calculated by a simplified method to represent the response of the indirectly affected part. The rotations of the supports in the substructure are omitted, which may contribute to the difference in the obtained deflection. Two iterations of yield design calculation and one intermediate non-linear analysis have been done using the proposed coupling strategy with a total computation time 𝑡 26 𝐶𝑆 = 697.0𝑠, while the non-linear analysis on the whole structure requires 𝑡 26 𝑁𝐿 = 8,620.0𝑠. Hence, for this example one decreases computation cost by 91.9 %. -25 -

Illustration of local failure scenario C3

The third scenario is the notional loss of the columns # 1 and 6 (Figure 15). The ultimate loads of the static and kinematic approaches of the first iteration of the yield design calculation are 𝑄 𝑢,𝑠 𝐶3,1 = 11.9 𝑘𝑁 𝑚 ⁄ and 𝑄 𝑢,𝑘 𝐶3,1 = 12.5 𝑘𝑁 𝑚 ⁄ . As 𝑄 𝑢,𝑠 𝐶3,1 < 𝑊 𝑎 (31.6 𝑘𝑁 𝑚 ⁄ ) , the structure cannot resist the gravity loads, where the failure mechanism presented in Figure 16 occurs when the bending moment reaches the bending resistance limit of the beams. Furthermore, the failure mechanism identified in Figure 16 indicates the loss of stability of the directly affected part, where there is no possibility of an alternative functioning. Therefore, the affected part is identified automatically by the displacement vector calculated by the kinematic approach of yield design, and it is removed for the next iteration of the yield design calculation (Figure 17). Then, the next iteration indicates that the ultimate loads with the new structural configuration are 𝑄 𝑢,𝑠 𝐶3,2 = 152 𝑘𝑁 𝑚 ⁄ and 𝑄 𝑢,𝑘 𝐶3,2 = 159.2 𝑘𝑁 𝑚 ⁄ . Hence, 𝑊 𝑎 = 31.6 𝑘𝑁 𝑚 ⁄ < 𝑄 𝑢,𝑠 𝐶3,2 , and the propagation of the failure stops at this stage with a collapsed part (sum of beams length) equal to 50𝑚 , which is equal to the initial affected part. Therefore, this scenario corresponds to the category C3 of scenarios for which one observes a collapse of the directly affected part.

Figure 17. Layout of the structure for the second iteration of the yield design calculation after the loss of columns # 1 and 6.

As a comparison, a non-linear static analysis is applied on the whole structure. Figure 18 shows the maximum vertical deflection 𝛿 1 on the top of the removed column # 1 according to the uniform load 𝑄. As no alternative function can be developed, the load capacity of the structure cannot ensure equilibrium and its maximum value is reached for 𝛿 1 > 0.13𝑚. Hence, the ultimate load is 𝑄 𝑢,𝑁𝐿 𝐶3 = 12 𝑘𝑁 𝑚 ⁄ , where 𝑄 𝑢,𝑠 𝐶3,1 < 𝑄 𝑢,𝑁𝐿 𝐶3 < 𝑄 𝑢,𝑘 𝐶3,1 . x

x: divergence of calculation performed when using the proposed strategy with computation time 𝑡 𝐶3 𝐶𝑆 = 22𝑠, while the nonlinear analysis on the whole structure requires 𝑡 𝐶3 𝑁𝐿 = 1,157.0𝑠.

Illustration of local failure scenario C4

The fourth scenario is the notional loss of the columns # 11 and 16 (Figure 19). The first iteration of the yield design calculation indicates that 𝑄 𝑢,𝑠 𝐶4,1 = 14. The failure mechanism (Figure 20) indicates that there is a possibility of developing an alternative functioning by the tensile membrane action due to the continuity of the beams on both sides of the affected part. Therefore, a non-linear analysis is performed on the substructure of the affected part (Figure 21). The stiffness of each spring is presented in Table 2. The second iteration of yield design calculation indicates that 𝑄 𝑢,𝑠 𝐶4,2 = 21.3 𝑘𝑁 𝑚 ⁄ and 𝑄 𝑢,𝑘 𝐶4,2 = 22.4 𝑘𝑁 𝑚 ⁄ . After the large displacement, the ultimate loads of the structure increase compared to the first iteration ( 𝑄 𝑢,𝑠 𝐶4,2 > 𝑄 𝑢,𝑠 𝐶4,1 and 𝑄 𝑢,𝑘 𝐶4,2 > 𝑄 𝑢,𝑘 𝐶4,1 ), while 𝑄 𝑢,𝑠 𝐶4,2 < 𝑊 𝑎 = 31.6 𝑘𝑁 𝑚 ⁄ .

Hence, the indirectly affected part of the structure is not able to resist the forces developed during the development the tensile membrane action. The computation time of the non-linear analysis on the whole structure is 𝑡 𝐶4 𝑁𝐿 = 12,351.0𝑠.

The use of the proposed coupling strategy indicates that three iterations of yield design calculation and one iteration of non-linear analysis on the substructure of the directly affected part are performed, where the total computation cost is 𝑡 𝐶4 𝐶𝑆 = 1,105.0𝑠.

CONCLUSIONS

An original structural modelling strategy is proposed and based on an iterative coupling between the yield design approach and a non-linear analysis, to simulate progressive collapse.

The adoption of the yield design theory allows a simplification in the analysis of the structural response. It is a direct approach that significantly saves computation time and mitigates It should be noted that several assumptions were made for simplification and that further work should be considered before considering the illustrative example for wider applications. Indeed, the affected part was isolated each time for non-linear analysis with a simplified modelling of the joints at the edges. As the objective of this work was not to develop a macro-model of connections but rather to focus on a strategy to iteratively follow the failure propagation, this assumption was adopted herein. Some further development are obviously required as the omission of the rotations of the supports in the sub-structure can cause an error in the prediction of the deflection. Joint behaviour strongly influences progressive collapse (strength, stiffness and deformation capacity) and an accurate modelling is needed to cater for important features of real behaviour. Also, including the displacement of the indirectly affected part should be considered to properly take into account P-delta effects. For the risk of buckling, the Euler load formulation was used as a first approximation and an adequate interaction relationship between axial compression and bending should be used in future studies for beam-column elements.

Several research works in the literature have already developed macromodels for modelling in an accurate way the response of supports in submodels [START_REF] Bao | Macromodel-Based Simulation of Progressive Collapse: RC Frame Structures[END_REF][START_REF] Sadek | Robustness of Composite Floor Systems with Shear Connections: Modeling, Simulation, and Evaluation[END_REF][START_REF] Vlassis | Progressive collapse of multi-storey buildings due to sudden column loss-Part II: Application[END_REF][START_REF] Vidalis | Improving the resistance to progressive collapse of steel and composite frames[END_REF] and integration of these developments in the proposed framework represents an interesting perspective. Also, the comparison with other approaches for numerical modelling should be considered [START_REF] Li | Analysis of robustness of steel frames against progressive collapse[END_REF][START_REF] Zhang | Robustness of steel moment frames in multi-column-removal scenarios[END_REF]. Some analytical methods (Demonceau, 2008;[START_REF] Huvelle | Complete analytical procedure to assess the response of a frame submitted to a column loss[END_REF] are proposed in the literature to predict the response of a frame structure submitted to a column loss with isolating the directly affected part. As these methods enable to identify the geometrical deformation under the applied loads, they can replace the iteration of the non-linear analysis in the proposed iterative strategy, which could again reduce the computation cost.

The methodology was applied to steel framed structures. An application to other materials such as reinforced concrete elements should be considered to investigate the potential of coupling between the yield design theory and non-linear analyses. Future research is needed to test other non-linear finite element models (with different formulations, Lagrangian, updated Lagrangian and corotational) and/or adapt the iterative coupling strategy for different types of materials.

The comparison with previous studies, both numerical and experimental, should be considered for the implementation of the proposed approach and reinforced concrete structures [START_REF] Parisi | Progressive collapse assessment of gravity-load designed European RC buildings under multi-column loss scenarios[END_REF][START_REF] Kiakojouri | Progressive collapse of framed building structures: Current knowledge and future prospects[END_REF][START_REF] Yu | Effect of boundary conditions on progressive collapse resistance of RC beam-slab assemblies under edge column removal scenario[END_REF][START_REF] Qiang | Kinked rebar configurations for improving the progressive collapse behaviours of RC frames under middle column removal scenarios[END_REF].

This work aims to be used in the future, by providing some recommendations on design for robustness, in link with the evolution of standards (revision of Eurocodes related to robustness, development of technical recommendations for robustness for fib Model Code 2020).

( 1 )

 1 Figure 1. Proposed structural modelling strategy for progressive collapse analysis.

  Figure 2. Characterization of springs of the substructure.

  Figure 3. The bay dimensions are 6𝑚 or 5𝑚 in the x direction and 4𝑚 in the y direction. The constitutive structural elements are beams with cross-section IPE360, and columns with cross-section HEB500. Material properties are those of S355 steel grade, where the yield limits is 355 MPa and the elastic Young Modulus (E) is 210 GPa. The floors consist of 25 cm thick reinforced concrete slabs carried by the principal steel beams. The beam/column and column/footing connections are considered as rigid joints as no additional bracing is used.

Figure 3 .

 3 Figure 3. Layout of steel-frame building (dimensions in meter).

  by the catenary action developed in the directly affected part),  C3 with collapse of the directly affected part: the structure does not succeed to find a second line of defence, and the failure mechanism leads to a partial collapse without further propagation beyond the first failure mechanism,  C4 expanded collapse: the indirectly affected part of the structure cannot support the redistribution of loads in the new structural configuration and the collapse propagates out of the directly affected part.

  shows the loading mode of the structure under this local failure scenario, and it shows the amplification of load on the directly affected part by the factor 1.5 to represent the dynamic effect. The first iteration of the yield design calculation indicates that 𝑄 𝑢,𝑠 𝐶1,1 = 37.7 𝑘𝑁 𝑚 ⁄ and 𝑄 𝑢,𝑘 𝐶1,1 = 39.2 𝑘𝑁 𝑚 ⁄ . As 𝑊 𝑎 (31.6 𝑘𝑁 𝑚 ⁄ ) < 𝑄 𝑢,𝑠 𝐶1,1 (36.7 𝑘𝑁 𝑚 ⁄ ), the structure can resist the loss of the column # 6 without initiation of any failure mechanism (category C1).

Figure 6 .

 6 Figure 6. Notional loss of column # 6.

Figure 7 .

 7 Figure 7. Deflection 𝛿 6 -load diagram of non-linear static analysis on the whole structure after the loss of column # 6.

Figure 8 .

 8 Figure 8. Notional loss of column # 26.

Figure 9 .

 9 Figure 9. Failure mechanism identified by the first iteration of the yield design calculation after the loss of column # 26.

Figure 10 .

 10 Figure 10. Substructure of the initial damaged part after the loss of column # 26.

𝐾

Figure 11 .

 11 Figure 11. Deflection-load diagram of non-linear static analysis of the initial damaged part after the loss of column # 26.

Figure 12

 12 Figure12Normal force -Bending moment curve at the joint 𝑁 1 after the loss of column # 26.

Figure 13 .

 13 Figure 13. Deformed model of the structure for the second iteration of yield design calculation after the loss of column # 26.

Figure 14 .

 14 Figure 14. Deflection-load diagram of non-linear static analysis on the whole structure after the loss of column # 26.

Figure 15 .

 15 Figure 15. Notional loss of columns # 1 and 6.

Figure 16 .

 16 Figure 16. Failure mechanism identified by to first iteration of the yield design calculation after the loss of columns # 1 and 6 (the displacement are not representative).

Figure 18 .

 18 Figure 18. Deflection 𝛿 1 -load diagram of non-linear static analysis on the whole structure after the loss of columns # 1 and 6.

  𝑄 𝑢,𝑠 𝐶4,1 < 𝑊 𝑎 = 31.6 𝑘𝑁 𝑚 ⁄ , the failure propagates according to the failure mechanism presented in Figure 20.

Figure 19 .

 19 Figure 19. Notional loss of columns # 11 and 16.

Figure 20 .

 20 Figure 20. Failure mechanism identified by the first iteration of the yield design calculation after the loss of columns # 11 and 16.

Figure 21 .

 21 Figure 21. Substructure of the initial damaged part after the loss of columns # 11 and 16.

Figure 22

 22 Figure 22 presents the diagram of the deflection 𝛿 𝑁11 on the top of the removed column # 11 and the applied load 𝑄. The load capacity of the structure increases until one reaches the applied

Figure 22 .

 22 Figure 22. Deflection-load diagram of non-linear static analysis of the initial damaged part after the loss of columns # 11 and 16.

Figure 23 .

 23 Figure 23. Normal force -Bending moment curve at the joint 𝑁 1 after the loss of columns # 11 and 16.

Figure 26 .

 26 Figure 26. Layout of the structure for the third iteration of the yield design calculation after the loss of columns # 11 and 16.

Figure 27 .Figure 28 .

 2728 Figure 27. Deflection 𝛿 11 -load diagram of non-linear static analysis on the whole structure after the loss of columns # 11 and 16.

Figure 29 .

 29 Figure 29. Deflection 𝛿 16 -normal force diagram at base supports after the loss of columns # 11 and 16.

  One main drawback of yield design theory is that one does not identify the capacity of the structure to find a new behaviour (catenary action) under large displacements as this behaviour is not investigated by this theory. A non-linear analysis based on a corotational formulation is then used to analyze the structural response of the part concerned by the failure mechanism. One can analyze the development of an alternative functioning in the directly affected part, and simulate the geometrical and the material non-linearities. The successive iterations of the yield design approach with the deformed geometrical configuration allow to check the ability of the structure to develop a second line of defence.The illustrative example shows the capacity of the proposed coupling strategy to model the failure propagation by identifying the initial and the final states of collapse. Besides, the comparison of the computation time shows the capacity of this strategy to decrease this time compared to a full non-linear analysis on the whole structure. One can investigate in a reasonable computation time a large number of local failure scenarios, so a general assessment of structural robustness can be performed.

Table 2 .

 2 Stiffness of springs at each extremity of the substructure after the loss of columns # 11 and 16.

	𝐾 1	𝐾 2	𝐾 3	𝐾 4	𝐾 5
	76,726.0 𝑘𝑁/𝑚	15,401.3 𝑘𝑁/𝑚	6,673.7 𝑘𝑁/𝑚	3,852.4 𝑘𝑁/𝑚	2,478.0 𝑘𝑁/𝑚
	𝐾 16	𝐾 17	𝐾 18	𝐾 19	𝐾 20
	159,317.8 𝑘𝑁/𝑚	50,690.2 𝑘𝑁/𝑚	26,347.9 𝑘𝑁/𝑚	16,781.2 𝑘𝑁/𝑚	11,277.4 𝑘𝑁/𝑚

MAST-EMGCU, Univ Gustave Eiffel, IFSTTAR, F-77447 Marne-la-Vallée, France

* MAST-EMGCU, Univ Gustave Eiffel, IFSTTAR, F-77447 Marne-la-Vallée, France, corresponding author : E-mail : andre.orcesi@univ-eiffel.fr

MAST-EMGCU, Univ Gustave Eiffel, IFSTTAR, F-77447 Marne-la-Vallée, France

Laboratoire Navier, ENPC, Univ Gustave Eiffel, CNRS, F-77447 Marne-la-Vallée, France

indirectly affected part, as shown in Figure 25, where the bending moment at b1 and b2 reaches the bending limit of the column element. The failure mechanism indicates the loss of stability of the affected part, where it is removed for the next iteration of the yield design calculation, as shown in Figure 26. Finally, the ultimate loads identified are 𝑄 𝑢,𝑠 𝐶4,3 = 152 𝑘𝑁 𝑚 ⁄ and 𝑄 𝑢,𝑘 𝐶4,3 = 159.2 𝑘𝑁 𝑚 ⁄ , which means that the collapse stops at this stage as 𝑊 𝑎 = 31.6 𝑘𝑁 𝑚 ⁄ < 𝑄 𝑢,𝑠 𝐶4,3 . The beam total lengths of the initially affected and the collapsed parts for this scenario are equal to 85𝑚 and 110𝑚, respectively.

Therefore, the collapse expands out of the directly affected part and this scenario falls in category C4. EN 1990[START_REF] Nutal | Robustness of Steel and Concrete Structures[END_REF]. EN 1990