Matteo Bastico
email: matteo.bastico@minesparis.psl.eu

David Ryckelynck

Laurent Corté

Yannick Tillier

Etienne Decencière
email: etienne.decenciere@minesparis.psl.eu

A Simple and Robust Framework for Cross-Modality Medical Image Segmentation applied to Vision Transformers

Keywords:

When it comes to clinical images, automatic segmentation has a wide variety of applications and a considerable diversity of input domains, such as different types of Magnetic Resonance Images (MRIs) and Computerized Tomography (CT) scans. This heterogeneity is a challenge for cross-modality algorithms that should equally perform independently of the input image type fed to them. Often, segmentation models are trained using a single modality, preventing generalization to other types of input data without resorting to transfer learning techniques. Furthermore, the multi-modal or cross-modality architectures proposed in the literature frequently require registered images, which are not easy to collect in clinical environments, or need additional processing steps, such as synthetic image generation. In this work, we propose a simple framework to achieve fair image segmentation of multiple modalities using a single conditional model that adapts its normalization layers based on the input type, trained with non-registered interleaved mixed data. We show that our framework outperforms other cross-modality segmentation methods, when applied to the same 3D UNet baseline model, on the Multi-Modality Whole Heart Segmentation Challenge. Furthermore, we define the Conditional Vision Transformer (C-ViT) encoder, based on the proposed cross-modality framework, and we show that it brings significant improvements to the resulting segmentation, up to 6.87% of Dice accuracy, with respect to its baseline reference. The code to reproduce our experiments and the trained model weights are available at https://github.com/matteo-bastico/ MI-Seg and in the Supplementary Code.

Introduction

Medical image segmentation is one of the most challenging and studied research fields of deep learning and artificial intelligence in recent years [START_REF] Siddique | U-Net and Its Variants for Medical Image Segmentation: A Review of Theory and Applications[END_REF][START_REF] Suganyadevi | A review on deep learning in medical image analysis[END_REF]. The precision of segmentation achieved competitive results thanks to recent efforts in developing convolution [START_REF] Malhotra | Deep Neural Networks for Medical Image Segmentation[END_REF][START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF] and transformerbased neural networks [START_REF] Hatamizadeh | Swin UNETR: Swin Transformers for Semantic Segmentation of Brain Tumors in MRI Images[END_REF][START_REF] Hatamizadeh | UNETR: Transformers for 3D Medical Image Segmentation[END_REF]. These algorithms are gen-erally trained on a single medical imaging modality, such as T1-or T2-Weighted Magnetic Resonance Images (MRIs) or Computed Tomography (CT) scans. Therefore, they often suffer from data variability when tested on images different from those seen during training. Data variability is a common issue in the medical field and it is due to several uncontrollable factors that depend on the specific clinic or patient needs [START_REF] Suetens | Fundamentals of Medical Imaging[END_REF], e.g., different imaging methods, different scanners, different acquisition settings or patient limitations. In this scenario, the most straightforward approach would be to train a separate model for each domain, requiring a massive amount of different annotated training data and not exploiting inter-domain information.

Recently, to tackle this issue, some multi-modal [START_REF] Wang | Multimodal medical image segmentation using multiscale context-aware network[END_REF] and cross-modality [START_REF] Valindria | Multi-modal Learning from Unpaired Images: Application to Multi-organ Segmentation in CT and MRI[END_REF][START_REF] Li | Towards Cross-modality Medical Image Segmentation with Online Mutual Knowledge Distillation[END_REF][START_REF] Zhang | Translating and Segmenting Multimodal Medical Volumes with Cycle-and Shape-Consistency Generative Adversarial Network[END_REF][START_REF] Zhou | Generalizable Cross-modality Medical Image Segmentation via Style Augmentation and Dual Normalization[END_REF] medical image segmentation techniques have been proposed. The first group of methods aims to produce better segmentation by simultaneously exploiting information from multiple sources. In this case, it is good practice to stack several images of different types to generate a combined input for the segmentation model [START_REF] Zhou | A review: Deep learning for medical image segmentation using multi-modality fusion[END_REF]. Furthermore, multi-modality can also be leveraged to produce synthetic images of missing domains at inference time [START_REF] Dalmaz | ResViT: Residual vision transformers for multi-modal medical image synthesis[END_REF]. Here, different modalities are rather seen as separate inputs for the model that generates a common representation and produces synthetic images. However, in both scenarios, registered medical images [START_REF] Derek | Medical image registration. Medical image registration[END_REF] are needed. Although registration is relatively simple, the acquisition of several images of the same patient is often limited by local resources and time constraints.

The second group of methods, i.e. cross-modality methods, focus on producing segmentation independently of the data type provided in input. More specifically, an assistant modality, of which several annotated data are available, is often used to improve the segmentation performance on another modality with fewer annotated data, i.e. target modality. Fine-tuning [START_REF] Tajbakhsh | Convolutional Neural Networks for Medical Image Analysis: Full Training or Fine Tuning[END_REF] or transfer learning [START_REF] Raghu | Transfusion: Understanding Transfer Learning for Medical Imaging[END_REF] are straightforward ways of adapting deep learning models to target modalities after they have been trained on some specific assistant modality. However, these approaches do not fully exploit shared cross-modality information. One intuitive way to overcome this problem is joint training [START_REF] Valindria | Multi-modal Learning from Unpaired Images: Application to Multi-organ Segmentation in CT and MRI[END_REF], where a model is trained simultaneously with multiple modalities. Nevertheless, it is relatively hard to directly learn common features when the domain shift is significant, e.g MRIs and CTs. Therefore, to directly learn from multi-modality data with large appearance discrepancy, the use of several feature extractors, i.e., one per modality, has been first proposed in an X-shaped architecture [START_REF] Valindria | Multi-modal Learning from Unpaired Images: Application to Multi-organ Segmentation in CT and MRI[END_REF], implying a significant overhead on the segmentation model and requiring adjustment to generalize to other clinical tasks. To address this issue, several techniques based on synthetic image generation prior to segmentation have been introduced [START_REF] Chen | CrDoCo: Pixel-level Domain Transfer with Cross-Domain Consistency[END_REF][START_REF] Jiang | Tumor-aware, Adversarial Domain Adaptation from CT to MRI for Lung Cancer Segmentation. Medical image computing and computer-assisted intervention : MICCAI[END_REF][START_REF] Zhou | Generalizable Cross-modality Medical Image Segmentation via Style Augmentation and Dual Normalization[END_REF]. In many cases, the assistant modality is used to synthesize the target, or vice versa, and train the model on a mix of real and synthetic data [START_REF] Li | Towards Cross-modality Medical Image Segmentation with Online Mutual Knowledge Distillation[END_REF]. The additional pre-processing step required by these methods to make the clinical images segmentation-ready adds computational complexity, which is often critical in real-time applications such as computerassisted surgery [START_REF] Tanzi | Real-time deep learning semantic segmentation during intra-operative surgery for 3D augmented reality assistance[END_REF].

In this work, we propose a general framework, applicable to any encoder-decoder literature architecture, aiming to produce high-quality cross-modality segmentation without introducing overhead on the segmentation model nor needing registered clinical images for training. With our framework, medical images from different domains can be fed individually directly into a single modality-conditioned model, which generates the desired segmentation by selfadapting its encoder normalization layers. The adaptation is based on Conditional Instance Normalization (CIN), which has been originally proposed for the transfer of arbitrary artistic styles by Dumoulin et al. [START_REF] Vincent Dumoulin | A Learned Representation For Artistic Style[END_REF]. Thanks to that, and similarly to joint training, our models can be trained in an end-to-end way by randomly mixing different data of several modalities, in what we call interleaved mixed training fashion, without prior synthetic image style transfer. Furthermore, based on the proposed framework, we formally define the Conditional Vision Transformer (C-ViT) encoder architecture to build modality-agnostic ViTbased models [START_REF] Dosovitskiy | An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale[END_REF] for image segmentation or classification, as shown in Figure 1. We extensively evaluate our method on the Multi-Modality Whole Heart Segmentation (MM-WHS) 2017 Challenge [START_REF] Zhuang | Evaluation of Algorithms for Multi-Modality Whole Heart Segmentation: An Open-Access Grand Challenge[END_REF]. We show that C-ViT improves the baseline model and other multi-modal techniques and that our framework outperforms other cross-modality segmentation methods, when applied to the same baseline model. Our contributions are summarized as follows:

• We present a novel framework to enhance the quality of cross-modality segmentation, while removing expensive input pre-processing steps and the need of registered data for training. • Following the framework, we introduce the Conditional Vision Transformer encoder, which adapts its normalization layers based on the input modality and can be used for several tasks. • We evaluate our framework on the public whole heart and substructures segmentation challenge [START_REF] Zhuang | Evaluation of Algorithms for Multi-Modality Whole Heart Segmentation: An Open-Access Grand Challenge[END_REF], showing that it outperforms other cross-modality learning techniques both on UNet and ViT baselines.

Related Works

Multi-Modality Learning in Medical Imaging. In recent years, several deep learning architectures have been proposed for image segmentation, achieving extremely good performance [START_REF] Chen | TransUNet: Transformers Make Strong Encoders for Medical Image Segmentation[END_REF][START_REF] Hatamizadeh | Swin UNETR: Swin Transformers for Semantic Segmentation of Brain Tumors in MRI Images[END_REF][START_REF] Hatamizadeh | UNETR: Transformers for 3D Medical Image Segmentation[END_REF][START_REF] Milletari | V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation[END_REF][START_REF] Singh | UNet Architectures in Multiplanar Volumetric Segmentation -Validated on Three Knee MRI Cohorts[END_REF]. Among them, UNet [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF] is surely the most popular and is used as a baseline to create better performing models [START_REF] Siddique | U-Net and Its Variants for Medical Image Segmentation: A Review of Theory and Applications[END_REF]. More recently, some variation based on vision transformers (ViT) [START_REF] Dosovitskiy | An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale[END_REF] and swin-transformers [START_REF] Liu | Swin Transformer: Hierarchical Vision Transformer using Shifted Windows[END_REF], such as TransUNET [START_REF] Chen | TransUNet: Transformers Make Strong Encoders for Medical Image Segmentation[END_REF], UNETR [START_REF] Hatamizadeh | UNETR: Transformers for 3D Medical Image Segmentation[END_REF] and Swin-UNETR [START_REF] Hatamizadeh | Swin UNETR: Swin Transformers for Semantic Segmentation of Brain Tumors in MRI Images[END_REF], have been shown to outperform previous versions of UNet. In the clinical imaging field, multi-modality learning is a similar, yet challenging, task. Multiple medical image domains have been exploited for synthetic image generation based on Generative Adversarial Networks (GAN) [START_REF] Dalmaz | ResViT: Residual vision transformers for multi-modal medical image synthesis[END_REF][START_REF] Salman | Image Synthesis in Multi-Contrast MRI With Conditional Generative Adver-sarial Networks[END_REF][START_REF] Goodfellow | Generative Adversarial Networks[END_REF][START_REF] Sun | Synthesis of pseudo-CT images from pelvic MRI images based on an MD-CycleGAN model for radiotherapy[END_REF][START_REF] Zhang | Multi-Contrast MRI Image Synthesis Using Switchable Cycle-Consistent Generative Adversarial Networks[END_REF] or multimodal image segmentation [START_REF] Fidon | Scalable Multimodal Convolutional Networks for Brain Tumour Segmentation[END_REF][START_REF] Guo | Medical Image Segmentation Based on Multi-Modal Convolutional Neural Network: Study on Image Fusion Schemes[END_REF][START_REF] Wang | Multimodal medical image segmentation using multiscale context-aware network[END_REF][START_REF] Zhou | A review: Deep learning for medical image segmentation using multi-modality fusion[END_REF]. Interestingly, the two tasks have also been merged by Zhang et al. [START_REF] Zhang | Cross-Task Feedback Fusion GAN for Joint MR-CT Synthesis and Segmentation of Target and Organs-At-Risk[END_REF] in a cross-task feedback fusion GAN that first generates synthetic CT and then performs multi-modal segmentation. The latter uses cross-domain information to achieve better performances, but it is restricted to when registered images are available. To relax the need of aligned medical images and exploit inter-domain features during training, cross-modality segmentation approaches have recently gained attraction. Firstly, Zheng et al. [START_REF] Zheng | Cross-modality medical image detection and segmentation by transfer learning of shapel priors[END_REF] investigated the effectiveness of shape priors learned from an assistant modality to improve the segmentation on a target modality by marginal space learning. Valindria et al. [START_REF] Valindria | Multi-modal Learning from Unpaired Images: Application to Multi-organ Segmentation in CT and MRI[END_REF] developed dual-stream encoder-decoder models having different branches for each modality and implementing weights sharing techniques to extract cross-modality features. Similarly to our work, some researcher tried to play with the normalization layers of the models to improve their generalization capacities [START_REF] Fan | Adversarially Adaptive Normalization for Single Domain Generalization[END_REF][START_REF] Pan | Two at Once: Enhancing Learning and Generalization Capacities via IBN-Net[END_REF][START_REF] Segu | Batch Normalization Embeddings for Deep Domain Generalization[END_REF][START_REF] Seo | Learning to Optimize Domain Specific Normalization for Domain Generalization[END_REF][START_REF] Zhou | Generalizable Cross-modality Medical Image Segmentation via Style Augmentation and Dual Normalization[END_REF]. Among others, Pan et al. [START_REF] Pan | Two at Once: Enhancing Learning and Generalization Capacities via IBN-Net[END_REF] introduced the IBN-Net which simultaneously exploiting Instance and Batch Normalization to capture both appearance changes and content information. Segu et al. [START_REF] Segu | Batch Normalization Embeddings for Deep Domain Generalization[END_REF] proposed to collect prior domain-dependent statistics by training ad-hoc Batch Normalization (BN) to map the modalities on a shared latent space.

With the aforementioned advances in synthetic images generation, many works tried to tackle the challenge by assisting segmentation models with prior image translation [START_REF] Chen | CrDoCo: Pixel-level Domain Transfer with Cross-Domain Consistency[END_REF][START_REF] Hoffman | CyCADA: Cycle-Consistent Adversarial Domain Adaptation[END_REF][START_REF] Jiang | Tumor-aware, Adversarial Domain Adaptation from CT to MRI for Lung Cancer Segmentation. Medical image computing and computer-assisted intervention : MICCAI[END_REF][START_REF] Li | Towards Cross-modality Medical Image Segmentation with Online Mutual Knowledge Distillation[END_REF][START_REF] Zhang | Translating and Segmenting Multimodal Medical Volumes with Cycle-and Shape-Consistency Generative Adversarial Network[END_REF][START_REF] Zhou | Generalizable Cross-modality Medical Image Segmentation via Style Augmentation and Dual Normalization[END_REF]. Among them, Zhang et al. [START_REF] Zhang | Translating and Segmenting Multimodal Medical Volumes with Cycle-and Shape-Consistency Generative Adversarial Network[END_REF] tried to improve the segmentation for modalities with limited training samples by easing the appearance gap with other modalities using a GAN. Similarly, Li et al. [START_REF] Li | Towards Cross-modality Medical Image Segmentation with Online Mutual Knowledge Distillation[END_REF] proposed an Image Alignment Module to reduce the appearance gap between assistant and target modality and introduced a Mutual Knowledge Distillation scheme [START_REF] Hinton | Distilling the Knowledge in a Neural Network[END_REF] to exploit modality-shared knowledge. More recently, Zhou et al. [START_REF] Zhou | Generalizable Cross-modality Medical Image Segmentation via Style Augmentation and Dual Normalization[END_REF] proposed to simulate the possible appearance changes of a target domains by non-linear transformation to augment source-similar and source-dissimilar images.

Different from these methods, which mainly rely on image translation or prior trainings, our framework tries to shrink the model complexity and, at the same time, improve the segmentation accuracy extracting all the available crossmodality features by conditioning the segmentation models and training them jointly with assistant and target modality. Moreover, many of the previous techniques do not allow for arbitrary switch of the input modality, but they just exploit assistant modality information to improve the target segmentation.

Normalization Layers. Our work is also related with normalization layers used in neural networks, which may differ based on the spatial dimension to which they are applied. The four main classes of normalization layers are batch, layer, group, and instance normalization. Batch Normalization was the first technique introduced to deal with the variability of layers input during training by normalizing the features across mini-batches [START_REF] Ioffe | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift[END_REF]. Layer Normalization (LN) [START_REF] Lei Ba | Layer Normalization[END_REF], used mainly in Natural Language Processing (NLP) and Recurrent Neural Networks (RNNs), was introduced to relax the dependency on the batch, normalizing only across features. Similarly, Group Normalization [START_REF] Wu | Group Normalization[END_REF] computes the mean and standard deviations in groups of channels rather than using the entire feature space.

CIN(z z z, m) = γ m z z z -µ(z z z) σ(z z z) + β m (1)
where µ(z z z) and σ(z z z) are computed across spatial dimensions independently for each channel and each sample [START_REF] Huang | Arbitrary Style Transfer in Real-time with Adaptive Instance Normalization[END_REF]. It allows the sharing of all convolutional weights of a style transfer network across many image styles by only changing the learnable normalization parameters. Inspired by that, in this work, CIN is exploited to generate a shared latent space for different input modalities, which is used to obtain the segmentation output independently of the medical input image domain.

Method

The overview of the proposed cross-modality segmentation framework is shown in Figure 2. The segmentation model, which can be chosen among those available in the literature, is conditioned by the input data modality to adapt its normalization layers to the fed clinical image. Therefore, the aim is to produce the desired segmentation independently of the input data domain.

In the following, the general framework for creating conditional models is introduced and it is applied to the transformer encoder, to obtain the C-ViT encoder. Finally, more details about the framework training procedure, including the loss function, are reported.

Conditional Segmentation Models. Let any state-ofthe-art encoder-decoder segmentation architecture be considered as baseline for the proposed framework.

y y y = D(E(x x x m , m)) , 1 ≤ m ≤ M ( 2
)
This concept is easily extended to architectures with skip connections, e.g. UNet [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF], in which the encoder may be seen as a sequence of S sub-modules, each of them generating a common latent space C s , 0 ≤ s ≤ S, where S is also the number of connections between encoder and decoder. Therefore, the decoder will simply gather the data from all the shared latent spaces to generate a modality-agnostic segmentation output.

In practice, we propose to generate the conditional encoder E(•, m), starting from any simple encoder E m (•), by adding the data modality as input (often available from meta-data, without manual intervention) and replacing all its normalization layers with CIN. In such way, the only overhead on the segmentation model is given by the extra normalization learnable parameters γ m and β m , 1 ≤ m ≤ M , of Equation 1, which are negligible with respect to the rest of the architecture. Furthermore, we avoid any pre-processing phase, such as synthetic image generation through GANs, which can heavily impact on the network performances, and we let the model directly learn a common shared latent space for all the input modalities, trying to fully exploit the inter-modality information.

Conditional Vision Transformer. The proposed C-ViT encoder is shown in Figure 3. As originally proposed by Vaswani et al. [START_REF] Vaswani | Attention Is All You Need[END_REF], it has two sub-layers, the first is a multihead self-attention mechanism (MSA), and the second is a simple, position-wise fully connected Multi-Layer Perceptron (MLP). Both sub-layers are preceded by a normalization [START_REF] Dosovitskiy | An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale[END_REF][START_REF] Wang | Learning Deep Transformer Models for Machine Translation[END_REF] and a residual connection is employed around each of them. Here, we replace the Layer Normalization (LN) [START_REF] Lei Ba | Layer Normalization[END_REF], which has historically been used in transformers, Table 1: Quantitative comparison with other methods for cross-modality medical image segmentation on the target modality (CT). All the techniques have the same UNet basline [START_REF] Valindria | Multi-modal Learning from Unpaired Images: Application to Multi-organ Segmentation in CT and MRI[END_REF], are trained using 20 MRIs and 10 CTs and are evaluated on the test set of MM-WHS dataset. The mean DICE score is reported, as well as the ones for all the heart substructures.

Model

Avg In a general scenario with M different input modalities, let L, N and K be the number of stacked transformer blocks that make up a transformer encoder, the number of input patches and the dimensions of the embedding space, respectively. For a given input embedding of the i-th transformer block, z z z i-1 ∈ R N ×K , we have

ẑ z z i = MSA(CIN(z z z i-1 , m)) + z z z i-1 , 1 ≤ i ≤ L (3)
z z z i = MLP(CIN(ẑ z z i , m)) + ẑ z z i , 1 ≤ i ≤ L (4)
where 1 ≤ m ≤ M is the modality corresponding to the input and CIN(•, •) is defined in Equation 1. Note that z z z 0 corresponds to the linear projection of the input patches, i.e. non-overlapping sub-patches of dimension P × P × P of the input volume, {p p p j ∈ R P 3 : 1 ≤ j ≤ N } , into a K-dimensional embedding space using a projection matrix E E E ∈ R P 3 ×K [START_REF] Hatamizadeh | UNETR: Transformers for 3D Medical Image Segmentation[END_REF]. Additionally, learnable positional embeddings, E E E pos ∈ R N ×K , are included to code spatial information, finally obtaining

z z z 0 = [p p p 1 E E; . . . ; p p p J E E E] + E E E pos . (5)
The rationale for substituting LN, which is inherited from the applications of the first transformers in NLP [START_REF] Wolf | Hugging-Face's Transformers: State-of-the-art Natural Language Processing[END_REF], for CIN, or more generally IN, is also motivated on the zoom of Figure 3. LN computes the average and variance of the single input patches along the K dimensions of the embedding space. Therefore, when it comes to images, the latter produces an independent normalization for each image patch without considering the entire input data. IN addresses this issue by computing the metrics for normalization across all the input patches but separately for each embedding dimension. Intuitively, the model is able to generate more meaningful embeddings when the input volume is considered as a whole for normalization.

The technique presented here can also be generalized to the Swin-transformer model by replacing the MSA of Equation 3 with W-MSA or SW-MSA, denoting window-based multi-head self-attention using regular and shifted window partitioning configurations, respectively [START_REF] Liu | Swin Transformer: Hierarchical Vision Transformer using Shifted Windows[END_REF]. Finally, in this case, to have a fully conditional encoder, also the patch merging block, used to generate hierarchical representations of the input patches, must be adapted in the normalizations.

Loss function and Training. To train cross-modality segmentation models using our framework, we used a combination of Dice Loss [START_REF] Jadon | A survey of loss functions for semantic segmentation[END_REF], L Dice , and Focal Loss [START_REF] Lin | Focal Loss for Dense Object Detection[END_REF], L Focal , as

L = λ D L Dice + λ F L Focal ( 6
)
where λ D and λ F are scalar factors greater than 0 to adjust the contribution of each individual loss to the total. We preferred this version of the loss, rather than the standard combination of Dice loss and Cross-Entropy (CE), to tackle the unbalancing of clinical image segmentation, which is often significant between background and foreground. The focal loss, instead, reshapes the standard CE loss so that it downweights the loss assigned to well-classified examples. In this way, more weight will be given in the loss to hard classified samples, i.e., labels difficult to segment, improving the learning process. Differently from joint training [START_REF] Valindria | Multi-modal Learning from Unpaired Images: Application to Multi-organ Segmentation in CT and MRI[END_REF], in which training data is divided into batches of different modalities, conditional models are trained using an interleaved mixed data training fashion. This means that different modalities are randomly fed into the model during training, even in the same batch. Therefore, the overall loss will provide an overview of the goodness of the segmentation independently from the input modality, pushing the network to be fair with all different types of data.

Experiments

In this section we describe the experiments carried out to evaluate the proposed cross-modality framework with in-Table 2: Performance comparison on the target modality (CT) of our framework applied to the Swin-UNETR model [START_REF] Hatamizadeh | Swin UNETR: Swin Transformers for Semantic Segmentation of Brain Tumors in MRI Images[END_REF]. All the models are trained using 20 MRI and 10 CT volumes and are evaluated on the test set of MM-WHS dataset. The mean DICE score is reported, as well as the ones for all the heart substructures and Whole Heart (WH) segmentation. terleaved mixed data training. The experiment settings are described, including the dataset, the implementation details, and the algorithms to tune the models hyper-parameters. Finally, the results are shown and discussed.

Model

Experiment settings

Dataset. We evaluate the proposed method on the Multimodality Whole Heart Segmentation Challenge 2017 (MM-WHS 2017) dataset [START_REF] Zhuang | Evaluation of Algorithms for Multi-Modality Whole Heart Segmentation: An Open-Access Grand Challenge[END_REF], which contains non-registered 20 MRI and 20 CT volumes for training and the groundtruth (GT) annotations of 7 cardiac substructures including the Left Ventricle blood cavity (LV), the Right Ventricle blood cavity (RV), the Left Atrium blood cavity (LA), the Right Atrium blood cavity (RA), the MYOcardium of the left ventricle (MYO), the Ascending Aeorta (AA), and the Pulmonary Artery (PA). Following previous cross-modality segmentation works on this dataset [START_REF] Li | Towards Cross-modality Medical Image Segmentation with Online Mutual Knowledge Distillation[END_REF], we used MRI as assistant modality and CT as target modality, since MRI has better contrast for soft tissue and ideally provide better information for heart substructures segmentation. We evenly and randomly split the CT data in order to perform a two-fold cross-validation and train the conditional models with 20 MRIs and 10 CTs at a time, to simulate data shortage for the target modality. Furthermore, MRI volumes are pre-processed using the N4 bias field correction algorithm to correct low frequency intensity non-uniformity present in MRI image data known as gain field [START_REF] Tustison | N4ITK: Improved N3 Bias Correction[END_REF]. Finally, all the volumes are re-sampled to an isotropic space and their values are normalized between 0 and 1.

Implementation Details. To provide a fair comparison, we first applied our framework to the same baseline used by other cross-modality previous works [START_REF] Li | Towards Cross-modality Medical Image Segmentation with Online Mutual Knowledge Distillation[END_REF][START_REF] Valindria | Multi-modal Learning from Unpaired Images: Application to Multi-organ Segmentation in CT and MRI[END_REF]. Namely, we implemented the network architecture with encoderdecoder structure proposed by Valindria et al. [START_REF] Valindria | Multi-modal Learning from Unpaired Images: Application to Multi-organ Segmentation in CT and MRI[END_REF], consisting of a 3D UNet [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF] with residual blocks with preactivation [START_REF] He | Identity Mappings in Deep Residual Networks[END_REF]. After that, we evaluated the efficiency of the proposed C-ViT encoder applied to the Swin-UNETR [START_REF] Hatamizadeh | Swin UNETR: Swin Transformers for Semantic Segmentation of Brain Tumors in MRI Images[END_REF] segmentation model, originally consisting of a Swintransformer encoder [START_REF] Liu | Swin Transformer: Hierarchical Vision Transformer using Shifted Windows[END_REF] and UNet style decoder, as shown in Figure 1. For all the models, we define an input size of 96 × 96 × 96 pixels and sub-patches, in the case of transformer-based architectures, of 16 × 16 × 16 pixels. Therefore, at each epoch, a fixed number of random crops of input size is extracted, for each volume. Note that, in this way, the effective batch size is given by N s × N c , where N s and N c are the number of samples and crops, respectively. For the transformer-based models, we follow the original implementation of Hatamizadeh et al. [START_REF] Hatamizadeh | Swin UNETR: Swin Transformers for Semantic Segmentation of Brain Tumors in MRI Images[END_REF], with L = 8 blocks and hidden size K = 768 in the encoder. Furthermore, data augmentation is applied to artificially increase training data and deal with inter-modality variations by randomly shifting and scaling the intensity of the input volumes, along with random flips in all three dimensions. Finally, to validate and test the models, sliding window inference is used with an overlap of 0.5.

The code is implemented in PyTorch, with the help of the MONAI library [START_REF] Cardoso | MONAI: An open-source framework for deep learning in healthcare[END_REF], and trained on a cluster equipped with multiple NVIDIA V100 GPUs. Optimization is performed using Adam with decoupled weight decay [START_REF] Loshchilov | Decoupled Weight Decay Regularization[END_REF] and a cosinedecay Learning Rate (LR) scheduler with warm-up, for a maximum of 2500 epochs. All the models are tuned to find the best set of hyper-parameters as described below.

Hyper-parameters Tuning. To take full advantage of the segmentation models, we tuned them using the Optuna framework [START_REF] Akiba | Optuna: A Nextgeneration Hyperparameter Optimization Framework[END_REF], following the guidelines from [START_REF] Godbole | Deep Learning Tuning Playbook[END_REF]. Therefore, the batch size is not tuned but is kept unchanged among trials as the maximum that can fit our memory availability, that is, 16 three-dimensional crops (N s = 4 and N c = 4). The optimal LR is found for each model using the cyclical learning rate technique [START_REF] Smith | Cyclical Learning Rates for Training Neural Networks[END_REF] and a logarithmic search space centered on that value is defined for tuning. The number of LR Warm-Up epochs is chosen among the values of 2, 4 and 6% of the maximum training epochs [START_REF] Izsak | How to Train BERT with an Academic Budget[END_REF] and the optimizer weight decay is also tuned. Finally, the loss function of Equation 6 is used with λ D = λ F = 1.

For each trial, the hyper-parameters are sampled form the search space using the Tree-structured Parzen Estimator (TPE) algorithm [START_REF] Bergstra | Algorithms for hyper-parameter optimization[END_REF][START_REF] Ozaki | Multiobjective Tree-Structured Parzen Estimator[END_REF]. This technique aims to find the best set of parameters by fitting two Gaussian Mixture Models (GMMs) to the good and bad trials and minimizing their ratio. Furthermore, Asynchronous Successive Halving (ASHA) [START_REF] Li | A System for Massively Parallel Hyperparameter Tuning[END_REF] is used to early-stop bad trials and save computational time and resources.

Experiment results

We report here the results of our experiments using the proposed framework on the MM-WHS dataset [START_REF] Zhuang | Evaluation of Algorithms for Multi-Modality Whole Heart Segmentation: An Open-Access Grand Challenge[END_REF], quantitative using the Dice similarity coefficient and qualitative by comparing segmentation images. In the following, we first compare our proposal with other techniques and then we evaluate the accuracy of the proposed C-ViT.

Comparison with Other Methods. The comparative quantitative results for heart substructures segmentation are shown in Table 1. In fine-tuning the model is first trained using the assistant modality only and then, the acquired knowledge is transferred to the target modality with a subsequent training. In join-training the baseline model is trained with both modalities simultaneously by alternating them on different batches. We also compare the accuracy with the X-shape architecture proposed by Valindria et al. [START_REF] Valindria | Multi-modal Learning from Unpaired Images: Application to Multi-organ Segmentation in CT and MRI[END_REF] and the online training with synthetic image generation approach of Zhang et al. [START_REF] Zhang | Translating and Segmenting Multimodal Medical Volumes with Cycle-and Shape-Consistency Generative Adversarial Network[END_REF]. Finally, a knowledge distillation cross-modality segmentation technique is also considered for comparison [START_REF] Li | Towards Cross-modality Medical Image Segmentation with Online Mutual Knowledge Distillation[END_REF]. As highlighted by the latter, fine-tuning, joint-training and the X-shaped architecture bring marginal improvements on the segmentation accuracy since they do not fully exploit cross-modality information. Online GAN-based synthetization with and without mutual knowledge distillation start to significantly improve the mean accuracy of target modality segmentation, up to 3.06%. Nevertheless, in both cases, significant overhead is introduced on the model, limiting its possible real-time applications. Our method achieves significant improvements in the segmentation accuracy, while reducing the complexity of both training and inference. We observe an enhancement of 0.65% in the mean Dice and up to 2.71% on the single heart substructures, namely in the RV. However, a small decrease in performances is registered for AA and PA, probably due to the fact that the manual segmentation of these structures generally covers beyond the real size of the vessels while the test tool cuts them to a limited length [START_REF] Zhuang | Evaluation of Algorithms for Multi-Modality Whole Heart Segmentation: An Open-Access Grand Challenge[END_REF].

Evaluation of C-ViT and Ablation Study. We compare the performance of the C-ViT-based segmentation model with its baseline [START_REF] Hatamizadeh | Swin UNETR: Swin Transformers for Semantic Segmentation of Brain Tumors in MRI Images[END_REF] and the result obtained by finetuning and joint training the same model (with interleaved mixed fashion). Table 2 summarizes the quantitative results in terms of Dice accuracy of Whole Heart Segmentation (WHS) and single substructures segmentation. Furthermore, Figure 4 shows a qualitative comparison on single slices of the segmentation obtained on CT by using different methods, and Figure 5 directly compares the 3D segmentation of a whole sample. Our framework enhances of 4% the mean Dice of heart substructures, with significant improvements in each of them, and 6.87% the accuracy of WHS, with respect to the baseline model. Qualitative, we can see important improvements when using C-ViT with respect to other methods, that may also result in refinement of the GT (e.g. the segmentation of PA on the second row of Figure 4). In the 3D comparison, we observe the improvement in the segmentation produced by C-ViT which does not contain false positives outside the region of the GT. Although in our cross-validation the ViT-based conditional model performs better than the UNet one (91.18% ± 0.68 vs 90.76% ± 0.55 of mean Dice), in the test set, the latter seems to be slightly better likely due to the fact that transformers usually need more data for finer generalization [START_REF] Park | How Do Vision Transformers Work?[END_REF].

Differently from some other related works [START_REF] Jiang | Tumor-aware, Adversarial Domain Adaptation from CT to MRI for Lung Cancer Segmentation. Medical image computing and computer-assisted intervention : MICCAI[END_REF][START_REF] Li | Towards Cross-modality Medical Image Segmentation with Online Mutual Knowledge Distillation[END_REF] our framework do not just exploit assistant modality to improve the segmentation of the target one, but it keeps the ability of producing high-quality segmentation for the assistant modality by switching the input domain. To confirm that, In Table 3, WHS and Mean Dice of MRI segmentation is reported, where the baseline is considered to be the model trained only with assistant modality and used to fine-tune the target modality. Also the assistant modality segmentation accuracy benefits of our framework, as well as of joint-training, improving with respect to the baseline model. Moreover, Figure 6 also shows some qualitative improvements of the MRI segmentation, especially from baseline to C-ViT. As previously mentioned, transformers are "datahungry" [START_REF] Wang | Towards Data-Efficient Detection Transformers[END_REF], in the sense that they suffer from significant performance drops on small-size datasets. We have shown that, with our conditional framework applied to ViT, it is possible to relax the need of an huge dataset for training by gathering several small datasets of different modalities.

Conclusions

In this work, we proposed a simple framework that aims to segment different types of medical images, while reducing the model overhead and the need of registered data for training, using a single cross-modality conditional model trained with interleaved mixed data. We introduced a general definition of a conditional model, based on CIN, that can be applied to all state-of-the-art medical image segmentation architectures. Furthermore, we developed a new C-ViT encoder that can be used to create conditional ViTbased models. We validated our framework using a public multi-modal dataset for heart substructures segmentation, which collects both MRI and CT data. Our framework achieves new state-of-the-art performance for the crossmodality medical image segmentation with assistant and target modality. We have shown that not only the first one helps the learning on the latter when few annotated data are available, but also the segmentation of the assistant modality benefits of our framework. We believe that our work can be an important contribution for developing robust crossmodality medical image segmentation methods, to tackle cases in which the acquisition of a specific kind of image is limited by environmental or patient-specific restrictions. Furthermore, in the future, the proposed framework may be extended in an Unsupervised Domain Adaptation (UDA) fashion, to make possible the training on a single labeled modality with adaptation to unlabeled domains.

Figure 2 :

 2 Figure 2: Overview of the proposed conditional framework: the conditional encoder, E(•, m), generates S common latent spaces, {C 1 , . . . , C S }, one for each encoder-decoder connection, adapting itself to the input modality m and the decoder, D(•), is unique and does not need adaptation.

  Finally, Instance Normalization (IN) has been introduced, in the imaging context, to prevent instance-specific mean and covariance shift, simplifying the learning process [50]. Huang et al. [22] proposed a further variation of IN, called Adaptive IN to perform arbitrary image style transfer, that is, synthetic image generation, by adjusting the mean and variance of the content input to match those of the desired style input. The latter does not have learnable parameters, but it computes them directly from the input style. In our framework, we propose to use a similar variation of IN recently proposed by Dumoulin et al. [10], called Conditional Instance Normalization (CIN), which learns a different set of parameters γ m and β m for each style m ∈ [1, M ]. Given a threedimensional volume with C channels, z z z ∈ R C×H×W ×D , of modality m, the CIN is calculated as

  In a straightforward case, for a given data modality m ∈ [1, M ], let E m (•) : I m → H m and D m (•) : H m → O be the two functions that describe the encoder and decoder modules of the chosen architecture, respectively. The role of E m (•) is to transform the input, x x x m ∈ I m , into a latent feature space, H m , which can better represent the information carried by raw data. On the other hand, the purpose of D m (•) is to transform this representation into the final segmentation y y y ∈ O. In general, the space H m is learned ad-hoc by the model, based on the type of data used during training. To make the model independent of the input modality, we propose to create a general conditional encoding module, E(•, m) : I m → C, which transforms any input x x x m ∈ I m , 1 ≤ m ≤ M , into a common shared latent feature space C. In this way, a single decoder D(•) : C → O is needed to generate the segmentation of all the modalities. The relation between any input x x x m and the output y y y in a conditional model can then be expressed as

Figure 3 :

 3 Figure 3: Conditional Vision Transformer Encoder. The multi-head self-attention (MSA) mechanism is followed by a MLP and each of them is preceded by CIN. The latter consists of a switch mechanism to activate the IN corresponding the to modality flag in input. A focus on the different normalizing direction of IN and LN is also shown in the zoom.

Figure 4 :

 4 Figure 4: Qualitative comparison of the segmentation result on the CT target domain of our framework applied to the Swin-UNETR model [16] on the MM-WHS dataset. Our framework is compared with the baseline model, trained on the target modality, fine-tuning and joint-training with interleaved mixed fashion.

Figure 5 :Figure 6 :

 56 Figure 5: Comparison of the 3D segmentation of heart substructures from CT using different cross-modality adaptations.

Table 3 :

 3 Performance comaprison on MRIs, used as assistant modality, of our framework applied to the Swin-UNETR model[START_REF] Hatamizadeh | Swin UNETR: Swin Transformers for Semantic Segmentation of Brain Tumors in MRI Images[END_REF]. The baseline is the model used to do fine-tuning on the target modality.

		WH	Avg Dice	MYO	LA	Dice of substructures of heart LV RA RV	AA	PA
	Baseline	83.11	85.32	84.79	88.12	89.80	81.04	84.08	77.58	76.36
	Fine-tune	86.33	86.09	82.57	90.03	87.92	83.29	85.51	90.28	83.06
	Joint-training	89.03	87.99	86.25	92.07	91.97	85.35	87.98	89.55	82.76
	C-ViT [ours]	89.98	89.33	88.13	91.64	92.39	86.36	89.33	93.39	84.05
	Model		WH	Mean Dice
	Baseline		83.32	81.87
	Joint-training		86.65	84.82
	C-ViT [ours]		85.57	84.30

Acknowledgements

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 945304-Cofund AI4theSciences hosted by PSL University.

This work was granted access to the HPC/AI resources of IDRIS under the allocation 2022-AD011013902 made by GENCI.


Cyril Douthe
email: cyril.douthe@univ-eiffel.fr

Ioannis Stefanou

In-situ experiment on long-term behaviour of GFRP pipes under permanent bending

Keywords: Glass Fibre Reingorced Polymers (GFRP), residual curvature, prestress, ageing, durability, relaxation

The present study aims at studying the long-term behaviour of composites (GFRP) subjected to permanent bending. The specificity of this study largely rests on the original conditions of in-situ ageing of the specimens within an experimental structure of elastic gridshell, built in 2006 and disassembled in 2014.

This prototype was made of rectilinear tubes which were elastically deformed into a 3D stable shape. The structural members constituted hence a unique set of relaxation tests in ambient conditions where a wide variety of curvatures were imposed by the 3D shape. After dismounting, the tubes were sorted according to their permanent curvature in the structure. They were then subjected to three kinds of tests: recovery, longitudinal bending and transverse bending tests. The results were then compared with those obtained during the construction of the prototype, before ageing, and with members aged under the same conditions but not mechanically loaded.

The experimental results confirm the usual recommendations for using composites in civil engineering, which suggest limiting the flexural stresses to 30% of the tensile strength to avoid damage or creep problems. At the same time, they confirm the safety and long-term robustness of the concept of double curvature structures obtained by elastic deformation of a composite plane grid.

Introduction

Key element in the use of Glass Fibre Reinforced Composites (GFRP) in Civil Engineering applications, such as buildings, bridges and facades, among others, is the long-term evolution of the mechanical properties of GFRP's. In this type of applications, GFRP's are inevitably subjected to permanent loads.

As the applications of GFRP in construction is relatively limited up to date, the exact knowledge of the time-dependency of the fundamental mechanical parameters of GFRP is still not completely known. Laboratory experiments are frequently performed in order to improve the current state of knowledge.

However, the duration of these experiments is usually limited, due to practical reasons. Moreover, theoretical analyses [such as 1, 2, 3, 4, among others, and references therein] provide useful insight and information, but still have to be validated using real, long-term experimental data.

The importance and novelty of this work relies on the use of experimental data from a real GFRP, elastic gridshell structure, which was built in 2006 and dismantled in 2014. Therefore, our experimental data involve time-dependent and ageing phenomena of 8 years under real environmental conditions. To our knowledge, these experimental data are unique and complete previous ones (see [START_REF] Keller | Long-term performance of a glass fiberreinforced polymer truss bridge[END_REF][START_REF] Keller | Effect of natural weathering on durability of pultruded glass fiber–reinforced bridge and building structures[END_REF]) in the sense that the structure is in a state of permanent bending under high stresses, which are characteristic in gridshell structures.

The gridshell structure was installed in Marne-la-Vallée in France (coordinates 48 • 50 31.6 N, 2 • 35 19.1 E), where detailed meteorological data exist and can be extracted from several databases. The overall dimensions of the gridshell were approximately 13m × 6m and it was made of long, rectilinear GFRP pultruded tubes of circular cross-section. The tubes were designed to deform elastically under large displacements and rotations [START_REF] Douthe | Gridshell structures with glass fibre reinforced polymer[END_REF]. Consequently, a wide variety of curvatures were developed due to the the three-dimensional (3D) shape of the gridshell and the imposed geometrical constraints (Figure 7). More details about the gridshell structure are given in Section 2.

Dismantling the gridshell structure after 8 years of service, gave us access to the time-dependent behaviour of its GFRP constituents and enabled us to study their strength reduction and stiffness change (creep/relaxation). It is worth noticing, that the available experimental data for the time-dependent behaviour of GFRP's is mainly based on accelerated creep tests of tubes of limited length.

More specifically, starting from the 1950s a large amount of experimental data exists regarding the creep and fatigue response of GFRP [see 8, and references therein]. As the duration of most of the existing laboratory experiments are limited in time, accelerated characterisation techniques are frequently used in order to calibrate linear and non-linear viscoelastic theories and predict the long-term performance of composites under sustained loads. The majority of these experimental studies on creep have been performed in the aerospace and naval fields, and less in Civil Engineering, where the use of GFRP's seems to be a promising material [START_REF] Clarke | Structural Design of Polymer Composites: Eurocomp Design Code and Background Document[END_REF][START_REF] Ascione | An experimental study on the long-term behavior of CFRP pultruded laminates suitable to concrete structures rehabilitation[END_REF][START_REF] Ascione | Prospect for New Guidance in the Design of FRP[END_REF].

Creep behaviour

Temperature and humidity are central factors for creep as they influence the viscosity of the GFRP matrix. Daniali [START_REF] Daniali | Short-term and long-term behavior of two types of reinforced plastic beams[END_REF] did an extensive parametric study of the short-term and long-term behaviour of GFRP T-shape beams varying the types of matrices (polyester/vinylester), the temperature (ambient temperature/54 • C) and the geometry of beams (solid/cavity web). The author showed that the beams with vinylester matrix present less strength decrease at high temperature, longer creep life and smaller deflection compared with those of polyester matrix. However all specimens fully recovered after the removal of the applied load and no permanent deformation or damage was reported. An additional study was later conducted by Abdel-Magid et al. [START_REF] Abdel-Magid | Flexure creep properties of e-glass reinforced polymers[END_REF] on GFRP composites, with polyurethane matrix (PU) and epoxy matrix (EP). They observed that the shear properties of the matrix and the interfacial adhesive force between the matrix and the fibres is critical for the creep-failure strength. These works, among others, corroborate the importance of humidity and especially of temperature on the creep behaviour of GFRP.

In parallel, analytical models were proposed by Bank and Mosallam [START_REF] Bank | Creep and failure of a full-size fiber-reinforced plastic pultruded frame[END_REF],

based on experiments on plane portal frame structures made of FRP pultruded profiles. Following Findley's power [START_REF] Findley | 26-year creep and recovery of poly(vinyl chloride) and polyethylene[END_REF] for the prediction of the viscoelastic behaviour, the time-dependent response of the FRP elements and the contribution of shear force were considered. They found a reduction of 35% for the long-term effective flexural modulus and a reduction of 46% for the long-term effective shear modulus. Regarding the short-term tests, the behaviour of the structure was linear-elastic under low load level, and the deflection was completely recoverable. However, when the load increased, the deformation was no longer recoverable, and non-linear effects were observed due to damage accumulation. These results were confirmed by Shao and Shanmugam [START_REF] Shao | Deflection creep of pultruded composite sheet piling[END_REF], who also used the simplified Findley power law to estimate the tensile and shear modulus in a Timoshenko beam model. They found that the values of the time exponent in Findley's model for tensile creep, shear creep and creep deflection were almost equal and that the average could be used as a constant for the material studied.

More recently Sá et al. [START_REF] Sa | Creep behavior of pultruded gfrp elements -part 1: Literature review and experimental study[END_REF][START_REF] Sa | Creep behavior of pultruded gfrp elements -part 2 analytical study[END_REF] compared empiric and phenomenological formulations and found that Findley's power law, Bruger-Kelvin model and Prony-Dirichlet series presented good accordance between them. They have showed that the elastic strains increased by 30% after 50000 hours under a load level of 20% of the ultimate stress. They showed also that the power law, combined with the Euler-Bernoulli beam theory, gave a accurate prediction of deflections under bending loads up to 40% of the ultimate strength (slightly higher than the recommended normal service conditions where the load level is one third of the ultimate strength).

Similar approaches were then used for the investigation of pultruded GFRP on flexural creep by Yang et al [START_REF] Yang | Flexural creep tests and long-term mechanical behavior offiber-reinforcedpolymeric composite tubes[END_REF], on creep buckling by Harries et al [START_REF] Harries | Creep and creep buckling of pultruded glass-reinforced polymermembers[END_REF] and even on creep flexural-torsional buckling by Boscato et al [START_REF] Boscato | Creep effects in pultruded frp beams[END_REF]. In the three cases, Findley power law models were calibrated on experimental tests and showed good agreement with experimental results.

Relaxation and recovery

The effect of hybridisation on the creep and the relaxation of pultruded composites (glass fibre and carbon fibre with epoxy matrix) was evaluated by Barpanda and Raju [START_REF] Barpanda | Effect of hybridization on the creep and stress relaxation characteristics of pultruded composites[END_REF]. The analysis of the data showed that the creep compliance and the relaxation modulus of flexure were independent of the type of fibres and the configuration of the composites. Zaoutsos et al. [START_REF] Zaoutsos | On the non-linear viscoelastic behaviour of polymer-matrix composites[END_REF] and Guedes et al. [START_REF] Guedes | Analytical and experimental evaluation of nonlinear viscoelastic-viscoplastic composite laminates under creep, creep-recovery, relaxation and ramp loading[END_REF] studied the modelling of the non-linear behaviour of composites, in tension and in bending, respectively. The formulation developed by Schapery was used in those researches and the established models were in good accordance with the test results. Then Al-Haik et al. [START_REF] Al-Haik | Viscoplastic analysis of structural polymer composites using stress relaxation and creep data[END_REF] investigated the mechanical behaviour of carbon/epoxy composites. They set up a model with the parameters obtained from the relaxation tests and used them for the prediction of creep behaviour.

The prediction was found consistent with the experimental data in short-term at low temperature.

More recently, advanced models tend to go beyond Findley power law and to differentiate the behaviour of the matrix and the fibres like in Cardoso and Harries [START_REF] Cardoso | A viscoelastic model for time-dependent behavior of pultruded gfrp[END_REF], which captures well both phenomenon creep and relaxation and seems a very promising direction for numerical modelling. Nedjar et al [START_REF] Nedjar | Modeling of unidirectional fibre-reinforced composites under fibre damage[END_REF] for example used a fibre breakage equivalent into a continuous damage model and observed that creep in fibre-reinforced composites is essentially due to the matrix constituent whose role is to deform and support stresses primarily in shear.

Environmental ageing

Nkurunziza et al. [START_REF] Nkurunziza | Effect of sustained load and environment on long-term tensile properties of glass fiber-reinforced polymer reinforcing bars[END_REF] studied the effects of sustained loads and environmental ageing on the tensile properties of GFRP reinforcement bars. The residual strength of the bars exposed to de-ionized water stayed almost the same. However, alkaline solution reduced the strength to 88% and 68% of the initial value under the stress level of 25% and 38% of the ultimate strength, respectively. On the contrary the modulus was not much influenced by the applied conditions.

Furthermore, an experimental study was realised by Sousa et al. [START_REF] Sousa | Durability of glass fibre reinforced polymer pultruded profiles comparison between quv accelerated exposure and natural weathering in a mediterranean climate[END_REF] in order to identify the durability of GFRP pultruded profiles comparing accelerated ageing and environmental ageing under Mediterranean climate. They observed that artificial ageing and environmental ageing could be correlated regarding their influence on aesthetic properties and also on mechanical properties, but in a less obvious manner. Carra and Carvelli [START_REF] Carra | Ageing of pultruded glass fibre reinforced polymer composites exposed to combined environmental agents[END_REF] carried on the comparison of the different matrices: isophthalic polyester, orthophthalic polyester and vinylester.

They confirmed that the effects of the artificial ageing are comparable with those of the environmental ageing but that it is hard to set up accurately a correlation between the two types of ageing.

Purpose and content of this work

According the previous literature review, there are few experiments combining creep or relaxation tests with environmental ageing and high bending stresses. From the separated effects, it seems that, if one follows the recommendations for civil engineering applications recommending stress levels below 30% of the limit strength, then long term deformation will stay recoverable and limited to approximately 40% of the instantaneous deformation. Moreover the effect of natural ageing will be limited. The purpose of the present study is thus to investigate the validity of these hypotheses and enable safe applications of GFRP composites in Civil Engineering.

The paper is structured as follows. Section 2 describes the original set up of the in-situ relaxation experiment. Three sets of tests are conducted. More specifically, recovery tests are presented in section 3, longitudinal bending tests in section 4 and transverse bending tests in section 5. A conclusion and a summary of the main results on the durability of structures made of pultruded tubes under pronounced bending closes the paper.

Methodology

Description of the prototype

The prototype of structure used in the present study builds on a structural concept of elastic gridshell magnified in 1975 by the work of Frei Otto et al.

for the Bundesgartenschau in Mannheim [START_REF] Burkhardt | Multihalle Mannheim, Erfahrung am Bauwerk[END_REF][START_REF] Happold | Timber lattice roof for the Mannheim Bundesgartenschau[END_REF]. This prototype is made of GFRP instead of wood and in that sense is the first GFRP elastic gridshell in the world. It had been presented in Douthe et al [START_REF] Douthe | Gridshell structures with glass fibre reinforced polymer[END_REF] (see Figure 1) where the advantages of GFRP compared to other Civil Engineering materials are detailed. It is made of a two way flat grid whose members are tubular pultruded profiles (with a 42mm outer diameter and a 35mm inner diameter) connected with swivel scaffolding elements that insures that the whole grid has no shear rigidity. Then as the tubes are very slender (the longest reach 13.2m), the grid can be deformed elastically by bending into a three dimensional shape (about 3.5m high for a 6.7m span). After deformation, this shape has been stabilised through the adding of a third direction of members that braces the grid and confers it its shell-like behaviour. The displacements occurring during the life time of the structure are thus approximately two orders of magnitude lower than those required for the forming of the grid. It can therefore be considered that this highly hyperstatic structure constitutes a unique but complex relaxation test on initially straight members that are forced to remain in given curved shapes. This assertion will be verified later in section 3.4 by simulating long term deformation of the shape according to the evaluated creep parameters.

Description of environmental conditions

The gridshell has been built in February 2006 and remained uncovered until March 2008 where a PVC textile cover was installed. The product used is a Serge Ferrari Precontraint 502 which has a 10% solar transmission and filters 100% of UVB. The gridshell was dismounted in September 2009: the cover was removed, then the third direction of beams, the hinges at supports were unfasten, while the grid was sustained in the air with a crane, and finally flatten (see Figure 2). In summary, during eight years, the gridshell members were subjected to:

• 24 months of permanent bending with UV and rain;

• 6 months of UV and rain without bending stresses;

• 64 months of permanent bending protected from UV and rain.

The gridshell shape was chosen, so that initial bending stresses are all below 35% of the limit strength of the members. Their evolution during time has not been monitored, but, from the visual aspect of the textile cover, where no wrinkling was observed, it can be considered that the overall form did not change significantly during the life-time of the structure (again this is assessed a posteriori in section 3.4).

Outside temperature and air humidity were not registered but can be considered as very similar to that of Paris Montsoury meteorological station which is about 15km away from the campus and which has all data in open-access on infoclimat.fr.

Identification of members curvature

Considering the elastic gridshell prototype as a relaxation test, it is of first importance to have access to the actual curvatures of members in the structure.

In [START_REF] Douthe | Gridshell structures with glass fibre reinforced polymer[END_REF], it had been shown that the numerical tool used for the simulation of the forming process was reliable. Reference values for "in-situ" curvature are thus taken from the numerical model of the structure. Figure 3 shows their distribution relatively to the maximum admissible curvature. Immediately after dismounting of the structure, the beams were laid on the ground. They were in this state considered to be stress free, except the friction forces developed between the beams and the ground (concrete), which were low. They were not straight anymore which evidenced stress relaxation in the structure during the life-time of the building. Their arched geometry was thus measured through a distance to the chord between the two ends. The precision of the measurement were about ±2mm which is acceptable considering the fact that, for a 13m beam, the residual deflection was about 200mm at mid-span.

For estimating the residual curvature, this precision is however not sufficient.

It was thus decided to extrapolate the measurements with a 6 th degree polynomial (with forced null values at the beams ends: P (0) = P (L) = 0). It can be seen on Figure 4 that the interpolation is very close from the measurements (the distance between the two curves is always below 1mm). As a result, the residual curvature can be calculated at any point with a good reliability using the approximated polynomial of each deformed shape of beam. Obviously the proposed method with the beams laying on the ground cannot capture any out-of-plane residual curvature. Yet the analysis of the 3D model of the gridshell structure reveals that the curves described by the beams centrelines are 3D-curves which are very close to planar curves. Indeed the curves are smooth, slowly varying and the maximum distance to the best fitting plane is generally below 10cm (for 13m beams with a height of around 3m). The out-of-plane curvature is hence generally lower than 2% of the total curvature, especially in the areas where curvature is maximum (see Figure 5). Therefore, the reference "in-situ" curvatures of the beams will be those of the 3D curves.

Note that these curvatures are estimated from the numerical model and that their approximations are based on the inverse of the radius of the circumscribed circle of three successive nodes of the beam [START_REF] Douthe | Form-finding of a grid shell in composite materials[END_REF].

Experimental campaign

Hence, for this in-situ experiment on long-term behaviour of GFRP pipes under permanent bending, the raw material is a set of tubes whose instantaneous characteristics had been characterised at time T 0 in [START_REF] Douthe | Gridshell structures with glass fibre reinforced polymer[END_REF] and which were subjected to a long term relaxation test. A wide variety of curvatures is explored thanks to the complexity of the shape of the elastic gridshell (see figure 3). The objectives of the following experimental campaign are:

• to determine the amplitude and nature (reversible or not) of the residual deformations of the members and to relate them with the "in-situ" curvatures;

• to determine whether mechanical properties (longitudinal and transverse) have been influenced by ageing and the level of "in-situ" curvature.

To this end, residual curvatures have been measured twice (first on the dismounting day and then five months later) following the procedure described in section 2.3. The evolution of these residual curvatures is compared to reference creep tests in section 3. Afterwards, the members are sorted according to the level of curvature they were subjected to and submitted to bending tests in the longitudinal direction in section 4 and in the transverse direction in section 5.

Both residual stiffness and strength are investigated and compared to the initial characteristics of the material.

Recovery tests

Reference test

In a creep/recovery or relaxation/recovery test, when the load is removed, there is an immediate recovery of elastic deformation, followed by some gradual recovery of creep deformation. Depending on the nature of creep (elastic or plastic), the long term recovery can be partial or total if the material was not damaged. In the present case, no continuous recording was available for the follow-up of residual deformation of the members and the determination of the nature of creep. It was thus decided to estimate the time-dependent recoverable deformation from a reference test made on the same material: same producer, same formulation with polyester resin and almost the same fibre content (≈ 70% of fibre) but different cross-section. The reference test is made on a pultruded rectangle 40x16mm while the members were made of circular hollow section. This reference creep test is actually a test conducted in the laboratory on a device dedicated to creep test under four points bending, the same as the one depicted in [START_REF] Douthe | Gridshell structures with glass fibre reinforced polymer[END_REF]. The 1m long specimen is loaded during a period of one month with a maximal moment equal to 30% of the specimen strength, which corresponds to a stress level similar to that of the gridshell members. Creep deformations are measured in the central part of the specimen which is under pure bending (the instantaneous elastic part is not shown.) The measured creep deflections are presented in Figure 6.

Findley's power law is then used to estimate the deformation after five years.

The time dependant deformation are hence looked for in the following form: m • t n , where t stands for time in hour and where n and m are two material

Residual curvature at day 1

After the dismounting of the structure, the 34 members were measured one by one in at least ten points per length to estimate the residual curvature through a sixth order polynomial as explained in section 2.3. Then, the "insitu" curvatures are calculated at characteristic points of the numerical model.

From the curvilinear abscissas of those points and the polynomial model, the residual curvature at corresponding points is then calculated. The scatter plot of residual curvature relatively to "in-situ" curvature is shown in Figure 7. In a first approximation, one could state that the residual curvature represents about 6.3% of the "in-situ" curvature. However, considering the fact that the relation between the two curvatures seems not strictly linear, one may try to find an empiric relation between them. To this end, Findley's power law is used here, as it is generally well adapted to predict the creep deformation of composites. For a given time, the deformation can be expressed in the form of a hyperbolic sine function of the imposed stress. Inspired by that, a hyperbolic sine model passing through the origin is fitted to the data of Figure 7 with a relatively satisfying coefficient of determination (R 2 = 0.89).

Estimation of recovery characteristics

On June 12th, 2014, 5 months after the dismantling of the structure, the pipes were thus measured again following the same method as at day 1. The results are shown in Figure 8, where obviously the scatter is higher than at day 1 because of the relative size of uncertainties. The residual curvature represents now about 4.0% of the "in-situ" curvature. Compared with the results at day 1, the relative decrease of residual curvature is equal to 37% of the curvature at day 1. Trying then to fit a hyperbolic sine function with the same argument as the function at day 1 (κ residual = C 5months sinh(8.3κ in-situ )), it can be seen that the amplitude of residual deformation passes from 0.0057 to 0.0032 which correspond to a reduction of 44%. Both estimations are close enough to the value obtained in the reference test (42%). It can thus be concluded that, probably, the creep deformations in the prototype will be totally recovered and that the permanent curvature and ageing have no major influence on the material behaviour, as will be confirmed in the following.

Discussion on the long term stability of the shape

The preceding measurements allow for the estimation of the long term behaviour of the members used in the prototype. Indeed, dependency in the curvature level was characterised in previous section 3.2 as a hyperbolic sine function of the curvature ( sinh(8.3κ)). Time dependency was also characterised in section 3.1 at 30% of the limit curvature as a Findley power law with characteristic values m ≈ 0, 026 and n ≈ 0, 24. Combining both and following the engineering approach of Sà et al [START_REF] Sa | Creep behavior of pultruded gfrp elements -part 2 analytical study[END_REF], one can obtain an estimate of the variation of the longitudinal characteristics with time:

E(t) = E 0 1 + 0.026 sinh(8.3κ) sinh (8.3κ 30% ) • t 0.24 -1 (1)
Replacing the linear elastic material law by this roughly estimated material law in the software used for the simulation of the prototype, one is able to get the evolution of the shape and of the stresses through time. Figure 9 shows hence the displacements induced by creep deformations after 10 years. Maximum displacements are located around the opening which is the known weakest point of the structure, they remain however very limited (< 15mm) and it can thus safely be concluded that the curvatures in the members did not change during the life time of the prototype. Interestingly, if one then compares the bending moments in the structure at the time of completion (see Figure 10a) and after 10 years (see Figure 10b), one notices that not only the maximum value drops of about 30% but also that, due to the stress dependency of the material law, the distribution of bending moments has become more uniform. From the structural designer point of view, the stress relaxation in the present case is hence in favour of safety, provided that it is not associated with a strength diminution, which is exactly investigated in the following.

Longitudinal characteristics

Experimental set-up

The tests on the long term longitudinal characteristics were realised in the laboratory facilities on an MTS testing machine with a capacity of 100 kN.

They consist in four points bending tests in which a frame made of a double HEB140 steel profiles is fixed on the machine platform to adjust the span between the supports. Isostatic conditions are realised for the support and the loading element through steel cylinders with a diameter of 30 mm. The whole experimental set-up is illustrated in Figure 11. The length of the specimen is 1000 mm with an effective span of the specimens of 840 mm which corresponds to 20 times the diameter of the pipes. This ratio was considered sufficient for the contribution of shear stresses to the total deformation to be neglected (in the spirit of ISO standard for flexural properties of composites [START_REF]Fiber-reinforced plastic composite -Determination of flexural properties[END_REF]).

Two series of tests are conducted: one for the identification of the elastic properties where the load is kept relatively low and one for the identification of the specimen ultimate strength. During the first series for the flexural modulus characterisation, the specimen is in direct contact with the supports and the loading elements, so that for the displacement measure no bias is introduced from the elasticity of the supports. The specimens are loaded up to 2 kN, which corresponds to approximately 20% of the estimated failure load and then unloaded. The load path was completely reversible.

On the contrary, during the tests for the ultimate strength characterisation, two layers of silicone were placed between the specimen and the supports as well as between the specimen and the loading elements to prevent local crushing (see Figure 12). In order to further prevent such failure, short steel pipes were also

Measuring devices

Before the tests, the dimensions (diameter and thickness) of every specimen were measured three times with the help of a vernier caliper. The averages of the three measurements were used as the final dimensions.

For the determination of the longitudinal bending modulus, displacement measurements are based on four linear variable differential transformers (LVDT), corresponding to the maximum number of channels available on the machine.

For the ultimate strength, only the displacement of the machine is recorded.

Two settings of LVDTs are used. In the first one, there is one LVDT below the specimen at mid-span, one below the loading frame in the South and two below the loading frame in the North in order to identify rigid body motion of the loading frame (see Figure 13). In the second setting, only 3 LVDTs are set, all below the specimen, one at mid-span and the other two at locations corresponding to the loading frame (see Figure 14). In this way, the displacements are measured on the lower fibre, the relative displacement between sides f load and mid-span f max is not biased by the ovalisation of the cross-section (contrary to the first setting). The longitudinal flexural modulus E L is then easily calculated by standard Euler beam theory from the applied load P , the length l of the specimen and cross section inertia

I: E L = 3P l 3 648(f max -f load )I ( 2
)

Sorting of specimens

The proper identification of longitudinal bending properties required a specimen length of 1m. Yet the lengths of the pipes in the gridshell range from 4m to 13m. It is thus necessary to define a methodology to cut off the specimens from the whole members, so that the "in-situ" curvature of the specimen is as homogeneous as possible. We hence start identifying on the polynomial approximated model of the "in-situ" members where the point with the maximum curvature is located. The tubes are then cut symmetrically around that position into pieces of 1m with the help of a hacksaw.

Finally, for each resulting 1m specimen, the average curvature and the curvatures at both ends are evaluated. The comparison of this average value with the extreme values gives a measure of the curvature homogeneity along the insitu specimens. It allows for a sorting of specimens into 4 classes according to the degree of variation of curvature:

1. The difference between the average curvature and the curvature at both ends does not exceed 10%.

2. The difference between the average curvature and the curvature at one end is comprised between 10% and 20%, while the difference between the average curvature and the curvature at the other end does not exceed 20%.

3. The difference between the average curvature and the curvature at one end is comprised between 10% and 20%, while the difference between the average curvature and the curvature at the other end does not exceed 30%.

4. The difference between the average curvature and the curvature at one end exceed 30%.

The specimens are finally sorted in groups according to their average curvature or equivalently to their in-situ permanent stress relatively to the initial ultimate stress. The number of specimens of each class in each group is shown in Table 1. In each group, 8 specimens are selected in the highest possible class, 5 for the identification of the residual longitudinal flexural modulus, 3 for identification of the ultimate strength. These specimens are selected, so that their average curvature is as close as possible (less than 1% of the ultimate strength)

from six target values: 0%, 15%, 20%, 25%, 30% and 35% of the ultimate strength.

Residual longitudinal bending stiffness

The complete campaign was conducted with the two LVDT settings which gave very similar results, with about 2% difference for each group which is about the uncertainty of the measure. Both settings can be considered to have the same accuracy, although the standard deviation of measurements with the second one is more homogeneous (close to 2%). The second setting avoids the bias of tube ovalisation, it is therefore chosen for the rest of the study. The average longitudinal flexural moduli calculated with the second setting (Figure 14) of each group of specimens is shown in Figure 15. Looking into the results of Figure 15, it is observed that the residual longitudinal bending modulus decreases with the permanent stress level until 30% of the ultimate strength (about 4%). This conclusion is similar to that of Khan and Kim [START_REF] Khan | Time-dependent and residual behavior of pultruded gfrp beams subjected to sustained intensities and cold temperature[END_REF]. However, the average modulus of the last group, the one with a highest stress is almost equal to that of the reference specimen (the 0% group).

A quick ANOVA on the whole set of data confirms that, from a statistical point of view, a few percent decrease is probable with the relative level of prestress defined by the ratio initial stress/strength.

The average modulus of each group is then compared with the initial value before ageing [START_REF] Douthe | Etude de structures élancées précontraintes en matériaux composites, application à la comception des gridshells[END_REF] (values presented here are slightly different from those of the initial publication as they have been corrected taken into account the actual characteristics of the cross sections and not those of the manufacturer). These values are summarised in Table 2. There is thus no effect of ageing (at that time scale and in those use conditions) and, statistically no significant effect of permanent stresses under those levels of bending stresses on the longitudinal stiffness.

Failure modes and the ultimate bending strength

During the ultimate strength tests, micro-cracking can be heard from a load of 2 kN (corresponding to about 20% of the limit strength), and the intensity of the sound increases with the load level, similarly to the large scale bending test on the initial material [START_REF] Douthe | Etude de structures élancées précontraintes en matériaux composites, application à la comception des gridshells[END_REF]. Since two layers of silicone are placed between the specimen and the supports as well as between the specimen and the loading elements, it can reasonably be assumed that the cracks are due to the damage by bending of the material. As the failure load approached, the sound emissions become louder, which may be due to the local crushing or the generation and the propagation of the micro-cracks. A typical load-displacement curve is shown in Figure 16. As expected, the behaviour is linear elastic with a brittle fracture.

The non-linear part at the beginning is due to the deformation of the silicone. When the failure load is reached, the stress state in the specimen is very complex, especially due to its tubular nature: longitudinal stresses due to bending moments (Figure 17b), transverse compression at supports and loads due to shear forces (Figure 17a) and transverse stresses due to ovalisation (Figure 17c).

Among the 18 specimens, three failure modes are observed: crushing at loading points (10 specimens), failure at mid-span (5 specimens) and longitudinal split-ting (3 specimens). An illustration of each failure mode is shown in Figure 17. The average failure loads of each group of specimens and of each failure mode are presented in Figures 18 and19. Figure 18 shows that there is no obvious relation between the failure load and the "in-situ" stress level, especially because the scattering of values increases with the stress level. This scattering has also to be related to the importance of bifurcation phenomena in crack propagation and failure of structural systems where small different imperfections might lead to very different failure modes. To complete the analysis of longitudinal properties, like for the stiffness, the ultimate longitudinal stresses of the aged specimens are compared to the initial ultimate stress [START_REF] Douthe | Etude de structures élancées précontraintes en matériaux composites, application à la comception des gridshells[END_REF]. Table 3 summarises the results of the comparison.

It can be seen that the ultimate strength of the unloaded members are similar: in average, ageing has no influence on the tube capacity. Some correlation is however observed between a loss of capacity and permanent bending stresses (at most 8.8%). This is in good agreement with previous observations and the fact that micro-cracking could be heard during the tests starting from 2 kN corresponding to 20% of the ultimate strength.

Transverse characteristics

In order to understand the failure mechanism of the gridshell members, it is necessary to investigate their transverse characteristics. The stiffness and strength in the transverse direction are thus studied in the following.

Experimental set-up

Like longitudinal characteristics, transverse characteristics are investigated through bending tests at structure scale, through radial compression of cross sections of tubes as shown in Figure 20. This experimental setting is similar in proportion and procedure to the ones used by Gupta and Abbas [START_REF] Gupta | Lateral collapse of composite cylindrical tubes between flat platens[END_REF] and Li et al [START_REF] Li | On lateral compression of circular aluminum, cfrp and gfrp tubes[END_REF], who have proved reliable in the identification of crushing properties of GFRP.

The same MTS testing machine as in section 4 is used, but this time with a load cell of 10 kN capacity. The alignment of the loading plates is guaranteed by the manufacturer with an unknown precision and there is no possibility for adjustment. The tests are displacement driven with a speed of 2 mm/min. All the specimens are loaded until rupture. All the tests are carried out at ambient temperature ( 26 ± 1 • C). Notice that all the specimen are stored in the laboratory for more than 48 hours before the test. Notice also that the displacement measure is not "pure" and blends deformation of the experimental set up with deformation of the specimen (which are of a higher order of magnitude). It may however be supposed that the experimental set up displacements are independent of the specimen and time. Therefore, the results of the investigations on transverse characteristics presented here should be understood as relative rather than absolute.

Measurement devices

Before the tests, the dimensions (diameter and thickness) of every specimens are measured three times with the help of a vernier caliper. The averages of the three measurements are used as the final dimensions. During the tests, the total force P and the cross-head displacement δ are recorded (see Figure 21). Considering then the slenderness of the cylinder walls compared to the cylinder radius, the specimen can be considered as a circular beam in bending (stresses induced by normal and shear forces are negligible). Simple calculations of strength of materials allow for the evaluation of the transverse modulus E T by [START_REF] Douthe | Etude de structures élancées précontraintes en matériaux composites, application à la comception des gridshells[END_REF]:

E T = P δ 3R 3 t 3 L π - 8 π ( 3
)
where t is the thickness of the tube, L is its length and R the average radius.

The maximum transverse bending stress σ rupt T is then obtained at the crown where the load is applied (point A in Figure 21) from the maximal load P rupt by:

σ rupt T = 6 π P rupt R t 2 L (4)
The stress on the edge of the tube (point B in Figure 21) is significantly lower, about 43% and given by:

σ B T = 6P rupt R t 2 L 1 2 - 1 π (5)

Specimen

Following the sorting of the 1m specimens exposed in section 4.3, one specimen in each group is chosen with an in-situ curvature of 0%, 15%, 20%, 25%, 30% and 35% of the ultimate strength. These specimen are then cut into small cylinders with two lengths, 20 mm and 40 mm (3 specimens each), as in [START_REF] Douthe | Etude de structures élancées précontraintes en matériaux composites, application à la comception des gridshells[END_REF].

The idea is to find a compromise between the probability of damaging the specimen during cutting (which decreases with the length) and the influence of the non-parallelism of the loading plates (which increases with the length). From the campaign conducted here, it seems that the 40 mm specimens are more reliable as their average strength is 15% higher that that of 20 mm specimen. Both coefficients of variation are however equally high (about 30% of the average value).

Residual transverse flexural modulus

The average residual transverse flexural modulus of each group of specimens is shown in Figure 22, together with their standard deviation. No obvious relation can be found between the level of in-situ stress and the residual transverse modulus, and from a statistical point of view, considering standard deviation, one could conclude that there is no influence of permanent stresses. Comparing then these values with the initial values [START_REF] Douthe | Etude de structures élancées précontraintes en matériaux composites, application à la comception des gridshells[END_REF] (see Table 4), one can see that in average, ageing has no influence on the transverse stiffness and that there is no coupling between ageing and permanent bending stress (at least for the considered level of stresses).

Residual transverse flexural strength

A typical force-cross-head displacement curve is shown in Figure 23. It can be seen from Equation ( 4) and ( 5) that the moment at the top/bottom of the specimen is superior to that at the two lateral points. Failure therefore starts from the top/bottom of the specimen, but after the first hinge is formed, the section is still stable (with an hinge at point A, the structure in Figure 21 becomes isostatic) and it can resist additional load until new hinges form on the side. This residual capacity of the cross-section is clearly visible in Figure 23:

the first drop of the curve corresponds to the rupture at the top/bottom of the specimen, it is accompanied by important sound, but the force can continue to increase until the failure of the lateral points. These results are in agreement with existing literature [START_REF] Gupta | Lateral collapse of composite cylindrical tubes between flat platens[END_REF][START_REF] Li | On lateral compression of circular aluminum, cfrp and gfrp tubes[END_REF]. for each group. The corresponding initial values [START_REF] Douthe | Etude de structures élancées précontraintes en matériaux composites, application à la comception des gridshells[END_REF] on the cross section before ageing have also been added for comparison. The spread of residual strength is much higher than that of residual stiffness, but so was also the spread of initial strength compared to initial stiffness. However for each criterion (mean, minimum and maximum, deviation), initial and residual values are comparable: there is thus no significant influence of ageing or permanent bending on the transverse bending strength. [MPa] 6.9 5.1 3.9 1.9 8.1 5.9 6.7 10.5

6. Conclusions and perspectives

Summary of results

In this research, the long-term behaviour of GFRP pipes was studied based on an original in-situ experiment provided by the construction of the first GFRP elastic gridshell in the world. An empiric relation was established between the initially imposed curvature and the residual one. The specimens were then tested to obtain the residual mechanical properties of the material. By comparing the residual mechanical properties with the initial values, the effects of environmental ageing combined with permanent bending stress were evaluated.

The main conclusions of this experimental campaign are given below.

• There has been a recovery of approximately 40% in the first 5 months, which indicates that, for the stress level tested in the prototype, deformations "remain" elastic and are fully recoverable.

• The residual modulus is always inferior to the initial value, with a difference ranging up to 4%. There is a little reduction of the longitudinal flexural modulus due to the environmental ageing combined with permanent bending stress, but it can be neglected considering the coefficient of safety during the design of the structure. It is however hard to establish a relation between the reduction of the longitudinal flexural modulus and the level of prestresss.

• No significant influence of ageing on the residual longitudinal strength was observed with a decrease of at most 8%.

• Similar observations were made for the transverse properties.

Perspectives for GFRP applications in Civil Engineering

This study confirms thus the feasibility and durability of GFRP structures for practical civil engineering applications, even when submitted to permanent bending stresses. Indeed, provided that the stress level remains below 30% of the ultimate strength, deformations remain recoverable and no significant damage occur. One may even say that in the specific case of elastic gridshells, relaxation is advantageous as part of the permanent stresses are released during the lifetime of the structure without stiffness or strength capacity loss.

Figure 1 :

 1 Figure 1: View of the uncovered elastic gridshell.

  The grid was then stored flat outdoor for six months and exposed to UV and rain . The gridshell was then rebuilt in March 2010 and covered again. It was definitively dismounted following the former procedure in January 2014 at the beginning of the present experimental campaign.

Figure 2 :

 2 Figure 2: Dismounting and unfolding of the structure.

Figure 3 :

 3 Figure 3: Initial ratio of curvatures in the prototype relatively to the maximum admissible curvature.

Figure 4 :

 4 Figure 4: Measured residual heights and extrapolated model for beam n. 8.

Figure 5 :

 5 Figure 5: Comparison between 3D curvature and 2D curvature on the best fitting plane for beam n. 8.

Figure 6 :Figure 6

 66 Figure 6: Reference creep test and extrapolation with Findley's power law supposing total recovery.

Figure 7 :

 7 Figure 7: Residual curvature versus "in-situ" curvature at day 1.

Figure 8 :

 8 Figure 8: Residual curvature versus "in-situ" curvature after five months.

Figure 9 :

 9 Figure 9: Displacement in mm induced by creep deformation under self stress after 10 years.

Figure 10 :

 10 Figure 10: Evolution of the bending moments [Nm] in the prototype with time.

Figure 11 :

 11 Figure 11: Four point bending test: scheme of experimental set-up.

Figure 12 :

 12 Figure 12: Disposals to prevent local fracture: silicone layer and inner cylinder for supports.

Figure 13 :

 13 Figure 13: First setting of LVDTs on the four points bending tests.

Figure 14 :

 14 Figure 14: Second setting of LVDTs on the four points bending tests.

Figure 15 :

 15 Figure 15: Residual longitudinal bending stiffness.

Figure 16 :

 16 Figure 16: Typical brittle load-displacement curve of bending tests.

Figure 17 :

 17 Figure 17: Three types of failure modes in bending.

Figure 18 :

 18 Figure 18: Failure load depending on the level of permanent stress.

Figure 19 :

 19 Figure 19: Failure load depending on the failure mode.

Figure 20 :

 20 Figure 20: Experimental set-up for transverse compression tests.

Figure 21 :

 21 Figure 21: Scheme for the calculation of transverse characteristics.

Figure 22 :

 22 Figure 22: Residual Modulus in the transverse direction depending on the level of permanent stress.

Figure 23 :

 23 Figure 23: Load path of cross-section under transverse bending (Only the response of the sample for a force higher than 50N is reported, due to high tolerances of the apparatus and of the acquisition system in the beginning of the measurements.)

Table 1 :

 1 Inventory of specimens according to curvature level and homogeneity.

	Class	5% 5-10% 10-15% 15-20% 20-25% 25-30% 30-35%	35%
	1	0	0	0	2	15	61	56	3
	2	0	1	3	25	20	14	2	0
	3	0	2	13	18	5	0	0	0
	4	25	34	17	4	1	0	0	0

Table 2 :

 2 Effect of ageing and permanent bending on the longitudinal bending stiffness.

	Percentage of ult. strength (%)	0	15	20	25	30	35
	Residual long. modulus (GPa) 28.09 27.77 27.63 27.32 27.04 28.01
	Initial long. modulus (GPa)	28.11
	Difference (%)	0	-1.2	-1.7	-2.8	-3.8	-0.4

Table 3 :

 3 Effect of ageing and permanent bending on the longitudinal ultimate stress.

	Percentage of ult. strength (%)	0	0.15 0.2 0.25 0.3 0.35
	Residual ultimate stress (MPa) 349 351 329 359 322 319
	Initial ultimate stress (MPa)	350
	Difference (%)	-0.4 0.1 -6.0 2.6 -8.1 -8.8

Table 4 :

 4 Effect of ageing and permanent bending on the transverse modulus.

	Group	0	0.15 0.2 0.25 0.3 0.35
	Residual trans. modulus (GPa) 5.89 6.81 6.63 5.89 5.83 6.01
	Initial value (GPa)	6.05
	Difference (%)	-3.4 11.6 8.8	-3.4 -4.5 -1.4

Table 5 :

 5 Effect of ageing and permanent bending on the transverse modulus.

	Group	Init	0% 10% 15% 20% 25% 30% 35%
	σ mean T	[MPa] 16.1 15.3 22.2 26.6 18.4 17.9 16.9 18.2
	σ min T	[MPa]	8.8	9.5	15.9 25.2	9.8	8.4	6.5	8.2
	σ max T	[MPa] 28.7 18.5 26.6 30.2 31.8 24.1 25.4 29.4
	σ stdv T

Acknowledgement

The authors would like to thank the master student Mr Geng Shuai for performing most of the experiments and his work on this topic.