Ngoc-Tien Le

Benjamin Lecouteux

Laurent Besacier

Joint ASR and MT Features for Quality Estimation in Spoken Language Translation

This paper aims to unravel the automatic quality assessment for spoken language translation (SLT). More precisely, we propose several effective estimators based on our estimation of transcription (ASR) quality, translation (MT) quality, or both (combined and joint features using ASR and MT information). Our experiments provide an important opportunity to advance the understanding of the prediction quality of words in a SLT output that were revealed by MT and ASR features. These results could be applied to interactive speech translation or computer-assisted translation of speeches and lectures. For reproducible experiments, the code allowing to call our WCE-LIG application and the corpora used are made available to the research community.

Introduction

Automatic quality assessment of spoken language translation (SLT), also named confidence estimation (CE), is an important topic because it allows to know if a system produces (or not) user-acceptable outputs. In interactive speech to speech translation, CE helps to judge if a translated turn is uncertain (and ask the speaker to rephrase or repeat). For speech-to-text applications, CE may tell us if output translations are worth being corrected or if they require retranslation from scratch. Moreover, an accurate CE can also help to improve SLT itself through a second-pass N-best list re-ranking or search graph re-decoding, as it has already been done for text translation in [START_REF] Bach | Goodness: A method for measuring machine translation confidence[END_REF] and [START_REF] Luong | Word confidence estimation for smt n-best list re-ranking[END_REF], or for speech translation in [START_REF] Besacier | Spoken language translation graphs redecoding using automatic quality assessment[END_REF]. Consequently, building a method which is capable of pointing out the correct parts as well as detecting the errors in a speech translated output is crucial to tackle above issues.

Outline The outline of this paper goes simply as follows: section 2 reviews the state-of-the-art on confidence estimation for ASR and MT. Our word confidence estimation (WCE) system using multiple features is then described in section 3. The experimental setup (notably our specific WCE corpus) is presented in section 4 while section 5 evaluates our joint WCE system and finally, section 6 concludes this work and gives some perspectives.

Related Work on Confidence Estimation for ASR and MT

Several previous works tried to propose effective confidence measures in order to detect errors on ASR outputs. Confidence measures are introduced for Out-Of-Vocabulary (OOV) detection by [START_REF] Asadi | Automatic detection of new words in a large vocabulary continuous speech recognition system[END_REF]. [START_REF] Young | Recognition confidence measures: Detection of misrecognitions and out-of-vocabulary words[END_REF] extends the previous work and introduces the use of word posterior probability (WPP) as a confidence measure for speech recognition. Posterior probability of a word is most of the time computed using the hypothesis word graph [START_REF] Kemp | Estimating confidence using word lattices[END_REF]. Also, more recent approaches [START_REF] Lecouteux | Combined low level and high level features for out-of-vocabulary word detection[END_REF] for confidence measure estimation use side-information extracted from the recognizer: normalized likelihoods (WPP), the number of competitors at the end of a word (hypothesis density), decoding process behavior, linguistic features, acoustic features (acoustic stability, duration features) and semantic features.

In parallel, the Workshop on Machine Translation (WMT) introduced in 2013 a WCE task for Machine Translation. [START_REF] -F. Han | Quality estimation for machine translation using the joint method of evaluation criteria and statistical modeling[END_REF] [9] employed the Conditional Random Fields (CRF) [START_REF] Lafferty | Conditional random fields: Probabilistic models for segmenting et labeling sequence data[END_REF] model as their machine learning method to address the problem as a sequence labelling task. Meanwhile, [START_REF] Bicici | Referential translation machines for quality estimation[END_REF] extended their initial proposition by dynamic training with adaptive weight updates in their neural network classifier. As far as prediction indicators are concerned, [START_REF] Bicici | Referential translation machines for quality estimation[END_REF] proposed seven word feature types and found among them the "common cover links" (the links that point from the leaf node containing this word to other leaf nodes in the same subtree of the syntactic tree) the most outstanding. [START_REF] -F. Han | Quality estimation for machine translation using the joint method of evaluation criteria and statistical modeling[END_REF] focused only on various n-gram combinations of target words. Inheriting most of previously-recognized features, [START_REF] Luong | LIG system for WMT13 QE task: Investigating the usefulness of features in word confidence estimation for MT[END_REF] integrated a number of new indicators relying on graph topology, pseudo reference, syntactic behavior (constituent label, distance to the semantic tree root) and polysemy characteristic. The estimation of the confidence score uses mainly classifiers like Conditional Random Fields [START_REF] -F. Han | Quality estimation for machine translation using the joint method of evaluation criteria and statistical modeling[END_REF][START_REF] Luong | LIG System for Word Level QE task at WMT14[END_REF], Support Vector Machines [START_REF] Langlois | Loria system for the wmt12 quality estimation shared task[END_REF] or Perceptron [START_REF] Bicici | Referential translation machines for quality estimation[END_REF]. Some investigations were also conducted to determine which features seem to be the most relevant. [START_REF] Langlois | Loria system for the wmt12 quality estimation shared task[END_REF] proposed to filter features using a forward-backward algorithm to discard linearly correlated features. Using Boosting as learning algorithm, [START_REF] Luong | Towards accurate predictors of word quality for machine translation: Lessons learned on french -english and english -spanish systems[END_REF] was able to take advantage of the most significant features.

Finally, several toolkits for WCE were recently propo-sed: TranscRater for ASR [START_REF] Jalalvand | Transcrater: a tool for automatic speech recognition quality estimation[END_REF] 1 , Marmot for MT2 as well as WCE toolkit [START_REF] Servan | An Open Source Toolkit for Word-level Confidence Estimation in Machine Translation[END_REF] 3 that will be used to extract MT features in the experiments of this paper.

To our knowledge, the first attempt to design WCE for speech translation, using both ASR and MT features, is our own work [START_REF] Besacier | Word confidence estimation for speech translation[END_REF][START_REF] Besacier | Spoken language translation graphs redecoding using automatic quality assessment[END_REF] which is further extended in this paper submission.

Building an Efficient Quality Assessment (WCE) System

The WCE component solves the equation:

q = argmax q {p SLT (q|x f , f, e)} (1)
where x f is the given signal in the source language, spoken language translation (SLT) consists in finding the most probable target language sequence ê = (e 1 , e 2 , ..., e N ) ; f = (f 1 , f 2 , ..., f M ) is the transcription of x f ; q = (q 1 , q 2 , ..., q N ) is the sequence of quality labels on the target language and q i ∈ {good, bad}4 . This is a sequence labelling task that can be solved with several machine learning techniques such as Conditional Random Fields (CRF) [START_REF] Lafferty | Conditional random fields: Probabilistic models for segmenting et labeling sequence data[END_REF]. However, for that, we need a large amount of training data for which a quadruplet (x f , f, e, q) is available. In this work, we will use a corpus extended from [START_REF] Besacier | Word confidence estimation for speech translation[END_REF] which contains 6.7k utterances. We will investigate if this amount of data is enough to evaluate and test a joint model p SLT (q|x f , f, e).

As it is much easier to obtain data containing either the triplet (x f , f, q) (automatically transcribed speech with manual references and quality labels infered from word error rate estimation) or the triplet (f, e, q) (automatically translated text with manual post-editions and quality labels infered using tools such as TERp-A [START_REF] Snover | Terp system description[END_REF]) we can also recast the WCE problem with the following equation:

q = argmax q {p ASR (q|x f , f ) α * p M T (q|e, f ) 1-α } (2)
where α is a weight giving more or less importance to W CE ASR (quality assessment on transcription) compared to W CE M T (quality assessment on translation). It is important to note that p ASR (q|x f , f ) corresponds to the quality estimation of the words in the target language based on features calculated on the source language (ASR). For that, what we do is projecting source quality scores to the target using word-alignment information between e and f sequences. This alternative approach (equation 2) will be also evaluated in this work even if it corresponds to a different optimization problem than equation 1. In particular, the choice of α is only set a priori in our experiments to 0.5 which is probably not the best option.

In both approaches -joint (p SLT (q|x f , f, e)) and combined (p ASR (q|x f , f ) + p M T (q|e, f )) -some features need to be extracted from ASR and MT modules. They are more precisely detailed in next subsections.

WCE Features for Speech Transcription (ASR)

In this work, we extract several types of features, which come from the ASR graph, from language model scores and from a morphosyntactic analysis. These features are listed below (more details can be found in [START_REF] Besacier | Word confidence estimation for speech translation[END_REF]):

• Acoustic features: word duration (F-dur).

• Graph features (extracted from the ASR word confusion networks): number of alternative (F-alt) paths between two nodes ; word posterior probability (Fpost).

• Linguistic features (based on probabilities by the language model): word itself (F-word), 3-gram probability (F-3g), log probability (F-log), back-off level of the word (F-back), as proposed in [START_REF] Fayolle | Crf-based combination of contextual features to improve a posteriori word-level confidence measures[END_REF],

• Lexical Features: Part-Of-Speech (POS) of the word (F-POS),

• Context Features: Part-Of-Speech tags in the neighborhood of a given word (F-context).

For each word in the ASR hypothesis, we estimate the 9 features (F-Word ; F-3g ; F-back ; F-log ; F-alt ; F-post ; Fdur ; F-POS ; F-context) previously described.

In a preliminary experiment, we will evaluate these features for quality assessment in ASR only (W CE ASR task). Two different classifiers will be used: a variant of boosting classification algorithm called bonzaiboost [START_REF] Laurent | Boosting bonsai trees for efficient features combination : application to speaker role identification[END_REF] (implementing the boosting algorithm Adaboost.MH over deeper trees) and the Conditional Random Fields [START_REF] Lafferty | Conditional random fields: Probabilistic models for segmenting et labeling sequence data[END_REF].

WCE Features for Machine Translation (MT)

A number of knowledge sources are employed for extracting features, in a total of 24 major feature types, see Table 1.

It is important to note that we extract features regarding tokens in the Machine Translation (MT) hypothesis sentence. In other words, one feature is extracted for each token in the MT output. So, in the Table 1, target refers to the feature coming from the MT hypothesis and source refers to a feature extracted from the source word aligned to the considered target word. More details on some of these features are given in the next subsections.

Internal Features

These features are given by the Machine Translation system, which outputs additional data like N -best list. 5. Using this kind of feature is controversial, however we observed that such features are available in general use case scenarios, so we decided to include them in our experiments. Contrastive results without these 2 features will be also given later on.

A very similar feature set was used for a simple W CE M T task (English -Spanish MT, WMT 2013, 2014 quality estimation shared task) and obtained very good performances [START_REF] Luong | Word confidence estimation and its integration in sentence quality estimation for machine translation[END_REF].

In this paper, we will use only Conditional Random Fields [START_REF] Lafferty | Conditional random fields: Probabilistic models for segmenting et labeling sequence data[END_REF] (CRFs) as our machine learning method, with WAPITI toolkit [START_REF] Lavergne | Practical very large scale crfs[END_REF], to train our WCE estimator based on MT and ASR features.

Experimental Setup

Dataset

The dev set and tst set of this corpus were recorded by french native speakers. Each sentence was uttered by 3 speakers, leading to 2643 and 4050 speech recordings for dev set and tst set, respectively. For each speech utterance, a quintuplet containing: ASR output (f hyp ), verbatim transcript (f ref ), English text translation output (e hypmt ), speech translation output (e hyp slt ) and post-edition of translation (e ref ), was made available. This corpus is available on a github repository 6 . The total length of the dev and tst speech corpus obtained are 16h52, since some utterances were pretty long.

ASR Systems

To obtain the speech transcripts (f hyp ), we built a French ASR system based on KALDI toolkit [START_REF] Povey | The kaldi speech recognition toolkit[END_REF]. Acoustic models are trained using several corpora (ESTER, REPERE, ETAPE and BREF120) representing more than 600 hours of french transcribed speech.

We propose to use two 3-gram language models trained on French ESTER corpus [START_REF] Galliano | Corpus description of the ester evaluation campaign for the rich transcription of french broadcast news[END_REF] as well as on French Gigaword (vocabulary size are respectively 62k and 95k). The ASR systems LM weight parameters are tuned through WER on the dev corpus.

Table 2 presents the performances obtained by two above ASR systems.

These WER may appear as rather high according to the task (transcribing read news). A deeper analysis shows that these news contain a lot of foreign named entities, especially in our dev set. This part of the data is extracted from French medias dealing with european economy in EU. This could also explain why the scores are significantly different between dev and test sets. In addition, automatic postprocessing is applied to ASR output in order to match requirements of standard input for Machine Translation.

SMT System

We used moses phrase-based translation toolkit [START_REF] Koehn | Moses: Open source toolkit for statistical machine translation[END_REF] to translate French ASR into English (e hyp ). This mediumsize system was trained using a subset of data provided for IWSLT 2012 evaluation [START_REF] Federico | Overview of the IWSLT 2012 evaluation campaign[END_REF]: Europarl, Ted and News-Commentary corpora. The total amount is about 60M words. We used an adapted target language model trained on specific data (News Crawled corpora) similar to our evaluation corpus (see [START_REF] Potet | The lig machine translation system for wmt 2010[END_REF]). This standard SMT system will be used in all experiments reported in this paper.

Obtaining Quality Assessment Labels for SLT

After building an ASR system, we have a new element of our desired quintuplet: the ASR output f hyp . It is the noisy version of our already available verbatim transcripts called f ref . This ASR output (f hyp ) is then translated by the exact same SMT system [START_REF] Potet | The lig machine translation system for wmt 2010[END_REF] already mentioned in subsection 4.3. This new output translation is called e hyp slt and it is a degraded version of e hypmt (translation of f ref ).

At this point, a strong assumption we made has to be revealed: we re-used the post-editions obtained from the text translation task (called e ref ), to infer the quality (G, B) labels of our speech translation output e hyp slt . The word label setting for WCE is done using TERp-A toolkit [START_REF] Snover | Terp system description[END_REF] between e hyp slt and e ref . This assumption, and the fact that initial MT post-edition can be also used to infer labels of a SLT task, is reasonable regarding results (later presented in Table 4 andTable 5) where it is shown that there is not a huge difference between the MT and SLT performance (evaluated with BLEU).

The remark above is important and this is what makes the value of this corpus. For instance, other corpora such as the TED corpus compiled by LIUM 7 contain also a quintuplet with ASR output, verbatim transcript, MT output, SLT output and target translation. But there are 2 main differences: first, the target translation is a manual translation of the prior subtitles so this is not a post-edition of an automatic translation (and we have no guarantee that the good/bad labels 7. http://www-lium.univ-lemans.fr/fr/content/ corpus-ted-lium extracted from this would be reliable for WCE training and testing) ; secondly, in our corpus, each sentence is uttered by 3 different speakers which introduces speaker variability in the database and allows us to deal with different ASR outputs for a single source sentence.

Final Corpus Statistics

The final corpus obtained is summarized in Table 3, where we also clarify how the WCE labels were obtained. For the test set, we now have all the data needed to evaluate WCE for 3 tasks:

• ASR: extract good/bad labels by calculating WER between f hyp and f ref ,

• MT: extract good/bad labels by calculating TERp-A between e hypmt and e ref ,

• SLT: extract good/bad labels by calculating TERp-A between e hyp slt and e ref .

Table 4 and Table 5 summarize baseline ASR, MT and SLT performances obtained on our corpora, as well as the distribution of good (G) and bad (B) labels inferred for both tasks. Logically, the percentage of (B) labels increases from MT to SLT task in the same conditions.

Experiments on WCE for SLT

SLT Quality Assessment Using only MT or ASR Features

We first report in Table 6 the baseline WCE results obtained using MT or ASR features separately. In short, we evaluate the performance of 4 WCE systems for different tasks:

• The first and second systems (WCE for ASR / ASR feat.) use ASR features described in section 3.1 with two different classifiers (CRF or Boosting).

• The third system (WCE for SLT / MT feat.) uses only MT features described in section 3.2 with CRF classifier.

• The fourth system (WCE for SLT / ASR feat.) uses only ASR features described in section 3.1 with CRF classifier (so this is predicting SLT output confidence using only ASR confidence features !). Word alignment information between f hyp and e hyp is used to project the WCE scores coming from ASR, to the SLT output.

In all experiments reported in this paper, we evaluate the performance of our classifiers by using the average between the F-measure for good labels and the F-measure for bad labels that are calculated by the common evaluation metrics: Precision, Recall and F-measure for good/bad labels. Since two ASR systems are available, F-mes1 is obtained for SLT based on ASR1 whereas F-mes2 is obtained for SLT based on ASR2. For the results of Table 6, the classifier is evaluated on the tst part of our corpus and trained on the dev part. Concerning WCE for ASR, we observe that F-measure decreases when ASR WER is lower (F-mes2<F-mes1 while W ER ASR2 < W ER ASR1 ). So quality assessment in ASR seems to become harder as the ASR system improves. This could be due to the fact that the ASR1 errors recovered by bigger LM in ASR2 system were easier to detect. The effect of the classifier (CRF or Boosting) is not conclusive since CRF is better for F-mes1 and worse for F-mes2.

Data

As can be seen from the results of WCE for SLT, we can see that F-measure is better using MT features rather than ASR features (quality assessment for SLT more dependent of MT features than ASR features). Again, F-measure decreases when ASR WER is lower (F-mes2<F-mes1 while W ER ASR2 < W ER ASR1 ).

In the next subsection, we try to see if the use of both MT and ASR features improves quality assessment for SLT.

SLT Quality Assessment Using both MT and ASR Features

We now report in Table 6 WCE for SLT results obtained using both MT and ASR features. More precisely we evaluate two different approaches (combination and joint):

• The first system (WCE for SLT / MT+ASR feat.) combines the output of two separate classifiers based on ASR and MT features. In this approach, ASR-based confidence score of the source is projected to the target SLT output and combined with the MT-based confidence score as shown in equation 2 (we did not tune the α coefficient and set it a priori to 0.5).

• The second system (joint feat.) trains a single WCE system for SLT (evaluating p(q|x f , f, e) as in equation 1 using joint ASR features and MT features. All ASR features are projected to the target words using automatic word alignments. However, a problem oc-curs when a target word does not have any source word aligned to it. In this case, we decide to duplicate the ASR features of its previous target word. Another problem occurs when a target word is aligned to more than one source word. In that case, there are several strategies to infer the 9 ASR features: the average value for F-post, F-log, F-back and the maximum value for F-3g, F-alt, F-dur. For the other features, we generate the values of the first source word aligned to it. The results of Table 6 show that joint ASR and MT features do not improve WCE performance: F-mes1 and F-mes2 are slightly worse than those of Table 6 (WCE for SLT / MT features only). We also observe that simple combination (MT+ASR) degrades the WCE performance. This latter observation may be due to different behaviors of W CE M T and W CE ASR classifiers which makes the weighted combination ineffective. Moreover, the disappointing performance of our joint classifier may be due to an insufficient training set (only 2643 utterances in dev !). Finally, removing Occu-rInGoogleTranslate and OccurInBingTranslate features for Joint lowered F-mes between 1% and 1.5%.

These observations lead us to investigate the behaviour of our WCE approaches for a large range of good/bad decision threshold.

While the previous tables provided WCE performance for a single point of interest (good/bad decision threshold set to 0.5), the curves of figures 1 show the full picture of our WCE systems (for SLT) using speech transcriptions systems ASR1 and ASR2, respectively. We observe that the classifier based on ASR features has a very different behaviour than the classifier based on MT features which explains why their simple combination (MT+ASR) does not work very well for the default decision threshold (0.5). However, for threshold above 0.75, the use of both ASR and MT features is slightly beneficial. This is interesting because higher thre- sholds improves the F-measure on bad labels (so improves error detection). Both curves are similar whatever the ASR system used. These results suggest that with enough development data for appropriate threshold tuning (which we do not have for this very new task), the use of both ASR and MT features should improve error detection in speech translation (blue and red curves are above the green curve for higher decision threshold 8 ).

8. Corresponding to optimization of the F-measure on bad labels (errors).

Conclusion 6.1. Main Contributions

In this paper, we introduced a new quality assessment task: word confidence estimation (WCE) for spoken language translation (SLT). A specific corpus, distributed to the research community 9 was built for this purpose. We formalized WCE for SLT and proposed several approaches based on several types of features: Machine Translation (MT) based features, automatic speech recognition (ASR) based features, as well as combined or joint features using ASR and MT information. The proposition of a unique joint classifier based on different feature types (ASR and MT features) could allow to operate feature selection in the future and analyze which features (from ASR or MT) are the most efficient for quality assessment in speech translation. Our experiments have shown that MT features remain the most influential while ASR features can bring interesting complementary information. In all our experiments, we systematically evaluated with two ASR systems that have different performance in order to analyze the behavior of our quality assessment algorithms at different levels of word error rate (WER). This allowed us to observe that WCE performance decreases as ASR system improves. For reproducible research, most features 10 and algorithms used in this paper are available through our toolkit called WCE-LIG. This package is made available on a Gi-tHub repository 11 under the licence GPL V3. We hope that the availability of our corpus and toolkit could lead, in a near future, to a new shared task dedicated to quality estimation for speech translation. Such a shared task could be proposed in avenues such as IWSLT (International Workshop on Spoken Language Translation) or WMT (Workshop on Machine Translation) for instance.

Other Perspectives

In addition to re-decode SLT graphs, our quality assessment system can be used in interactive speech translation scenarios such as news or lectures subtitling, to improve human translator productivity by giving him/her feedback on automatic transcription and translation quality. Another application would be the adaptation of our WCE system to interactive speech-to-speech translation scenarios where feedback on transcription and translation modules is needed to improve communication. On these latter subjects, it would also be nice to move from a binary (good or bad labels) to a 3-class decision problem (good, asr-error, mt-error). The outcome material of this paper (corpus, toolkit) can be definitely used to address such a new problem.

Figure 1 -

 1 Figure 1 -Evolution of system performance (y-axis -F-mes1 -ASR1 and F-mes2 -ASR2) for tst corpus (4050 utt) along decision threshold variation (x-axis) -training is made on dev corpus (2643 utt).

Table 1 -

 1 List of MT features extracted. Word Posterior Probability (WPP) and Nodes features are extracted from a confusion network, which comes from the output of the Machine Translation N -best list. WPP Exact is the WPP value for each word concerned at the exact same position in the graph. WPP Any extracts the same information at any position in the graph. WPP Min gives the smallest WPP value concerned by the transition and WPP Max its maximum.

	1 Proper Name	10 Stop Word	19 WPP max
	2 Unknown Stem	11 Word context Alignments	20 Nodes
	3 Num. of Word Occ.	12 POS context Alignments	21 Constituent Label
	4 Num. of Stem Occ.	13 Stem context Alignments	22 Distance To Root
	5 Polysemy Count -Target	14 Longest Target N -gram Length 23 Numeric
	6 Backoff Behaviour -Target 15 Longest Source N -gram Length 24 Punctuation
	7 Alignment Features	16 WPP Exact
	8 Occur in Google Translate	17 WPP Any
	9 Occur in Bing Translator	18 WPP min
	3.2.2. External Features

Below is the list of the external features used: • Proper Name: indicates if a word is a proper name (same binary features are extracted to know if a token is Numerical, Punctuation or Stop Word). • Unknown Stem: informs whether the stem of the considered word is known or not. • Number of Word/Stem counts the occurrences of a word/stem in the sentence. • Alignment context features: these features (#11-13 in Table

1

) are based on collocations and proposed by

[START_REF] Bach | Goodness: A method for measuring machine translation confidence[END_REF]

. • Longest Target (or Source) N -gram Length: we seek to get the length (n + 1) of the longest left sequence (w i-n ) concerned by the current word (w i ) and known by the language model (LM) concerned (source and target sides). We also extract a redundant feature called Backoff Behavior Target.

Table 2 -

 2 ASR performance (WER) on our dev and test set for the two different ASR systems.

	Task dev set tst set
	ASR1 21.86% 17.37%
	ASR2 16.90% 12.50%

6. https://github.com/besacier/WCE-SLT-LIG/

Table 3 -

 3 # dev utt # test utt # dev words # test words method to obtain WCE labels Overview of our post-edition corpus for SLT.

	f ref	881	1350	21 988	36 404
	f hyp1	881*3	1350*3	66 435	108 332	wer(f hyp1 , f ref )
	f hyp2	881*3	1350*3	66 834	108 598	wer(f hyp2 , f ref )
	e hypmt	881	1350	22 340	35 213	terpa(e hypmt , e ref )
	e hyp slt1 881*3	1350*3	61 787	97 977	terpa(e hyp slt1 , e ref )
	e hyp slt2 881*3	1350*3	62 213	97 804	terpa(e hyp slt2 , e ref )
	e ref	881	1350	22 342	34 880
		Task	ASR (WER) MT (BLEU) % G (good) % B (bad)
		MT	0%		49.13%	76.93%	23.07%
		SLT (ASR1) 21.86%		26.73%	62.03%	37.97%
		SLT (ASR2) 16.90%		28.89%	63.87%	36.13%

Table 4 -

 4 MT and SLT performances on our dev set.

Table 5 -

 5 MT and SLT performances on our tst set.

		Task	ASR (WER) MT (BLEU) % G (good) % B (bad)
		MT	0%	57.87%	81.58%	18.42%
		SLT (ASR1) 17.37%	30.89%	61.12%	38.88%
		SLT (ASR2) 12.50%	33.14%	62.77%	37.23%
	task	WCE for ASR WCE for ASR WCE for SLT WCE for SLT WCE for SLT	WCE for SLT
	feat. type ASR feat.	ASR feat.	MT feat.	ASR feat.	MT+ASR feat.	Joint feat.
		p(q|x f , f )	p(q|x f , f )	p(q|f, e)	p ASR (q|x f , f ) p ASR (q|x f , f ) α	p(q|x f , f, e)
						* p M T (q|e, f ) 1-α
		(CRFs)	(Boosting)		projected to e
	F-mes1	68.71%	64.27%	60.55%*	49.67%	52.99%	60.29%**
	F-mes2	59.83%	62.61%	59.83%*	44.56%	48.46%	59.23%**

Table 6 -

 6 WCE performance with different feature sets for tst set (training is made on dev set) -After removing OccurInGoogle-Translate and OccurInBingTranslate features: * for MT feat, lead to 59.40% and 58.11% for F-mes1 and F-mes2 respectively ; ** for joint feature, lead to 59.14% and 57.75% for F-mes1 and F-mes2 respectively.

	63					63
	60					60
	57					57
	54					54
	51					51
	0.00	0.25	0.50	0.75	1.00	0.00	0.25	0.50	0.75	1.00
		Favg(all) of WCE-SLT using MT feature				Favg(all) of WCE-SLT using MT feature
		Favg(all) of WCE-SLT using ASR feature				Favg(all) of WCE-SLT using ASR feature
		Favg(all) of WCE-SLT using MT+ASR feature sets			Favg(all) of WCE-SLT using MT+ASR feature sets
		Favg(all) of WCE-SLT using joint feature sets 1			Favg(all) of WCE-SLT using joint feature sets 1

https://github.com/hlt-mt/TranscRater

https://github.com/qe-team/marmot

https://github.com/besacier/WCE-LIG

q i could be also more than 2 labels, or even scores but this paper only deals with error detection (binary set of labels).


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Deflections of inflatable fabric panels at high pressure

Keywords: Inflatable beams, High-pressure, Following forces, Fabric structures

Inflatable structures made of modern textile materials with important mechanical characteristics can be inflated at high pressure (up to a several hundreds of kPa). They can be used as strong building elements thanks to their mechanical strength. The aim of the paper is to present experimental and analytical studies on the behaviour of inflated fabric panels at high pressure and submitted to bending loads. It is shown that inflatable structures cannot be viewed as ordinary plates or beams, because their deformation pattern is quite different. Experiments show that their behaviour depends on the applied load, the inflation pressure, and the constitut-ive law of the fabrics. Equilibrium equations are written in the deformed state to take into account the influence of geometrical stiffness and the following forces. A Timoshenko's beam theory must be used because sections of the panels do not satisfy the usual Bernoulli's beam theory. A new inflatable beam theory is developed. Wrinkling loads are derived from equilib-rium equations. Deflections satisfy the fact that the compliance of the inflatable panel is the sum of the beam compliance and of the yarn compliance. Comparisons between the results of our modelling and experimental results are shown and prove the accuracy of this theory on the mechanical strength of inflatable structures at high pressure.

Introduction

This paper presents results from research on the mechanics of inflatable structures at high pressure. Such structures are not new and a lot of inflatable structures are used by industry: temporary buildings, boats, pools, space antennas, life jackets, etc… Inflated structures have many interesting properties: they are light, easily folding and present reversible behaviour after 'failure' (they come back to their initial position after unloading). Inflation causes tension prestressing in the walls and in the yarns of the structures. This prestressing is proportional to the pressure and ensures an important mechanical strength. Generally, these structures are used with an inflation pressure lower than 100 kPa because they are built with low strength materials. Although inflatable structures are not a recent development, no studies have been conducted on this subject for high pressure values. A few papers can be found on the behaviour of inflated tubes. Leonard et al. [START_REF] Leonard | Structural considerations of inflatable re-entry vehicles[END_REF], Comer and Levy [START_REF] Comer | Deflections of an inflated circular cylindrical cantilever beam[END_REF] and Main et al. [START_REF] Main | Load-deflection behaviour of space-based inflatable fabric beams[END_REF] have studied the case of cantilever inflated fabric tubes at low pressure. They have shown that the wrinkling and/or collapse load is proportional to the applied pressure. This result has been verified at high pressure ( [START_REF] Wielgosz | Experimental study and numerical modelling of inflated fabric panels[END_REF], [START_REF] Wielgosz | Behaviour of inflated fabric beams at medium pressures[END_REF]) and comparisons between theoretical and experimental wrinkling loads of panels and tubes show a good accuracy between theory and experiments. We must therefore use a high pressure if we want a good mechanical strength of inflatable structures.

Unfortunately there are few results on the deflections of inflatable structures. Main et al. [START_REF] Main | Load-deflection behaviour of space-based inflatable fabric beams[END_REF] have studied inflatable cantilever fabric tubes and calculated the deflections of these structures by using the usual beam theory. Experimental and theoretical results were compared, and they have proved that this theory can be applied to inflatable tubes. In fact they have used fabrics with low characteristics materials (20 times lower than our fabrics) and their study was done at a pressure almost five times lower that the scale which is used here. We think that these studies should be extended. The obvious reason is that in a beam theory, the values of the deflections depend only on the flexural rigidity of the beam. The pressure does not appear in the beam solution, and it seems clear that an inflated beam at an extremely low pressure can have very large deflections. We have tried to apply a yarn theory to inflatable panels and tubes ( [START_REF] Wielgosz | Experimental study and numerical modelling of inflated fabric panels[END_REF], [START_REF] Wielgosz | Behaviour of inflated fabric beams at medium pressures[END_REF]) and in this case the result is in reverse order: the deflection values are close to experimental ones when the pressure is low, but deviate greatly from reality at high pressure. Moreover, in a yarn theory deflections are independent of the material constitutive law and this is irregular. The only way to obtain good results on the deflection values was to use numerical modelling, but the studies were tedious ( [START_REF] Wielgosz | Experimental study and numerical modelling of inflated fabric panels[END_REF], [START_REF] Wielgosz | Behaviour of inflated fabric structures at medium pressures[END_REF]). We will show that in fact the deflections of inflatable structures submitted to bending loads are a linear set of yarn and beam deflections. Given that these deflections are inversely proportional to the constitutive law and to the applied pressure, we have a second reason to greatly inflate the structures and to use high characteristics materials.

In the first section of the paper we will describe experimental work and show that inflatable panels cannot be viewed as ordinary plates or beams, because their deformation pattern is quite different. Experiments show that they behave like yarns when the pressure is low, like beams at high pressure and once again like yarns or mechanisms when the applied load is near their wrinkling load. A new inflated beam theory is therefore constructed in the second section in order to provide simple analytical formulas for the wrinkling load and for the deflection values of simply supported inflated panels submitted to inflation pressure going up to 300 kPa. Equilibrium equations are written in the deformed state to take into account the geometrical stiffness of the beam and the following forces. A Timoshenko's beam theory is used because sections of the panels do not satisfy the usual Bernoulli's beam theory. It will be shown that the compliance of inflated fabric beams is only the sum of yarn compliance and beam compliance. Analytical results are close to experimental ones.

Experimental results on inflatable panels

The tested panels are prototypes constructed by Tissavel Inc. They are made of two parallel-coated woven fabrics connected by yarns and a cross section of these panels is shown in Fig. 1. The yarn density is enough to ensure the flatness of the fabric structure. Fabrics are made with high strength polyester.

The panel's behaviour depends on the inflation pressure p that leads the fabrics and the yarns to be prestressed and then to support local compression loads. The panels are simply supported beams loaded by a concentrated force F. The width and height are b, h and l is the half span of the beam. The shapes of the deformed panel are shown in Fig. 2 for two values of the inflation pressure (50 and 300 kPa).

One can see that the bending shape depends on the pressure. When the pressure is low, large straight parts appear between points of loading and the free ends of the panel remain almost straight. When the pressure is higher, straight parts are shorter and a wider curved zone appears. In both cases we also take notice of light discontinuities of the rotation near loading areas (supports and loading point). This will be called local effects. The panel therefore has a behaviour that is similar to that of a tensioned yarn when the pressure is low and to that of a beam when the pressure is high. Experimental deflection curves for low and high values of the 3. When the load applied to the panel is near to its wrinkling or limit load, the shape is once again similar to that of a tensioned yarn as shown in Fig. 4.

All these experimental results prove that our inflatable panels behave successively like yarns and beams, and that their deformation pattern depends mainly on the inflation pressure and on the applied load.

From an engineering point of view, we must answer the following two questions: what is the limit load of an inflatable panel, and what is the maximum value of the deflection for a given load? We have shown [START_REF] Wielgosz | Experimental study and numerical modelling of inflated fabric panels[END_REF] that we can easily answer the first question. Theoretical limit loads can be calculated with limit analysis because the deformation pattern at the limit load is similar to the yield mechanisms of beams. We have compared theoretical and experimental limit loads for the panels. Theoreti-Fig. 3. Deflection curves for low and high pressure Fig. 4. Shape of an inflatable panel at its limit load (300 kPa) cal values of the limit load are close to experimental results and are obtained within 20% error.

Unfortunately we have not yet found an easy answer to the second question. Some comparisons on the maximum values of the deflection at low pressure with the tensioned yarn theory were given in [START_REF] Wielgosz | Experimental study and numerical modelling of inflated fabric panels[END_REF]. Theoretical values are close to experimental ones only for pressures lower than 50 kPa. Elsewhere a beam theory is not accurate because of the decreased curvature of the panels (where straight lines appear). In these areas the curvature is near zero and the bending momentum is linear; we cannot define a flexural rigidity. The second argument, which prevents the use of a usual beam theory, is due to the fact that in a beam theory the pressure is not taken into account. The only way to obtain correct values on the deflection was to use numerical modelling [START_REF] Wielgosz | Experimental study and numerical modelling of inflated fabric panels[END_REF], but the computation times were tedious, because we had to take into account large displacements and following forces.

We will now develop the mechanical theory of inflatable panels. This theory is inspired by our experiments, and we will consider the panel as a Timoshenko'beam. Fig. 5 presents a zoom on the section's rotation, and one can see that the section is not orthogonal to the neutral fibre. We will show that we are able to explain all these behaviours of the panels and give simple analytical formulae for the deflections.

Mechanics of inflatable panels

The mechanical strength of inflatable panels is obtained by use of its equilibrium equations, the constitutive law of the fabrics, and kinematical assumptions on its deformation pattern.

The theory will be developed in the case of a simply supported panel, loaded by a concentrated force in the middle of the structure. Fig. 6 gives an outline of the studied problem with the main notations.

Equilibrium equations: wrinkling load

Let us consider a small element of the panel (Fig. 7) with length dx, width b, and height h. Equilibrium equations are written for this element in its deformed position. The pressure effects are supposed to be replaced by forces normally applied to the membranes, because they are in fact following forces.

The angle between the membranes and the horizontal axis is named q and v is the deflection (q ϭ dv / dx). N i and N s denote the resultant stresses respectively in the lower and upper membrane; T is the shear stress. We must use a Timoshenko's beam theory for this case of thin-walled structures, because the influence of the shear stress cannot be neglected, as in Bernoulli's beam theory. The straight section of the panel is supposed to remain straight after deformation, but not orthogonal to the neutral fibre, as seen in the last section. a denotes this rotation. All the theory is based upon the hypothesis that the structure's behaviour depends mainly on the values of N i and N s .

Equilibrium equations are given by :

d(N i ϩ N S ) dx cosq ϭ 0 (1) dT dx ϩ d(N i ϩ N S ) dx sinq ϭ 0 (2) T ϩ h 2 d(N i ϪN S ) dx Ϫpbhsin(qϪa) ϭ 0 (3)
The first two equations show that N i ϩ N S and that the shear stress are constant.

The bending momentum results from the difference between the stresses in the two membranes :

M ϭ h 2 (N i ϪN S ) (4)
The third equation can also be written as (sin(qϪa))Ϸ(qϪa):

T ϩ dM dx Ϫpbh(qϪa) ϭ 0 (5)
One can notice the influence of the following forces in this equation.

Stresses are given by the global equilibrium of a sheet of the panel (Fig. 8). The Fig. 8. Loads on a sheet of the panel equilibrium equation of the air included in the two membranes shows that the length of the lower membrane must be greater than the length of the upper membrane. The equilibrium of this sheet gives the following equations:

N S ϩ N i ϭ pb 2 h b ϩ h (6) T ϭ F 2 Ϫpbhsin(qϪa) (7) h(N S ϪN i ) ϩ F(2lϪx) ϭ 0 (8)
Eq. ( 6), [START_REF] Fichter | A theory for inflated thin-wall cylindrical beams[END_REF] give the variation of the resultant stresses in the membranes :

N S (x) ϭ pb 2 h 2(b ϩ h) Ϫ F 2h (2lϪx) and N i (x) ϭ pb 2 h 2(b ϩ h) ϩ F 2h (2lϪx) (9)
The symmetry of the structure and its loading gives the graph (Fig. 9) of the theoretical membrane stresses in the panel. These equilibrium equations show on the one hand that the difference of rotation terms qϪa ϭ dv dx Ϫa is constant, and on the other hand that the wrinkling load of the panel is given by (when the resultant stress in the upper fabric falls):

F w ϭ pb 2 h 2 l(b ϩ h) ( 10
)
Let us now define the kinematics of the panel (Fig. 10). If P is a point of the neutral fibre, the displacement field is supposed to satisfy the following relation:

u → (P) ϭ u(x) e → x ϩ v(x)e → y ( 11
)
where the horizontal displacement u(x) and the deflection v(x) are only functions of x.

If Qi and Qs are two points of the lower and upper membranes, their displacements are obtained by writing :

u → (Q) ϭ u → (P) ϩ ⍀ → ٙPQ J→ with ⍀ → ϭ a e → z ( 12
)
The local strains e i (x) and e s (x) in the two membranes are therefore:

e i (x) ϭ u ,x ϩ h 2 a ,x e s (x) ϭ u ,x Ϫ h 2 a ,x (13)
Resultant stresses are obtained from the constitutive law of the fabric. A prestressed material satisfies the following law (s 0 is the pretension):

s ϭ s 0 ϩ Ee (14)
The resultant stresses in the membranes are therefore the following :

N i (x) ϭ pb 2 h 2(b ϩ h) ϩ E * bh 2 a ,x N s (x) ϭ pb 2 h 2(b ϩ h) Ϫ E * bh 2 a ,x (15)
where E * is the membrane modulus (product of the Young's modulus E by the thickness e of the fabric). The comparison between these formulas and the values providing from the equilibrium equations gives the differential equation satisfied by the rotation of a point of the neutral fibre of the panel:

a ,x ϭ F E * bh 2 (2lϪx) (16)
This is the usual beam solution :

a(x) ϭ F E * bh 2 ͩ 2lxϪ x 2 2 ͪ ϩ a 0 ( 17
)
The boundary condition at the middle span of the panel implies that :

a(x) ϭ F E * bh 2 ͩ 2lxϪ x 2 2 Ϫ 3l 2 2 ͪ (18)
If we use the usual Timoshenko's theory, the relation between the shear stress and the rotation dv dx Ϫa is written as:

T ϭ KG * S * ͩ dv dx Ϫa ͪ ( 19
)
where K is the shear coefficient, G * ϭ Ge, S * ϭ S / e and G is the in-plane shear modulus of the panel. Let us use Cowper's formulation [START_REF] Cowper | The shear coefficient in Timoshenko's beam theory[END_REF] for a thin-walled box section with uniform thickness to calculate the shear coefficient. The graph presented in Fig. 11 shows the evolution of K as a function of Poisson's ratio, for a panel with width b ϭ 0.2 m and height h ϭ 0.055 m. We have also to consider the distribution of the shear stress in a section of the panel, and to take into account the influence of the connecting yarns, which transmit the greatest part of T. The yarns have been considered as stiff bars, with hinges at their tips, and their global behaviour produces a mechanism which ensures the flatness of the panel. The shear stress transmitted by the lateral walls is small compared to the shear stress which is transmitted by the yarns. We must add another coefficient K':

K' ϭ 1 (N ϩ 2) ( 20
)
where N is the number of yarns in a section.We can write So

T ϭ KK ' G * S * ͩ dv dx Ϫa ͪ ϭ F 2 Ϫpbh ͩ dv dx Ϫa ͪ (21)
dv dx ϭ F 2(KK ' G * S * ϩ pbh) ϩ a (22)
By using the boundary conditions v(2l) ϭ 0, we obtain the following formula for the deflection l

Ͻ x Ͻ 2l v(x) ϭ F 2(KK ' G * S * ϩ pbh) (xϪ2l) ϩ F E * bh 2 ͩ Ϫ x 3 6 ϩ lx 2 Ϫ 3 2 l 2 x ϩ l 3 3 ͪ (23)
This formula for the deflection of the panels can be compared to the result given by Fichter [START_REF] Fichter | A theory for inflated thin-wall cylindrical beams[END_REF] for inflated cylindrical beams. A simple analogy between tubes and panels shows that we have the same kind of results. The way to solve the inflated structure behaviour is equivalent; Fichter used a variational formulation on tubes and we have used local equilibrium equations.

Let us now estimate the influence of the values of KK ' G * S * and pbh. When the pressure is low (100 kPa), the total load pbh equals 1100 N. This load will increase with the pressure. Our panel presents almost 400 yarns/m. The shear modulus of the fabric is less than 200,000 Pa/m. In this case, the ratio between the values of KK ' G * S * and pbh is about 5%. Our aim is to develop a theory for panels inflated at a greater pressure. So, we will suppose that:

KK ' G * S * Ͻ Ͻ pbh (24)
With this approximation, the bending shape is finally given by l Ͻ x Ͻ 2l

v(x) ϭ F 2pbh (xϪ2l) ϩ F E * bh 2 ͩ Ϫ x 3 6 ϩ lx 2 Ϫ 3 2 l 2 x ϩ l 3 3 ͪ (25)
The same developpements applied to the first part of the beam leads to 0

Ͻ x Ͻ l v(x) ϭ Ϫ F 2pbh x ϩ F E * bh 2 ͩ x 3 6 Ϫ l 2 x 2 ͪ (26)
And the maximum value of the deflection is:

v(l) ϭ Ϫ F 2pbh lϪ F E * bh 2 l 3 3 ( 27
)
which is only the sum between the tight yarn and the beam deflections. In an other word, the compliance of the inflatable panel is the sum of the beam compliance and of the yarn compliance. This result is used to explain the behaviour of our panels. When the pressure is low and when the membrane modulus is high, the first expression of the right hand side of this equation is prevalent and the global behaviour is described by the tensioned yarn behaviour. When the pressure is increased, the influence of this expression decreases and the behaviour of the panel becomes a linear set of yarn and beam behaviours. Now when the membrane modulus is low, the influence of the yarn solution can be weak even when the pressure is low ( [START_REF] Main | Load-deflection behaviour of space-based inflatable fabric beams[END_REF]).

Comparisons between experimental and theoretical results

Results on the wrinkling loads

Panels have been tested by applying displacements with an extremely low strain rate. A load cell gives the applied load to the panel. We suppose that the wrinkling load is reached when the load-deflection curve becomes non-linear. A comparison between experimental and theoretical values of the wrinkling loads is given in Table 1. We can see that theoretical values are given with an error of approx. 10%.

Results on the deflection

The wrinkling loads depend only on the applied pressure and the geometry of the panels. We have seen that the deflections also depend on the constitutive law of the fabrics. A sheet of the membrane fabrics (without the connecting yarns and without the induction) has been submitted to axial loads and its response gives the membrane modulus. The results depend on the time of the experiment: the type of materials used in the conception of the panels have viscous properties. It is obvious that creep is not taken into account in our simplified theory for the modelling of these structures. Fig. 12 shows the evolution of the load when a displacement is applied to one of the panels.

We must also point out that the next value of the membrane modulus is obtained from data on a fabric sample at the same time used to present all our experimental results on the panels. In this case we obtained:

E * ϭ 650,000 Pa m Fig. 13 shows the comparisons between theoretical and experimental values of the deflections of one of the panels for four values of the pressure (varying from 50 to 300 kPa).

Conclusion

It is advisable to use modern textile materials with high mechanical characteristics for imagining the behaviour of inflatable beams at high pressure. The paper deals with the first experimental and analytical results on modelling of the behaviour of inflatable fabric panels. The main results are relative to the wrinkling loads and the deflections under bending loads. Equilibrium equations are written in the deformed state to take into account the geometrical stiffness and the following forces. A Timoshenko's beam theory must be used because sections of the panels do not satisfy the usual Bernoulli's beam theory. Wrinkling loads are obtained from equilibrium equations. Inflatable panels cannot be viewed as ordinary beams, because their deformation pattern is a set of tensioned yarn and beam behaviour. A new inflatable beam theory has been developed and comparisons between experimental and theoretical results prove the accuracy of this theory on the mechanical strength of inflatable panels at high pressure. Development of new types of structures making full use of performance characteristics of modern textile materials is possible because that theory of their behaviour is now available when the pressure reaches several hundred Fig. 13. Theoretical and experimental values of the deflection kPa. It is also possible to foresee the building of light, easily transportable and extremely strong fabric structures. Work is now required on the reliability of such structures in order to prove that they can be used by industry. Our results must be extended to other types of beams such as tubes. The influence of shear stresses must be improved. This paper is only concerned with isostatic beams. Work is in progress on the behaviour of hyperstatic panels and tubes.

Fig. 1 .

 1 Fig. 1. Cross section of the panels

Fig. 5 .

 5 Fig. 5. Rotation of the section

Fig. 7 .

 7 Fig. 7. Loads on an inflatable panel element

Fig. 9 .

 9 Fig. 9. Theorical membrane stress in the panel

Fig. 11 .

 11 Fig. 11. Shear coefficient for a panel (b=0.2 m; h=0.055 m)

Table 1

 1 Theoretical and experimental values of the wrinkling load

	Pressure	100 kPa	200 kPa	300 kPa
	Theorical wrinkling load	59.3 N	118 N	178 N
	Experimental wrinkling load	60 N	108 N	160 N

Fig. 12. Relaxation of the panel

Acknowledgements

The authors would like to thank Tissavel Inc for providing the panels and M. E. Leflaive for helpful discussions.


C Wielgosz

J C Thomas

An in atable fabric beam ÿnite element

Keywords: in atable beams, high pressure, following forces

In atable structures made of modern textile materials with important mechanical characteristics can be in ated at high pressure (up to several hundreds kPa). For such values of the pressure they have a strong mechanical strength. The aim of the paper is to construct a new in atable beam ÿnite element able to predict the behaviour of in atable structures made of beam elements. Experiments and analytical studies on in atable fabric beams at high pressure have shown that their compliance is the sum of the beam compliance and of the yarn compliance. This new ÿnite element is therefore obtained by the equilibrium ÿnite element method and is modiÿed into a displacement ÿnite element. The sti ness matrix takes into account the in ation pressure. Comparisons between experimental and numerical results are shown and prove the accuracy of this new ÿnite element for solving problems of in atable beams at high pressure.

INTRODUCTION

This paper presents results from research on the mechanics of in atable beams at high pressure. Such structures have many interesting properties: they are light, easily folding and present reversible behaviour after failure. In ation gives tension prestressing in the fabrics and imply an important mechanical strength when the pressure reaches several hundreds kPa. A high pressure is interesting because their limit load is proportional to the applied pressure and their de ections are inversely proportional to the constitutive law of the fabrics and to the applied pressure [START_REF] Wielgosz | De ections of in atable fabric panels at high pressure[END_REF]. Analytical developments have been done to calculate wrinkling loads and de ections of cantilever beams [START_REF] Comer | De ections of an in ated circular cylindrical cantilever beam[END_REF][START_REF] Main | Load-de ection behaviour of space-based in atable fabric beams[END_REF]. The pressure used in Reference [START_REF] Main | Load-de ection behaviour of space-based in atable fabric beams[END_REF] was less than 70 kPa. We have studied the case of simply supported beams [START_REF] Wielgosz | De ections of in atable fabric panels at high pressure[END_REF] for values of the pressure going up to 300 kPa. The results on the wrinkling or collapse load are directly connected to the applied pressure and independent on the materials characteristics. Analytical results on the de ections are only relative to isostatic in atable beams. The aim of this paper is to

y θ α N S N i ( ) h b 2 h pb 2 + F )) sin( 2 h x d ( pb α - θ - - + l )) sin( 2 h x d ( pb α - θ + - + l x - l d T N i + dN i N + dN S S T+dT ( ) h b 2 h pb 2 + pbdx pbdx Figure 1.
Local and global equilibrium of the panel.

construct an in atable beam ÿnite element able to give the values of the displacement ÿeld for hyperstatic in atable beams.

MECHANICS OF INFLATABLE BEAMS

The mechanics of in atable beams will be developed for in atable panels. These structures are prototypes constructed by Tissavel Inc. They are made of two parallel-coated woven fabrics connected by yarns. Their behaviour depends on the in ation pressure p that leads the fabrics and the yarns to be prestressed and then to support local compression loads. Experimental and analytical results on the behaviour of these structures can be found in Reference [START_REF] Wielgosz | De ections of in atable fabric panels at high pressure[END_REF].

Let us consider a cantilever panel submitted to a concentrated force F. The width and height are named b; h and ' is the length of the beam. The mechanical strength of the panel is obtained by use of its equilibrium equations, the constitutive law of the fabrics, and kinematics assumptions on its deformation pattern.

Equilibrium equations

Let us come back on the main results of [START_REF] Wielgosz | De ections of in atable fabric panels at high pressure[END_REF]. Equilibrium equations are written for an element in its deformed position to take into account the geometrical sti ness and the following forces. The initial position refers to the in ated beam, without any load. The panel is supposed to present large straight parts. We must use a Timoshenko's beam theory because the straight section of the panel does not stay orthogonal to the neutral ÿbre (see Figure 5. In Reference [START_REF] Wielgosz | De ections of in atable fabric panels at high pressure[END_REF]). Denotes the rotation of the straight section and  denotes the rotation of the neutral ÿbre. The pressure e ects are supposed to be replaced by forces normally applied to the membranes, because they are following forces. N i and N s denote the resultant stresses, respectively, in the lower and upper membrane; T is the shear stress. All the theory is based upon the hypothesis that the structure's behaviour depends mainly on the values of N i and N s . Figure 1 presents a sheet of the pressurized panel.

The local equilibrium equations allows to write N i + N S and that the shear stress T is constant. The global equilibrium equations are given for a cantilever beam:

N s + N i = pbh 2 b + h (1)
T = F -pbh(Â -) (2)
F(' -x) + h 2 (N S -N i ) = 0 (3)
And the stresses in the membranes are:

N i (x) = pb 2 h 2(b + h) - F h (' -x) (4) N s (x) = pb 2 h 2(b + h) + F h (' -x) (5)

De ections

If P is a point of the neutral ÿbre and if Q i and Q s are two points of the lower and upper membranes, their displacements are obtained by the following relations:

u(P) = u(x)e x + v(x)e y ; u(Q) = u(P) + ∧ PQ with = e z (6)
The horizontal displacement u(x) and the de ection v(x) are only functions of x. The local strains i (x) and s (x) in the two membranes are therefore:

i (x) = u ; x + h 2 ; x s (x) = u ; x - h 2 ; x (7)
Resultant stresses are obtained from the constitutive law of the fabric and are given by

N i (x) = pb 2 h 2(b + h) + E * bh 2 ; x and N s (x) = pb 2 h 2(b + h) - E * bh 2 ; x (8)
where E * is the membrane modulus (product of the Young modulus E by the thickness e of the fabric). E * is obtained from uniaxial traction experiment on a sample fabric. In fact, fabrics are orthotropic materials and the membrane moduli are di erent in warp and weft directions.

In this beam formulation, the warp direction is mainly concerned, and we will suppose that an isotropic constitutive law can be used to give a 'beam answer' to the engineering problem. Moreover, the viscous properties of the fabrics are not taken into account in our theory, hence all the measurements have been done after the creep has stopped. The comparison between formulas (4), ( 5) and (8) gives:

d dx = 2F E * bh 2 (' -x) (9)
By using Equation ( 2), and assuming that for these in atable panels, the shear stress can be neglected with respect to the in uence of the normal stress [START_REF] Wielgosz | De ections of in atable fabric panels at high pressure[END_REF], we can write:

dv dx = F pbh + ( 10
)
The boundary conditions at the clamped end give the closed form of de ection, where I * is equal to the second moment of area divided by the thickness:

v(x) = F pbh x + 2F E * bh 2 ' x 2 2 - x 3 6 = F pbh x + F E * I * ' x 2 2 - x 3 6 (11)
which is nothing but the sum between the tight yarn and the beam de ections. In an other word, the compliance of the in atable panel is the sum of the yarn compliance and of the beam compliance.

CONSTRUCTION OF THE INFLATABLE FINITE ELEMENT

Let us consider an in atable beam and denote by V and F the total displacement and load vectors:

V T = [v 1 1 v 2 2 ] ( 12
)
F T = [F 1 1 F 2 2 ] ( 13
)
The deÿnition of nodal unknowns is usual: v i and i denote displacement and rotation at node i, and F i and i denote load and torque at the same node. When this element is a cantilever in atable beam submitted to a load and a torque at node 2, its compliance matrix is simply obtained by adding the usual matrices of beam and yarn:

v 2 2 = F 2 2 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ' 3 3E * I * + p' S ' 2 2E * I * ' 2 2E * I * ' E * I * ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ F 2 2 (14)
where S is the area of the section of the extremity (S = pbh).

The global equilibrium equations are:

F 1 1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ -1 0 -1 -' 1 0 0 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ F 2 2 = B F 2 2 ( 15
)
The usual theory of the equilibrium ÿnite element method shows that the sti ness matrix K of the free ÿnite displacement element is obtained from the sti ness matrix of the reduced isostatic ÿnite element K r by using the following equations:

K = BK r B T (16)
where the reduced matrix K r is the inverse of the compliance matrix : The free sti ness matrix of the in atable fabric beam element is therefore:

K r = -1 (17)
K = 12E * 2 I * 2 pS ' 2 (12E * I * + pS' 2 ) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ' E * I * ' 2 2E * I * - ' E * I * ' 2 2E * I * ' 2 2E * I * ' 3 3E * I * + ' pS - ' 2 2E * I * ' 3 6E * I * - ' pS - ' E * I * - ' 2 2E * I * ' E * I * - ' 2 2E * I * ' 2 2E * I * ' 3 6E * I * - ' pS - ' 2 2E * I * ' 3 3E * I * + ' pS ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (18)
One can see that the pressure appears in the sti ness matrix.

COMPARISONS BETWEEN EXPERIMENTAL AND FINITE ELEMENT RESULTS

Figure 2 shows comparisons between ÿnite element modelling (FE) and experimental results (exp.) for a simply supported panel pressurized at 200kPa (b = 0:2m; h = 0:055m; ' = 1:6m). The usual beam solution (Euler assumptions) is presented too for 47 and 106 N and is inaccurate. The average value of E * is 650 000 Pa m. De ections are obtained for loads varying from 47 to 106 N, just lower than the wrinkling load.

The main advantage of a beam ÿnite element is to be used for solving problems of hyperstatic beams. A panel clamped at one end and simply supported at the other end has been tested up to its wrinkling load. Figure 3 shows one of the experiments made on this panel. Comparisons between experimental and ÿnite element results is shown in Figure 4. Even if the section of the panel is vertical at the clamped end, the angle of the neutral ÿbre is not equal to zero, in accordance with Timoshenko's beam theory. One can see that the results obtained with the in atable beam ÿnite element are close to the experimental ones. Moreover, values of the resultant stresses, according to formula (8), give the wrinkling load of the panel.

CONCLUSION

A new ÿnite element devoted to the study of in ated panels has been constructed taking into account the geometrical sti ness and the following forces. The sti ness matrix takes into account the internal pressure of the beam. Comparisons between experimental and numerical results for isostatic and hyperstatic panels prove the accuracy of this theory on the mechanical strength of in atable beams at high pressure and the e ciency of this in atable ÿnite element.

Figure 2 .

 2 Figure 2. De ection of the panel-isostatic case.

Figure 3 .Figure 4 .

 34 Figure 3. Experiment on in atable panel: hyperstatic case.


P Jouve

R Bouzidi

RESOLUTION OF ELASTOPLASTIC CONSTITUTIVE RELATIONS APPLICATION TO THE FIBER REINFORCED SAND

The aim of this paper is to introduce a new formulation for the resolution of elastoplastic constitutive relations. A variant of the initial stress method is proposed to reduce the number of increments and iterations and avoid a drift of results. A validation of this new algorithm is performed in two homogeneous tests : unreinforced sand and fiber reinforced sand in triaxial compression. For the sand, the basic elements used in the study are a double yield surface with isotropic hardening and a non-associated flow rule. Then, in order to check our computer program and the mechanical principle of superposition of the two continua (sand and fibers), a fmite element simulation of the construction of a wall under gravity forces is presented.

INIRODJJCTION

In the past decade, major efforts have been made to improve the elastoplastic constitutive relations of soils. In particular, the models describing the behaviour of granular soils are more and more efficient. Nowadays, most structures can be studied by using computers. However, in spite of the capacity of these computers, CPU time is often prohibitive. Clearly, numerical methods are not really efficient in computational time, because the numerical integration of the constitutive laws, connecting stress and strain increments, is not straightforward. This integration requires some tedious incremental and iterative methods, and their convergence in structural simulations is also very critical. Consequently, the industrial use of geotechnical software is still limited.

The aim of this study is to present a more cost-effective numerical method, one that will reduce the computational effort, or make it possible to study some ticklish problems. More precisely, this paper is concerned with a new formulation of the basic elements of the Newton-Raphson's method. A new relation for initial stress is expressed. This relation is introduced in the particular instance of plane strain. The applicability of the model is demonstrated for two homogeneous tests : unreinforced sand and fiber reinforced sand in triaxial testing. A complete structure (a sand-reinforced wall) is then dealt with using the finite element method.

ELASTOPLASTIC CONSTITUTIVE LAWS

Following plasticity theory, the strains are divided into an elastic part and a plastic part, respectively represented by superscripts e and p:

{E) = {£"} + {EI'} (1)
We assume the following non-linear elastic behaviour:

{£") = {L(a)} (2)
The elastic strain rates are related to stress rates by a symmetrical matrix :

(£e} = [oe]• 1 (cr} with: [oe] = [VaL]T}- 1 (3)
[ oe] is the elastic tangential matrix ; this matrix is non-singular.

We consider a model characterized by two yield surfaces with isotropic hardening. Let ri be the hardening parameter of mechanism j ; the elastic strains are contained within the current yield surface in stress space if :

Fi(a, d) ::;; 0.

When plastic yield occurs, the stresses are to be found on the following yield surface :

Fi(a, d) = 0 (4)
By differentiating this equation, we can write the consistency condition :

•. ( •)T(. ) ()pi • FJ = V 0 P a + -. rl = 0 drJ [START_REF] Villard | Mod6lisation du comportement mkanique du Texsol, Bulletip de Uaisop deS l&boratoires sies Popts et Cbauss¢es[END_REF] In this case, the description of the strain rate tensor is defined by means of the flow rule :

(f:Pi} = j_i (V 0 Gi) with: i_i ~ 0

where Gi : plastic potential of mechanism j,

).i : plastic multiplier of mechanism j.

The classical formulation assumes that the hardening parameter is defmed by the plastic strain rate : j-i = II £Pi II The norm II II will be determined later. Thus, we obtain : ti = Ni j_i with Ni = IIV aGill (7)

If mechanismj is active, relations (4) or (5), (6) and (7) stand for the corresponding plastic behaviour of the material. If we generalize these relations for all mechanisms, we obtain :

{F(a, r)} = {0} or [ V aF Y ( cr } + [ V rF Y {i ) = { 0) (8) (9)
The total plastic strain { £P} = { £P 1 } + ... + { £1' 0 }, may be expressed as :

( £P} = [VaG Jl i) with: { i) ~ {0} (10) In the case of a multimechanism, the hardening parameter tensor can be expressed by the following formula : {r}=[NJ{i_) (11) If all mechanisms are active, elastoplastic behaviour is entirely defmed by relations (1), [START_REF] Vermeer | A five-constant model unifying well-established concepts, Results of the lpternat Wor:ksho,p op Copstitutiye Relatiops for Soils[END_REF], ( 8), ( 10) and (11).

PROPOSED METHOD OF RESOLUTION Basic relations

Let us consider a structure loaded by nodal forces {P}, for which plastic yielding has occured. For this state (noted A in figure 1) we consider the "principal" variables at any point : { £}, { £P}, { r} and { cr}. Because { £"} is obtained directly by means of { cr} and equation ( 2), we call { ee} a "secondary" variable. The following relations have to be satisfied at each point : A similar expression is obtained for { F( cr+Acr, r+Ar)} = { 0}, which gives after expansion: By using { ~F}, expressed in equation ( 14), it follows :

~~F)+ ( V 0 F )T (Acr} + [ H] {AA.} = {0} with : [ H ] = ( V rF ] T [ N ] . ( 18
)
The elimination of {Acr} between the relations (17) and (18) gives the plastic multipliers:

(AA.) = [w]-1 ((V 0 Fy[D"]({Ae)+{&))+{~F)) (19) Let: and with : [ W] = [ V 0 F ]T [ ne] [VaG]-[ H] . [ DeP] = [ D")-[ D"J( VaG)[ W]-1 ( V 0 F )T[ D") (oo) = [D"][voa][w]-t(~F)-[nep](&).
Thus, the relation between changes of stress and strain is of the form :

{Acr) + {oo) = [ Dep) {Ae) (20) (21) (22)
For a given state and for a change of stress ( Acr} , ( Ae} can be calculated by means of equation ( 22) (see figure 1).

The matrices [ neP] and {00} are obtained by the matrices: [ ne], [ V ~ ], [ VrF] and [ V 0 0 ] . For each increment, there are two possible ways to calculate these matrices : a -we update them at the beginning of the stress increment (at point A in figure 1 ), b -we update them in the middle of the stress increment This last way is more accurate than the first, but it needs two computations for each increment.

1be above analysis is more rigorous than the usual ones because :

we take account of the approximation due to the expansion of { L( a)}, -we take account of the approximation of a point out of the yield surfaces, without having to project it on to these surfaces, -the matrix [ l)t'P) is non-symmetrical for a non-associated material like a soil, and we use this form for the computation of {AU} .

In a structural example, where {P} and (U} stand respectively for nodal forces and nodal displacements, we have the well known relations :

(e} = [B) (U} and (P} = L 1 [ B ]T (a}dV k vk
Within each load increment (AP} in elastoplastic behaviour, the incremental constitutive relation can be written as :

{AP} = L 1 [ B F {aa}dv = L 1 [ B F[ Dep] {ae}-{oo})dv k vk k vk . or (23) with { ~p} = L ( [ B F { 00} dV t Jvk and [ KeP] = L { [ B ]T [ Dep][ B ] dV t Jvk (24)
[ KeP) is the tangential stiffness matrix of the structure. We notice that, if [ oep] is not symmetrical, [ KeP) is not either.

General procedure for solutjop of structural problems

In a structural example we use an incremental procedure characterized by increments of load (AP}. Inside each increment we consider an iterative method based on the equations ( 22) and (23). The term (00} is considered as unbalanced stresses and the term (~P} as an initial load. The principal variables are known at the beginning of the step {AP} (or at the end of the previous step) and they are denoted as: (Uo}, {~}. . fork= 1, the system is:

{AP} + {3Po} = [ ~P]
{AuJ}, where {AP} is the load increment,

. fork> 1, the system is: and (at} = (o~r..I} + {Ao~r.) ; f -Return to "a" for a new iteration when the convergence criterion is not satisfied :

when mr 1 1 1 t~:-~il s ll. with 11 u i, ~I is the displacement length of the node i, and generally Jl is equal to 10-4.

This technique is a Newton-Raphson procedure, which satisfies the relations (12) at each step when convergence occurs. The imprecision of the calculation is mainly due to the incremental form for calculating the change of plastic strains during the iterations, the matrix [ V cP ] being updated at the beginning of each step. For limiting this imprecision, it is necessary to use small load increments { AP } .

For a more accurate procedure applied to the increment {AP}, the best way is to use a second iterative method with the matrix [ V cP ] updated at the mean values between the initial stresses and the fmal ones (obtained at the end of the frrst iterative process) : ( { cr 0 } + { crk}) I 2. This is a second order Runge-Kutta procedure which allows load increments greater than the first one with the same accuracy. This procedure is particularly interesting when there is some rotation of the principal directions of the stresses. For the examples presented here, it is not entirely the case. Thus, we only use the first procedure, [ V cP ] being updated at the beginning of each load increment

PRESENTATION OF THE CONSTITJJTIYE MODEL

1be incremental procedure for solution and its finite element formulation, described in the 3-dimensional case, are first used in a homogeneous test We then consider a structure of sand reinforced by continuous fibers. The Vermeer's model is adopted in this paper, to describe the elastoplastic behaviour of sand [START_REF] Vermeer | A five-constant model unifying well-established concepts, Results of the lpternat Wor:ksho,p op Copstitutiye Relatiops for Soils[END_REF]. This model has been deliberately chosen here because quite a nwnber of successful calculations for granular soils have been reported. In order to specify the version of the model used in this study, a brief description of the formulation is given below. the elastic volumetric strain at isotropic compression p 0 ,

:

the basic material constant.

This formulation involves no Poisson's ratio at all.

Plastic strajn

In the Venneer's model, the plastic strain is subdivided into two independent parts : a deviatoric part and a volumetric part Each of these strains is defined by means of a yield function F and a plastic potential function G.

Yield surfaces

Two surfaces act as the boundary of the elastic region in the stress space.

-1be first one is cone-shaped around the space diagonal. The yield surface is defined by the equation : friction related to IPp• Because of the isotropic hardening, the variable ~Pm is also a function of stress level, and the "generalized" plastic distorsion r 1 is a scalar acting as a hardening parameter. 1be rate of this parameter is defined by : r

• = _l_ "(~~-~~)2 + (~~-~)2 + (~~-£Pzi)2 + 6 [(~~)2 + (~n2 + (~~)2]
fi where £\' } is a plastic strain rate.

In the theory of isotropic hardening, the limit position of the expansion of this yield surface is the failure surface. The Vermeer's model uses the failure criterion of Matsuoka and Nakai expressed by : Fl( cr) = fo. Do+ lifo. A r {q,p) = 0 !pp is the peak friction angle, and A r (!pp) = 9 -sin 2 1Pp 1 -sin 2 ~Pp -The second yield surface is used to introduce the plastic volwnetric strain. This part is not fully explained by the first mechanism. We consider here the spherical yield surface expressed by : . The first one is formulated on the basis of the yield surface :

G 1 (cr, r 1 ) = ~I~+ ~A('Ifm)
where I;,., ~. IT~ are the three invariants of the stresses cr; j = Oi j -a Bi j , with ~j Kronecker's symbol and a a pseudo constant so that G 1 (cr, r 1 ) = 0. -Here we assume that the volumetric yield surface can be identified with the plastic potential: G2(a, r2) = F2(a, r2) .

SIMULATION OF TRIAXIAL TEST ON JJNREINFORCED SAND

To demonstrate the capability of the integration scheme described in the previous section, conventionally drained triaxial compression of unreinforced sand is simulated. This frrst example is studied with smooth contact between sample and platens. Consequently, homogeneous stress and strain states are generated in the whole sample. The aim is to test the procedure of resolution of elastoplastic equations, not the validity of the behaviour model. Three factors are examined and presented below :

-accuracy of solution when the law equations may be integrated by an analytical method, -convergence expressed by a number of iterations, -stability of the scheme with regard to the loading process.

In this study, three compression tests, for which numerical results obtained by Vermeer [START_REF] Vermeer | A five-constant model unifying well-established concepts, Results of the lpternat Wor:ksho,p op Copstitutiye Relatiops for Soils[END_REF] and Mestat [START_REF] Mestat | Validation n~rique du progiciel CESAR-LCPC pour les lois de comportement 6lastique non-linWre plastique[END_REF] are available, are performed:

-isotropic compression, -constant stress-ratio path, -conventional triaxial compression.

lsotropjc compressjop

According to the resolution procedure, the load is applied in several steps from initial state, defined by : 00 1 = 00 2 = 00 3 = 200 kPa To solve this problem, a quantitative parameter study with four loading processes is carried out to verify the numerical performance of this algorithm: 2, 5, 10 and 20 increments. For a given simulation, the variation of stress is constant. In order to describe the sand behaviour, the selected values are : Po = 200 kPa, ee 0 = 0.004, ec 0 = 0.0035 , ~ = 0.25 , tl >p = 43°, tl>cv = 32~ These basic material constants have been proposed at the Grenoble Workshop which fit with dense Karlsruhe sand [START_REF] Vermeer | A five-constant model unifying well-established concepts, Results of the lpternat Wor:ksho,p op Copstitutiye Relatiops for Soils[END_REF].

In this example, because of homogeneous stresses generated in the whole sample, the convergence criterion concerns the unbalanced stresses, and not the unbalanced nodal forces.

As a general rule for all the simulations in this application, the convergence is achieved when the norm of this vector is less than 0.01 Pa : llooll = 'V'(oo.) 2 +(oo2) 2 +(oo3) 2 < o.o1 Pa with { &Ji} : the unbalanced principal stress vector.

These results (stresses and strains) obtained during the loading process, can be verified.

Indeed, for such a stress path, the strain rate equation may be integrated analytically :

e = ~ ~ et [ (fo) ~ -1]
Table 1 and figure 2 show the numerical results for various loading process. In this test, we consider a constant stress-ratio path starting from the initial stress state : As we have already mentioned, this load is applied with different numbers of increments in order to test the reliability of the solution. The previous Vermeer's model parameters are also used in this example. To evaluate the hardening parameters, we consider a stress state on the yield surfaces. The compression test then starts with plastic strains. For this proportional loading and the simple plastic potential function of the second mechanism, the flow rule equation can be integrated This leads to the following relation [START_REF] Vermeer | A five-constant model unifying well-established concepts, Results of the lpternat Wor:ksho,p op Copstitutiye Relatiops for Soils[END_REF], [START_REF] Mestat | Validation n~rique du progiciel CESAR-LCPC pour les lois de comportement 6lastique non-linWre plastique[END_REF] :

E1 = Ei + G 1 + G 2 = S {a
••••t•••••••••••••••j•.;••:••:••••• o IOiacremealo I . • G I j. o• J I .O"j I 205 ••••••••••••••••-l••••••:•:••:•;j:••:••' ••: ........ l . . . . . . . . . . . . . . . . . . t-•••••••••••••••• !. •• ! i ! D • • ~• l l Axi~l Strain

Irjaxjal compression test : ~-copstapt test

To further evaluate the capability of the resolution procedure, the axial strain of the specimen subjected to constant lateral stress is predicted : 00 2 = 00 3 = 100 kPa , a 1 increasing from 100 kPa to failure. The Venneer' s model parameters are the same as the previous test but are applicable for a reference stress Po of 100 kPa. Only in terms of axial strains, this stress path involves a numerical process of resolution. So, we have made a comparison with the results of a classical resolution using the simple potential function of the first mechanism [START_REF] Mestat | Validation n~rique du progiciel CESAR-LCPC pour les lois de comportement 6lastique non-linWre plastique[END_REF]. The axial strain at 420 kPa obtained by the two methods is reported in table 3.

It was then decided to push the solution almost to the failure point. For four loading processes, the results of the numerical model are represented in the O"a-Ea and Ey-Ea diagrams (see figure 4). For all these cases, we note that the obtained solution is very regular until a maximum distortion of 15 %. This distortion has been compared with the analytical solution: This test is specific because the stress field is homogeneous, the loading is regular and the plastic flow direction is almost constant So, it cannot stand as a representative application of the algorithms performance. It stands as a first check of the computer program. However, this agreement in different simulations indicates the applicability of the algorithm. Contrary to the initial stress method with a constant elastic matrix and a simple formulation of the unbalanced stress, we note here a limited number of iterations. Furthermore, although the plasticity relations are incremental, it is interesting to note that large load increments yield close results. This would increase efficiency and economy of analysis in structural problems.

After the new algorithm of numerical integration is tested and to further evaluate the capability of the program presented here, the behaviour of a reinforced sand is predicted. It must be kept in mind that our computer program was originally written to solve the 2-dimensional problems of micro-reinforced structures.

400 i ••••••••••••••••••••••••••••••-~•- ; ••••••••••••••••••••••••••••+•••••••••••••••••••••••••••••••••••+••••••••••••••••••••••••••••••• l I ••••••••••••t•••••• o 5 increments

SIMULATION OF TRIAXIAL TEST ON REINFORCED SAND

1lte procedure of iteration and the constitutive model are now used to predict the stressstrain behaviour of sand reinforced by a network of flexible and continuous fibers. Like previously, the test consists in drained conventional triaxial compression with constant lateral stress. A similar analysis in uniform stress field has been performed experimentally and numerically by C. di Prisco and R. Nova in 1991 [START_REF] Prisco | A constitutive model for soil reinforced by continuous threads[END_REF]. Their approach will not be discussed here as it involves a different procedure. However, in this section, their numerical and experimental results are plotted on figure 7 and compared with the simulations yielded by our program.

The formulation of the constitutive law of sand reinforced by continuous threads is given in reference [START_REF] Villard | Mod6lisation du comportement mkanique du Texsol, Bulletip de Uaisop deS l&boratoires sies Popts et Cbauss¢es[END_REF]. Namely, the model proposed here may be considered as a continuum obtained by superimposing two continua. These continua are distinct but closely linked : solid medium and 30 network. It might be interesting to note that this assumption can reflect upon the simulation at failure state. Indeed, experimental testing indicates that the collapse of material can occur by disconnection of these continua or by failure of fibers in a shear band. These mechanisms are not considered here. The volumetric ratio of threads is expressed by p. The distribution of the directions of the threads can be fixed by the users. This particularity accounts for the anisotropy of the material and can favour a bedding plane, depending on the way in which it is put in place : see figure 5.b. In order to compare our results with those of C. di Prisco and R. Nova, we assume that the reinforcement acts isotropically. The following thread properties are assumed (see figure 6 The comparison between simulation and experimental axial strain is shown in figure 7. As can be seen from figure 7, the numerical solution shows a reasonable agreement with the experimental data. The axial strain is correctly predicted up to 8-10 %. Beyond this value, our previous assumptions concerning the constitutive law of the composite assembly, in particular the connection of sand and threads, are no longer valid. It is important to note that the discrepancy between the two predictions is due to an elastic perfectly plastic behaviour of materials assumed by C. di Prisco and R. Nova.

It may be noted that disagreements appear in the prediction of volumetric strain.

C. di Prisco and R. Nova pointed out the same divergent results with model test data, implicating the simple model and (or) the reliability of the measurements of volumetric strain. So, we are faced with the same problem. A new triaxial test will be needed to explain the reasons for this difference.

Before dealing with structures, we give some explanations about the mechanical principle of reinforcement. During the isotropic compression, the threads are compressed and can be ignored. In deviatoric loading, the deformation in a given direction induces a reaction of the threads in tension. To a lesser extent, this reaction acting in an angular sector increases with the dilatancy of the sand. Thus a fictitious confinement operates on the sand, allowing the deviatoric stress to increase : see figure 8 "Stress in sand of Texsol". The reinforcement can be explained by the Q-P diagram : according to previous remarks on modelling and model test data, we have considered here that failure occurs at an axial strain of 6,8 or 10 % depending on lateral pressure. Figure 9 indicates a "fictitious" cohesion (65 kPa), experimentally provided and used in reinforced sand structures design.

Deviatoric Stress (kPa) , , r---------~----------

800 •••••••••••••••••••••••••••••••••••••••e•• •••--.••-••••••--••-•••••••••••••••••• •••••••••••••••••••-•••••••-•••••••••••••• 0 ,

INITIAL STRESS DISTRIBUTION IN A REINFORCED SAND WALL

This problem involves the construction phase of a reinforced sand wall, assumed to be supported by an embedded foundation. The geometry of the selected wall is presented in figure 10. We study the sttess distribution of the structure subjected to the force of its own weight, prior to lateral earth pressure, under plane strain conditions. In order to estimate the weight sttesses, we propose a numerical procedure. According to the requirement of the resolution process, the calculation is performed in several increments, each increment corresponding to a percentage of the weight 0.8m During the construction of the wall, compaction and weight have led to a prestress of the fibers. Moreover, this prestress is necessary to start numerical resolution of this problem. According to a previous simulation, we consider here an isotropic initial stress (9kPa) in the sand corresponding to a prestress in the fiber induced by compaction. This initial "guess" is necessary for ensuring convergence of the iterative process of resolution, and small enough not to affect the fmal distribution of stress. The loading consists in solving the gravity stress field.

In order to avoid a stress state exceeding the yield condition at the early increments, we consider an arithmetic progression of gravity forces :

(Fv} = ~"!.1 ai {Fv} with ai = 2.i.:...l {Nf N : number of increments, { F v}: total gravity forces.

As mentioned earlier, the convergence criterion of the resolution process is expressed by the unbalanced sttess : llooll = V(oo.) 2 +(ooz) 2 +(oo3) 2 < o.olPa Vermeer's nxxlel and the hyperbolic nxxlel are used to model the sand and the threads. We have ajusted the constant parameters to the components of the geotechnical material originally conceived by LCPC : Texsol [START_REF] Khay | Texsol ouvrage de sou~nement[END_REF]. Using a least squares method in regression computation, the sand properties are taken as : Po = 200 kPa , Eeo = 0.004, Eco = 0.0035 , ~ = 0.25 , ' P = 42°, «Pcv = 32~

The following thread properties are assumed :

Eo= 12 400 MPa, Om= 550 MPa, p = 0.2%.

According to the observations on Texsol structures, it is assumed that the threads are deposited in a 3D angular sector of 30°. The bedding plane is horizontal. The mesh of the wall consists of 64 8-noded isoparametric elements with 3*3 integration (see figure 11). The results of the numerical evaluation are given in figure 12. This figure shows the settlement of the top of the wall for various loading processes. For all these cases, the obtained solution is very regular. Furthermore, a comparison of the loading considering 20, 30 and 50 increments, shows that the distributions are almost identical. For each increment in the last loading (50 increments), only five iterations are needed for convergence of the iterative procedure. It is important to note that a 10 increment loading leads to sizeable differences in the stress distribution. The behaviour of the wall subjected to lateral pressure could be affected by this different initial stress state : see figure 13. 10 increments 20 increments 30 increments 50 increments FIGURE 13: Reinforced sand wall: contours of vertical stress for various loading (kPa).

Figure 14 shows the horizontal stress distribution in the anisotropic medium equivalent to the network of fibers. This stress state resulting from gravity complements the initial stress due to the isotropic compression. Maximum value of horizontal component of tension in medium equivalent to fibers is then 17 kPa (9 kPa for initial stress and 8 kPa for weight). According to the density of fibers and spatial distribution, their maximum stress value is 8.5 MPa. This tension must be compared with the ultimate limit stress of the fibers : am = 550 MPa. This example demonstrates that the network of fibers is weakly activated. The aim of this simulation was to present a numerical initial stress state of the components of the reinforced sand, for a ground stress field. For the moment, we have no experimental data at our disposal, justifying this computation. However computational results are in agreement and indicate that the simulation using the previous resolution process is feasible. Furthermore, convergence and accuracy are assumed for a reasonable number of increments and iterations.

CONCLUSION

A former approach to geotechnical problems with a conventional resolution process, was not really encouraging. It was found that the resolution required very small steps to ensure convergence stability. In this paper, a new approach of analysing reinforced sand walls has been presented. The formulation, the implementation and the validation of the new resolution process have made it possible to acquire thorough knowledge of the plastic mechanisms and a full analysis of numerical problems. The good agreement between analytical and computed values for different loadings, demonsttates the validity of the new method which seems to be more economical than the conventional initial stress method.

Our program needs some improvements for the specific simulation of fiber reinfon:ed sand structures. However, in this version, it can be used for paramettic study. This kind of study will be needed in order to fully validate the assumptions of the mix of sand and fibers, and more particularly the interaction of these two components. It seems especially necessary to adjust the number and direction of the fibers acting in reinforcement, for the study of structures. This study requires a comparison between the simulations and measures carried out on the structures. This is our current aim taking as experimental references two retaining walls subjected to lateral pressure and fitted with measuring instruments by the Centre d'Ex¢rimentation Routi~re de Rouen (France).

FIGURE 1 :

 1 FIGURE 1 :Proposed constitutive terms for the incremental flow.

  {ro} and {ao}. It follows: {to}=[ B] {Uo}. {£0} = {L(ao)}. We note that the matices [ V aGo] and [ No) are calculated from {ao} and {ro}. At the iteration k of this loading step, we use the following technique : a-For each element and from {at-d and (rt_t}, calculate the matrices: [ Dk-1 ], [ V 0 Ft.J], [ VrFt.J], [ Wt-J] and[ D~1], and calculate the unbalanced stresses {~ak-1} from relations (13), (14) and (21); b -Calculation of the elastoplastic stiffness matrix [ ~~ 1 ] and calculation of the unbalanced nodal forces vector { 3P~r..J) : relation (24) ; c -Resolution of the linear equations system (non-symmetrical for a non-associated law) :

  {3P~r..J} = ( ~~1 ] {AU~r.}, updating of the nodal displacements {U~r.} = {U~r..I} + {AU~r.} ; d-For each element, determination of the change of total strain: {A£k} = [ B ]{AUk}, calculation of the change of plasticity multipliers { AA.k}, equation ( 19), calculation of the changes of plastic strain, hardening parameter and stress : {Ae~) = [ V aGo] {AA.~r.), {Ar~r.) = [ No) {AA.~r.) and {Acr~r.) = [ D~~1] {Ae~r.) ; e -Updating of the plastic and total strains, hardening parameters and stresses : {e~} = {eL) + {Ae~), {t~r.} = {t~r..d + {Ae~r.), (I"Jr.} = (r~r..d + {Ar~r.)

  CONSTITUTIVE RELATIONS APPLICATION TO THE FIBER REINFORCED SAND

Five

  parameters are used to characterize the double hardening behaviour of the sand : EfJe, EfJc, p, 'pet 'cv• Elastic strajp In contrast to some variants of the model, we consider a non-linear elastic constitutive law, expressed, in the 3-dimensional case, by: {te} = rlfs-{o} where: Gs = ~ ~~ {~;f"Ji} and On = ~} ~ oTi Po : the reference isotropic compression, ee 0 :

F 1 (

 1 o, rl) = Ia lla + lllaA(,m) = 0 where Ia' 11 0 and lll 0 are the three invariants of the stress tensor, and ' m is the mobilized 7/21 RESOLUTION OF ELASTOPLASTIC CONSTITUTIVE RELATIONS APPLICATION TO THE FIBER REINFORCED SAND

F 2 (

 2 cr r2) = ~(On)~-r2 = 0 ' Po with : p 0 : the reference isotropic compression, ec 0 : the plastic volumetric strain at reference isotropic compression p 0 , r2 : the second hardening parameter. r 2 = ,.j 3 l: (Erff Plastic potential functions -In the Vermeer's model, the cone-flow rule is non-associated.Here, in order to obtain accurate results for dense and loose sands, and to compare our results with some analytical studies, two plastic potentials are considered :

  cl l msin cllcv 'I'm ts the dilatance angle : sm 'I'm = , 1 -sin IPm sin IPcv and ~Pcv is the friction angle at constant volume.

  CONSTITUTIVE RELATIONS APPLICATION TO THE FIBER REINFORCED SAND . The second one is a simple plastic potential, proposed by Venneer, not involving the pseudo constant a : G 1 (a, r 1 ) = ~-t p sin 'I'm with J 25 the second invariant of the deviatoric stresses.

FIGURE 2 :

 2 FIGURE 2 : Isotropic compression test, loading from 200 kPa to 210 kPa.

00 1 =

 1 200 kPa 00 2 = 00 3 = 100 kPa . Then proportional stressing is applied : o 2 = o 3 = 0.5 o 1 .

  1 ~dl ~) with S = 1.00132.10-6 and ~ = 0.25, for current values of the Venneer's model parameters. The results of the numerical model are presented in table 2 and figure 3.

TABLE 2 ::

 2 Constant stress-ratio path, axial strain at 210 kPa for various loading processes. ~ ::::::::

FIGURE 3 :

 3 FIGURE 3 : Constant stress-ratio path, loading from 200 kPa to 210 kPa.

  CONSTITUTIVE RELATIONS APPLICATION TO THE FIBER REINFORCED SAND

q

  _ B C p .th B _ Po { p ) ~ C _ 6 sin ~ The percentage of error on total distortion at the last calculated load point is less than 1%.

FIGURE 4 :

 4 FIGURE 4 : Triaxial Compression Test, Effect of loading process on Distortion.

FIGURE 5 :FIGURE 6 :

 56 FIGURE 5 : Sample of Reinforced Sand.

  CONSTITUTIVE RELATIONS APPLICATION TO THE FIBER REINFORCED SAND -To describe the sand behaviour, we use the elastoplastic Vermeer's model, previously presented The five model constants are determined from the triaxial compression tests data of C. di Prisco and R. Nova, and by a least squares process. The best fit with the model test data, up to a distortion of 15%, are obtained with: Po = 200 kPa, ee 0 = 0.004, ec 0 = 0.0035, p = 0.45, lj)P = 41 ~ ljlcv = 31 o

1000 ;, 6

 10006 'f-~ -0 ~~----~-----500 ----r : ~:~~~~~:;;~

FIGURE 7 :

 7 FIGURE 7 : Comparison between mesured and predicted data, for a constant confining stress.

oFIGURE 8 :FIGURE 9 :

 89 FIGURE 8 : Mechanical principle of reinforcement, P-Q diagram of drained triaxial test on unreinforced sand and reinforced sand : ar = 100 kPa

  CONSTITUTIVE RELATIONS APPLICATION TO THE FIBER REINFORCED SAND

FIGURE 10 :

 10 FIGURE 10: Reinforced sand wall-geometry, initial stress and loading.

FIGURE 11 : 6 FIGURE 12 :

 11612 FIGURE 11 : Mesh of the reinforced sand wall.

  CONSTITUTIVE RELATIONS APPLICATION TO THE FIBER REINFORCED SAND

  FIGURE 14 : Reinforced sand wall : distribution of stress in medium of fibers (kPa).

TABLE 1 :

 1 Isotropic compression test axial, 212strain at 210 kPa for various loading processes.

TABLE 3 :

 3 Triaxial compression test, axial strain at 420 kPa for various loading processes.

					Axial strain at 420 kPa
	simulation	number of increments iterations average number of	elastic strain (10-3)	cone-plastic cap-plastic strain strain (10-~ (10-4)	total (I0-1
	1	5	6	1.4348	1.2415	4.2851	1.4278
	2	20	5	1.4348	1.2328	4.4247	1.4205
	3	80	4.8	1.4348	1.2274	4.4632	1.4155
	4	320	3.8	1.4348	1.2258	4.4730	1.4140
	program "Lois"	32000	-	-	-	-	1.4136

ACKNOWI.EDGEMENTS

The French program "GRECO ~oma~riaux" is gratefully acknowledged for its fmancial support.