N Bioret

G Moreau

M Servières
email: myriam.servieres@ec-nantes.fr

Urban Localization based on Correspondences between Street Photographs and 2D Building GIS Layer

Keywords: Augmented Reality, Geographic Information Systems, Pose computation, Vanishing points, Correspondences between image and model, 3D Model

We propose a localization approach in urban environments based on correspondences between street photographs and a 2D GIS. Assumption of vertical facades and use of vanishing points theory allows extracting a partial geometry from the images : angles and width ratios between facades. This information is then transformed into GIS queries, which result in a layer of every possible pose for the camera. Wrong poses are then discarded based on consistency criteria, using road layer information or building layer information that has not been used before. Eventually, combination of possible poses coming from several images of the same site result in a unique pose, the true one. We show results of our experiments based on real data of Nantes city, France.

Introduction

Onsite GIS applications meet an increasing demand in urban environments, for retrieving, creating or updating geographic information. Such applications always require a precise localization of the user. While geographic information is usually in 2 dimensions, 3D building models become available as well as facade textures and also street pictures if they can be registered in the GIS frame. Need for accurate and reliable 3D data becomes serious in the field of urban planning in a context of sustainable development. Moreover, tools for 3D virtual exploration of cities grow popular [START_REF] Snavely | Photo tourism : exploring photo collections in 3d[END_REF], but they are limited by 3D data low accuracy and coverage. Interaction with 3D data brings new challenges, to which Augmented Reality seems to be a promising solution. Geographic data is overlaid in real time on pictures of a scene, which demands a real time registration of a camera frame in a GIS frame, involving both 3D localization and orientation of a camera. In our approach, such registration is obtained by matching photographs taken in streets with geographic data. Since we work with a 2D building layer and a 2D road layer, we achieve a 2D localization and 3D orientation, assuming facades are vertical. As a feedback loop, our GIS-connected Augmented Reality application will insert new 3D data in the GIS, for instance facade textures and the associated semantic information. This article is structured as follows : part 2 presents a state of the art about outdoor localization based on computer vision. Part 3 explains how image data is matched with GIS data. Part 4 presents the results of our experiments and part 5 discusses the possible improvements and future work.

Existing work

Outdoor localization based on computer vision typically relies on a model on the viewed scenes. However, some applications get rid of any model and build a cloud of 3D points modeling the scene, but the registration of such a local model is still required [START_REF] Schindler | Towards featurebased building reconstruction from images[END_REF][START_REF] Snavely | Photo tourism : exploring photo collections in 3d[END_REF][START_REF] Bartczak | Extraction of 3D freeform surfaces as visual landmarks for real-time tracking[END_REF][START_REF] Wang | Simultaneous localization, mapping and moving object tracking[END_REF][START_REF] Moslah | Fitting uncalibrated photographs to geo-referenced urban 3d models[END_REF]. The first type of model data used for outdoor localization is street images, which have previously been registered in the GIS frame. Such approaches rely on image matching and triangulation as presented in [START_REF] Zhang | Image based localization in urban environments[END_REF]. The size of an image database can get huge as its coverage grows, leading to non real time computation. The first way traditionally explored to overcome that limitation is reduction of computation time through an optimized indexing algorithm [START_REF] Shao | Hpat indexing for fast object/scene recognition based on local appearance[END_REF][START_REF] Goedeme | Fast wide baseline matching for visual navigation[END_REF][START_REF] Fritz | Urban object recognition from informative local features[END_REF][START_REF] Zhang | Image based localization in urban environments[END_REF][START_REF] Steinhoff | How computer vision can help in outdoor positionning[END_REF]. The second way is reducing the database size with simplified textures [START_REF] Johansson | A system for automatic pose-estimation from a single image in a city scene[END_REF], spherical images [START_REF] Jong Weon Lee | Tracking with omni-directional vision for outdoor ar systems[END_REF] or region of interests rather than complete images [START_REF] Robertson | An imagebased system for urban navigation[END_REF]. Vector data, for instance 3D wireframe building models, are also used. Virtual tracking provides an accurate outdoor localization [START_REF] Behringer | Model-based visual tracking for outdoor augmented reality applications[END_REF][START_REF] Jiang | A robust hybrid tracking system for outdoor augmented reality[END_REF][START_REF] Ribo | Hybrid tracking for outdoor augmented reality applications[END_REF] but initialization of the camera pose is still a challenging issue. Posit algorithm [START_REF] David | Softposit : Simultaneous pose and correspondence determination[END_REF] aims at establishing automatic correspondences between images and 3D model, but no result have been shown in an outdoor environment. Use of GPS is interesting [START_REF] Haala | A multi-sensor system for positioning in urban environments[END_REF][START_REF] Sourimant | Gps, gis and video fusion for urban modeling[END_REF] but the accuracy in dense urban environment may not be enough for pose initialization. Most of the approaches mentioned above consider a 3D model of only 1 building and there-fore, research of the right building in a GIS is not considered. [START_REF] Robertson | Building architectural models from many views using map constraints[END_REF] uses a model made of a 2D map, containing several buildings, but correspondences between images and 2D model are found manually, which implies the user knows how to establish such correspondences. A third type of model, combination of vector data and textures, brings a better robustness and accuracy. Initialization of the camera pose is still required, and differential GPS is often considered [START_REF] Reitmayr | Going out : Robust model-based tracking for outdoor augmented reality[END_REF][START_REF] Reitmayr | Initialisation for visual tracking in urban environments[END_REF][START_REF] Cappelle | Localisation in urban environment using GPS and INS aided by monocular vision system and 3D geographical model[END_REF][START_REF] Volker Coors | Matching buildings : Pose estimation in an urban environment[END_REF] in association to a sweeping algorithm. However, this kind of approach faces the 2 main limitations we find in approaches based only on image data or only on vector data. On the one hand, creation of an image database with large coverage and accurate registration in big cities seems currently out of reach. On the other hand, no satisfying method has been found for pose initialization using a model containing several buildings.

3 Matching GIS and images

Geometry extraction from images

Our approach is based on facade extraction from the images, based on the assumption that facades are vertical. Vanishing points associated to the facades can be extracted using for instance [START_REF] Kalantari | Détection automatique des points de fuite et calcul de leur incertitude à l'aide de la géometrie projective[END_REF] or [START_REF] Kosecka | Video compass[END_REF]. The vertical vanishing point is the same for every facade, and each facade is associated to 1 horizontal vanishing point. Based on our previous work [START_REF] Bioret | Localization approach based on image/GIS correspondence using a simple 2d building layer[END_REF], we can extract the width ratio between 2 neighbor facades and the angle between 2 facades even if these are not neighbor facades. Each image measurement is associated to a measurement uncertainty, which we propagate to the angles and width ratios measurements. However, it is not directly possible to extract length information from the image, because of the unknown scale factor.

Segments queries in the GIS

Then this information is transformed into GIS queries. Our GIS contains a 2D building layer, which is made of 2D polygons representing building footprints, and a 2D road layer made of polylines representing the median axis of the roads. For 2 neighbor facades F 1 and F 2 , we extract an angle α and a width ratio r, associated with uncertainty measurements. Those uncertainties are given as standard deviations and are then transformed into tolerances δ α and δ r . Then, we query in the GIS every couple of segments (s 1 , s 2 ) such that : -s 1 and s 2 share a common extremity, -s 1 and s 2 are separated by an angle belonging to [αδ α , α + δ α ] -s 1 and s 2 are related by a length ratio belonging to [rδ r , r + δ r ] Figure 1 details how these queries work. We have then a set of every possible couple of segments in the GIS building layer, with correspondences between s i segments and F i facades. Therefore we can compute a set of every possible pose. This number of possible poses can be seriously reduced using consistency criteria. If more complex struc-tures are identified in the images such as collections of n neighbor facades, we can use more complex segments queries, which will result in fewer possible poses than with only 2 neighbor facades.

Figure 1 -Segments queries in the GIS building layer

Consistency queries in a GIS

First of all, if several unrelated angles and ratios are computed from an image, several sets of possible poses are generated and are then merged together. During the merging of n poses layers, we can remove every pose which do not belong to a group of n poses. In this query, as in every other consistency query, we use tolerances which prevent from deleting the true pose.

Secondly, we can tell from the image if there is a road between the pose and the viewed facades. For each computed pose and using GIS road layer, we can obtain the same information. If there is a contradiction between GIS and image measurements, the pose is discarded. Similarly, we can compute for each pose and using the building layer if there is an obstruction by a building located between the pose and the viewed facades. If such a situation occurs, the pose is discarded, which allows us to discard any pose located inside of a building. Figure 2 gives an obstruction example. Thirdly, more precise measurements can be made in the image, like azimuth measurements. For each possible pose, a viewing polygon is computed, defined as follows : this includes every point in the GIS that can be viewed from the pose, in a particular angular domain and within a given range distance (in our experiments this range distance was about 100 m, which corresponds to the limit of the image resolution to reliably distinguish vertical facades). This viewing polygon contains every azimuth angle of viewed vertical edges and every orientation of vertical facades, which can also be computed from the images. For every inconsistency between GIS and image measurements, the pose is discarded. Definition of tolerances for such a query is however difficult because it must take into account GIS modeling errors. Figure 3 details how these queries work.

Queries using several images

While consistency queries aim at cutting the number of possible poses down to 1, it is not always possible to achieve this goal only using 1 image. We have to combine information coming from different pictures taken at the same location, with varying orientations. One set of possible poses is generated for each image, and are merged using a distance criterion. For n images, we should observe groups of n poses located within a tolerance distance R, and every pose which do not meet this assumption are discarded.

Experimental results

Our experiments were based in real data from the city of Nantes, France. Our GIS data can be viewed on http:// www.geonantes.fr, and initially contains 7138 buildings, which corresponds to a 3km 2 area. We can remove from this GIS every segment that cannot be viewed from a street point of view, and as a result, we query a transformed building layer containing 2715 polygons. Moreover, we have taken a set of 28 street photographs, distributed on 7 different sites. On each site, the different poses are located within a circle of radius 10 m. Those photographs were taken with the same focal length, and therefore we consider that the internal camera parameters are known. Radial distortions in the image have been corrected based on OpenCV library. In our approach, vanishing points are extracted automatically but recognition of the vertical edges limiting facades, and definition of neighborhood relationship between 2 facades is done manually. One can notice that [START_REF] Micusik | Detection and matching of rectilinear structures[END_REF] gives an interesting approach for automatic segmentation of the image into facades. On the 28 images, 4 were found correctly positioned in the GIS. 4 other images have not been located because the number of possible poses was above 1. There were 20 failures.

-8 failures were due to a problem in GIS data, and more precisely 4 failures were due to a recent building, viewed on the image but not present in the GIS (an example is given in Figure 4), 3 errors were due to a wrong angle between 2 facades in the building layer (an example is given in Figure 5), and 1 error was due to a 3D structure which does not belong to a building but which was considered as part of the building. on an image, and therefore no pose can be computed. We also noticed that in the case there were no failures, position accuracy was less than 1 meter, which is very interesting because this is enough for pose initialization using visual tracking.

Discussion

Solving failures

Some improvements can be made to solve the failures listed in paragraph 4. First of all, use of multi-hypotheses approach could help find the right correspondences between GIS and images in ambiguous cases, although it could generate much more possible poses. In case of facades separated by a 180 degrees angle, the vertical edges limiting 2 neighbor facades are difficult to find because they cannot be found using vanishing points, only by texture analysis. In case of a wrong GIS modeling, wrong segment orientation or over segmentation of a polygon, multi-hypotheses approach could also help because such modeling errors occur in very special case that could be predicted. Secondly, one could also increase tolerances used in queries so as to take into account error modeling in GIS. Thirdly, we could also take advantage of a width modeling for the roads.

Reducing the number of possible poses

We can also explore several ways to reduce the number of possible poses. It depends on the number of considered polygons in the GIS building layer, but also on the relative orientations of the facades. For instance, unusual angles in a city are easier to find than angles of 90 or 180 degrees, which are extremely frequent in a city like Nantes. Use of an accurate GPS would help reduce the number of polygons to query, although GPS signals are hardly available in dense city centers. Use of absolute orientation sensors, even with a low precision, will help filter out many wrong poses. However, this will really be useful if the distribution of facade orientations in the city is isotropic.

Using 3D data

Use of 3D data allows highly selective pose filtering, but we face the problems of availability, accuracy and reliability of such data. Indeed, modeling of 3D data is much more difficult that modeling of 2D data. On the one hand, correspondences between images and 3D model will be ambiguous in some cases, which implies that human interaction will be mandatory to establish such correspondences. On the other hand, correspondences between images and 2D model are also ambiguous because we do not take into account building height in inter visibility computation. As a result, we plan for the future to work on the use of 3D data using interactive approach.

Conclusion

We have proposed in this article an outdoor localization approach for street images, based on GIS building layer. Automatic correspondences between GIS and geometry extracted from the images result in a layer of every possible pose. Wrong poses are then filtered out using consistency criteria with building and road layer and also merging the possible poses coming from several images. Experimental results have shown that this approach is relevant, but limi-tations appear like the ambiguity in correspondences between images and 2D GIS. Many ways can be explored to overcome these limitations, and chiefly multi-hypotheses approach, use of 3D data, and multi view localization. Moreover, it seems that interactive localization is more robust than automatic one. Therefore our future work also includes a better integration of the user role during the localization process.

Figure 2 -

 2 Figure 2 -Example of obstruction query in the GIS building layer

Figure 3 -

 3 Figure 3 -Viewing Polygon used in a visibility query

Figure 4 -

 4 Figure 4 -Missing building in the GIS

Figure 5 -

 5 Figure 5 -False angles in the GIS

Figure 6 -

 6 Figure 6 -Difficult correspondences between Images and GIS, due to 180 degrees angles


R Baleh

A Abdul-Latif

A Menouer

D Razafindramary

New experimental investigation of non-conventional dynamic biaxial plastic buckling of square aluminum tubular structures

Keywords: Biaxial plastic buckling, Loading complexity, Dynamic loading, Mean collapse load, Energy absorbed, Square aluminum tube

New experimental investigation of non-conventional dynamic biaxial plastic buckling of square aluminum tubular structures

A C C E P T E D M A N U S C R I P T

1 Highlights  This work deals with non-conventional energy-dissipating system of biaxial plastic buckling  The goal is to enhances the energy dissipating capacity of thin-walled square aluminum tubes.  Several the well-developed ACTP device was employed.  Biaxial loading paths (combined compression-torsion) with different are generated from a uniaxial compressive load via different inclination angles (30°, 37°, 45°, 53° and 60°).  The results show that the higher the biaxial loading complexity provided by the ACTP, the greater the interaction between the loading complexity and the strain rate, and the greater the energy absorbed by the structures.  The total energy absorbed is increased by more than 100% for of the most complicated loading path (i.e., Biaxial-60°). Abstract: Motivated by the biaxial plastic dynamic buckling behavior of tubular structures, the particular interest of this work was to develop a non-conventional energy-dissipating system. This system is based on the studying of the experimental behavior of square metallic tubular structures under dynamic complex loading paths. This non-conventional solution aims to enhance the strength properties of the used structures. Accordingly, the well-developed device (the ACTP: Absorption par Compression-Torsion Plastique) has been employed where biaxial loading paths (combined compression-torsion) with different complexities were generated from a uniaxial compressive load. Thus, five inclination angles (30°, 37°, 45°, 53° and 60°) were used to study their effect on the biaxial plastic buckling of square aluminum tubes. This new comprehensive experimental study aimed to investigate the effect of three loading boundary conditions (i.e., simple, intermediate and severe) on the tubes" behavior.

The dynamic deformation, mean collapse load (P m ) and energy absorbed are key parameters that require investigation. They are largely affected by the loading path complexity, applied strain rate and their interaction. The results showed that the higher the biaxial loading complexity provided by the ACTP, the greater the interaction between the loading complexity and the strain rate, and the greater the energy absorbed by the structures. Changes in local deformation mechanisms induced by the loading complexity thus lead to the energy absorbed enhancement. The latter was higher than 100%, in favor of the most complicated loading path (i.e., Biaxial-60°) compared to the classical uniaxial case for an axial deflection of 20 mm.

Simple mathematical equations were proposed predicting the P m of buckled tubes. These equations can describe the tubes" behavior fairly well.

Introduction

It is well-known that the improvement of the vehicular crashworthiness can be bolstered via passive safety systems called energy absorber devices for mitigating both passengers from severe injuries or fatalities and structures damaging during accidents. They can be set in many mechanical systems, like road vehicles, railway couches, aircrafts, ships, lifts, satellite recovery, aircraft soft-landing, etc. The objective of research programs in this area is usually to design specific geometric sections that minimize the amount of force transferred to the integrity of occupants and deform regularly during the crushing process. Hence, to have a better performance, the appropriate determination of the most important criteria such as a superior strength-to-weight ratio and a low cost-to-weight ratio, is persistently pursued. For a given geometry, many of the current developments are based on high-performance structures of high strength, high elastic moduli, low weight, and considerable toughness. Nowadays, reliable and safe components and structural systems receive significant attention especially in crashworthy design aiming to control the response of structures.

One of these passive safety devices concepts uses large deformation, which is the most realistic concept based on mechanical elements such as tubes, stiffeners, plates, shells, etc.

These energy absorbers are one-shot items, i.e., once having been plastically deformed, should be discarded and replaced. In the last few decades, numerous research programs related to this topic have been conducted (e.g., [START_REF] Johnson | Metallic energy dissipating systems[END_REF][START_REF] Jones | Structural Impact[END_REF][START_REF]Collapsible impact energy absorbers: an overview[END_REF][START_REF] Lu | Energy Absorption of Structures and Materials[END_REF][START_REF] Olabi | Metallic tube type energy absorbers: A Synopsis[END_REF][START_REF] Qiu | Some topics in recent advances and applications of structural impact dynamics[END_REF][START_REF] Baroutajia | On the crashworthiness performance of thin-walled energy absorbers: Recent advances and future developments[END_REF]). Since each device has its own features, the energy absorption patterns, their dependence on the deformation modes and the obtained failure during collapse are the key factors that classify the performance of an energy absorber.

The energy absorbed capacity depends, broadly speaking, on the method of loading and its magnitude, strain rates, deformation or displacement patterns and material properties (yield stress, strain hardening, fracture mechanisms) [START_REF] Johnson | Metallic energy dissipating systems[END_REF].

Thin-walled tubular structures offer the widest range of possible uses in such energy dissipating systems particularly under compressive loading. For tubes, the most used materials are metallic such as aluminum alloys, steel alloys, and fiber reinforced composites. The absorbed energy is different for metallic and composite structures. For metals, the absorbed energy is principally governed by their progressive plastic deformation due to their ductile nature; however, in the case of composite materials (which are very often brittle), the energy absorbed is mainly directed by several combined fracture mechanisms. Several mechanical loading configurations have been proposed leading to different deformation modes like tube

A C C E P T E D M A N U S C R I P T 4
inversion where the tube can be plastically turned inside-out or outside-in (e.g., [START_REF] Al-Hassani | Characteristics of inversion tubes under axial loading[END_REF]). Also, tubes can be made to split and curl up (e.g., [START_REF] Stronge | Long stroke energy dissipation in splitting tubes[END_REF][START_REF] Atkins | On the number of cracks in the axial splitting of the ductile metal tubes[END_REF][START_REF] Lu | An experimental study on tearing energy in splitting square metal tubes[END_REF]) or can be plastically flattened due to lateral compression (e.g., [START_REF] Derutz | Crushing of a tube between rigid plates[END_REF][START_REF] Reddy | Lateral compression of tubes and tube-systems with side constraints[END_REF][START_REF] Nesnas | Lateral plastic collapse of cylinders: Experiments and modeling[END_REF][START_REF] Abdul-Latif | Plastic collapse of cylinders under constrained sides and length conditions[END_REF]). Under axial compressive load, thin-walled tubes of different cross-sections (circular, square, rectangular, polygonal, etc.) can also be plastically buckled.

The first reported study regarding the plastic buckling of thin-walled tube has been pioneered by Mallock in 1908 [START_REF] Mallock | Note on the instability of tubes subjected to end pressure, and on the folds in a flexible material[END_REF]. Plastic buckling is the most extensive and efficient means for absorbing energy due to the average collapse load stability, the available stroke per unit mass, and the high percentage of tube material contribution in plastic deformation, i.e., the combination of bending and stretching strains (e.g., [START_REF] Lu | Energy Absorption of Structures and Materials[END_REF][START_REF] Olabi | Metallic tube type energy absorbers: A Synopsis[END_REF][START_REF] Guillow | Quasi-static Compression of Thin-Walled Circular Aluminum Tubes[END_REF][START_REF] Zhang | Energy Absorption of an Axially Crushed Square Tube with a Buckling Initiator[END_REF][START_REF] Zhang | Axial Crushing of Circular Tubes with Buckling Initiators[END_REF][START_REF] Sun | Energy absorption of aluminum alloy thin-walled tubes under axial impact[END_REF][START_REF] Abdul-Latif | Innovative solution for strength enhancement of metallic like-composite tubular structures axially crushed used as energy dissipating devices[END_REF]). Therefore, its specific energy absorbed capacity is approximately 10 times greater than that in laterally compressed tubes [START_REF]Collapsible impact energy absorbers: an overview[END_REF]. The recorded load-deflection curve shows three distinct phases. The first phase, which is the shortest one, corresponds to the initial maximum peak load, while during the second phase which is the longest one, the crushing force drops and oscillates around a mean value. Finally, a quick increase in the crushing load defines the third phase since a small amount of material can be deformed, ending the crushing operation. The mean crushing load with a first maximum peak load is considered in determining the crush force efficiency factor () as a parameter to evaluate device efficiency.

In the case of circular tubes, three principal collapse modes can be observed: axisymmetric mode, diamond mode and mixed mode. To have better crashworthiness characteristics, more promise has been observed in the axisymmetric mode than in the diamond or mixed one [START_REF] Abdul-Latif | Some improvements on the energy absorbed in axial plastic collapse of hollow cylinders[END_REF].

However, the thin-walled square tubes, which were used in this study, deform differently compared to circular ones; unless their force-displacement relationship features are similar. This is because both circular and square tubes undergo progressive collapse under axial crushing [START_REF] Zhang | Energy Absorption of an Axially Crushed Square Tube with a Buckling Initiator[END_REF][START_REF] Zhang | Axial Crushing of Circular Tubes with Buckling Initiators[END_REF][START_REF] Abramowicz | Dynamic axial crushing of square tubes[END_REF][START_REF] Jones | Static and dynamic axial crushing of circular and square tubes[END_REF][START_REF] Hanssen | Optimum design for energy absorption of square columns with aluminum foam filler[END_REF][START_REF] Zhao | On the strength enhancement under impact loading of square tubes made from rate insensitive metals[END_REF][START_REF] Zhao | An Experimental Study of Square Tube Crushing Under Impact Loading Using a Modified Large Scale SHPB[END_REF]. The geometrical ratio of square tube c/t (c is the side length and t is the thickness) has a significant part in determining the collapse mode of a tube wall. In fact, it can be plastically deformed by four different collapse modes as reported in [START_REF] Atkins | On the number of cracks in the axial splitting of the ductile metal tubes[END_REF]: (i) extensional mode with c/t < 7.5, (ii) inextenstional mode for c/t > 40.8, (iii) asymmetric mixed-mode with 7.5 ≤ c/t ≤ 40.8 [START_REF] Reid | Static and dynamic crushing of tapered sheet metal tubes of rectangular cross-section[END_REF], and (iiii) non-compact crushing mode for c/t = 100, i.e., an excessively thin-walled tube where the folds are not continuous and they are separated by slightly curved panels [START_REF] Tarigopula | Axial Crushing of Thin-Walled High-Strength Steel Sections[END_REF]. Globally, this mode is always considered relatively unstable with a tendency to Euler-type buckling, which is an undesirable energy-dissipating mechanism.

A C C E P T E D M

A N U S C R I P T 5

Structural and material features, strain rate, and inertia effects are important features that should be considered in structural crashworthiness design with different sections, mainly circular and square, using thin-walled high-strength steel sections. They strongly influence the dynamic tube-crushing process as shown by several studies [1-6, 17-19, 23-24, 26-30].

Several experimental and theoretical studies displayed the complexity of the plastic buckling phenomenon due to coupling between the inertia effect and plastic properties of the collapsed material. Note that the strain rate sensitivity phenomenon relevant to the crashworthiness topic can often affect the behavior of engineering materials by strengthening of thin-walled structures as in the current study. Therefore, it can be considered a beneficial phenomenon, once the material requires greater energy to be deformed dynamically [START_REF] Jones | Structural Impact[END_REF]. Consequently, the knowledge of the phenomena of highly nonlinear dynamic structural problems is essential to suitably predict their mean collapse load (P m ) and the corresponding energy absorption capacities. Three plastic instability types can be often recorded experimentally [START_REF] Jones | Several Phenomena in Structural Impact and Structural Crashworthiness[END_REF][START_REF] Karagiozova | Dynamic Elastic-Plastic Buckling of Circular Cylindrical Shells Under Axial Impact[END_REF][START_REF] Karagiozova | On Dynamic Buckling Phenomena in Axially Loaded Elastic-Plastic Cylindrical Shells[END_REF]: (i)

Euler mode (total bending of the structure), (ii) dynamic plastic buckling (buckling over the entire length of the cylinder with a moderate radial displacement), and, finally, (iii) dynamic progressive buckling (successive folding process). The latter type is of particular interest in this study. When a relatively low impact velocity is applied (lower than 30 m/s) with a ratio of striker mass / specimen mass > 14, the inertia effect can be neglected [START_REF] Wang | An Energy Criterion for Dynamic Plastic Buckling of Circular Cylinders Under Impulsive Loading[END_REF]. In this case, plastic flow occurs under sequential compression and bending phases constituting a complex load/unload path. Accordingly, the dynamic plastic buckling demonstrates large sensitivity to the loading path [START_REF] Karagiozova | Dynamic Effects on Buckling and Energy Absorption of Cylindrical Shells Under Axial Impact[END_REF]. Moreover, the effect of impact velocity and the tube wall thickness on the collapse mode has been examined in [START_REF]Mushrooming of circular tubes under dynamic axial loading[END_REF]. By controlling these two parameters, a mushrooming mode could be captured, and is associated with folding of the tubes at medium dynamic velocity, whereas a mushrooming associated with tube wrinkling has been observed at high velocity with large wall thicknesses.

Challenges and originality

Recently, design of shock absorber devices using thin-walled tubes has seen much progress through non-conventional solutions. In fact, a great deal of relevant research and development opens up new horizons in this area. They reveal that energy absorption performance can be drastically enhanced by solutions such as different shapes, composite materials and cellular materials (honeycombs, filler material like polymeric and metallic foams, etc.) (e.g., [START_REF] Abdul-Latif | Innovative solution for strength enhancement of metallic like-composite tubular structures axially crushed used as energy dissipating devices[END_REF][START_REF] Hanssen | Static and Dynamic Crushing of Circular Aluminum Extrusions with Aluminum Foam Filler[END_REF][START_REF] Song | Light-weight thin-walled structures with patterned windows under axial crushing[END_REF][START_REF] Zhang | Energy absorption of multi-cell stub columns under axial compression[END_REF][START_REF] Zhang | Axial crushing of tapered circular tubes with graded thickness[END_REF]). Moreover, since collapse modes very often play a key role on the energy absorption

A C C E P T E D M A N U S C R I P T 6
capacity, designing tapered thin-walled square tubes, with and without indentations, has been investigated for improving the energy absorption capacity during crushing [START_REF] Tarigopula | Axial Crushing of Thin-Walled High-Strength Steel Sections[END_REF][START_REF] Nagel | Dynamic simulation and energy absorption of tapered tubes under impact loading[END_REF][START_REF] Asanjarani | Multi-objective crashworthiness optimization of tapered thin-walled square tubes with indentations[END_REF]. The impact of different parameters on the energy absorbed has been also studied, such as the cross section, thickness, taper angle, number of tapers, web width and length of tubes. Nagel and Thambiratnam [START_REF] Nagel | Dynamic simulation and energy absorption of tapered thin-walled tubes under oblique impact loading[END_REF] concluded that the energy absorption capacity of tapered tubes is greater than that of straight tubes having an equivalent cross-section. Furthermore, the mean load and absorbed energy decrease significantly with an increase in the applied load angle due to a change in failure mode. It has been recently reported that the tapered thin-walled square tubes with indentations show better energy absorption performance compared to those without indentations [START_REF] Asanjarani | Multi-objective crashworthiness optimization of tapered thin-walled square tubes with indentations[END_REF].

Cross-sectional profiles also represent an appropriate track with which the improvement and optimization of shock absorbers could be achieved. This can be explained by means of the plastic buckling that has been carried out using a stepwise thickness of square extruded member or multi-cell profiles with a square element added to the corner part of a given crosssection [START_REF] Kim | Effect of the cross-sectional shape on crash behavior of a three dimensional space frame[END_REF][START_REF] Xie | Impact characteristics and crashworthiness of multi-cell, square, thin-walled, structures under axial loads[END_REF]. The related energy absorbed depends mainly on the number of angle elements in the cross-section. This is due to the severe deformation concentrated in these zones near to the corner providing higher energy absorption. Accordingly, these multi-cell profiles are weight-effective structures that generate exceptional gains in terms of absorbed energy up to 220% [START_REF] Zhang | Energy absorption of multi-cell stub columns under axial compression[END_REF].

As another non-conventional idea, the functionally graded thickness within thin-walled tubes has been proposed to control the plastic straining and its distribution. Such a concept can be applied to different shapes of structures like square and circular thin-walled tubes and frusta [START_REF] Zhang | Axial crushing of tapered circular tubes with graded thickness[END_REF][START_REF] Zhang | Axial crushing and optimal design of square tubes with graded thickness[END_REF][START_REF] Li | Comparison of functionally-graded structures under multiple loading angles[END_REF][START_REF] Li | A comparative study on thin-walled structures with functionally graded thickness (FGT) and tapered tubes withstanding oblique impact loading[END_REF]. The impact of this idea is evident in the enhancement of the energy absorption capacity with lower initial peak load [START_REF] Zhang | Axial crushing and optimal design of square tubes with graded thickness[END_REF]. Furthermore, the thickness of square tubes has been varied in such a way that more material is set in the corners due to the severe plastic strain occurring during crushing. This solution offers better energy absorption performance compared to uniform tube thickness with an enhancement between 30 and 35%.

The use of thin-walled tubes with triggers and cut-outs is also considered a non-conventional approach based on creation of geometrical imperfections like indentations in the sidewalls, holes and windows, dents [START_REF] Yuen | The energy-absorbing characteristics of tubular structures with geometric and material modifications: an overview[END_REF]. These triggers lead to a slight deviation of the structural geometry. Their effect as collapse initiators reduces the peak loads that encourage the symmetric deformation mode. Cut-outs in the form of circular holes have been made within aluminum circular tubes [START_REF] Gupta | Eff ect of annealing, size and cut-outs on axial collapse behaviour of circular tubes[END_REF]. This technique provides lower initial peak load and greater deformation stroke. Rectangle windows have also been tested within thin-walled square tubes.

A C C E P T E D M A N U S C R I P T

These windows reduce component weight and enhance the specific absorbed energy up to 54% while decreasing the initial peak load up to 63% [START_REF] Song | Axial crushing of thin-walled structures with origami patterns[END_REF]. The impact of cut-out introduction on the absorbed energy of tapered thin-walled tubes has been also demonstrated [START_REF] Guler | The eff ect of geometrical parameters on the energy absorption characteristics of thin-walled structures under axial impact loading[END_REF][START_REF] Taştan | Optimum crashworthiness design of tapered thin-walled tubes with lateral circular cutouts[END_REF].

Another non-conventional approach of origami patterns has been introduced to thin-walled tubes of square, hexagonal and octagonal cross-sections to improve energy absorption capacity [START_REF] Song | Axial crushing of thin-walled structures with origami patterns[END_REF]. These patterns lead to the reduction of the initial peak load and to minimization in the fluctuation of the force during crushing compared to the conventional tube response.

Moreover, a new non-conventional concept of metal like-composite has been developed. To produce such like-composite components, a hard phase is incorporated in thin-walled mild steel tubes made by case-hardening of 15% of tube outer surface with different geometrical shapes with a constant depth [START_REF] Abdul-Latif | Innovative solution for strength enhancement of metallic like-composite tubular structures axially crushed used as energy dissipating devices[END_REF]. It aimed to enhance plastic buckling resistance. The effect of case-hardened forms on the plastic buckling of thin-walled tubes has been investigated.

These forms are: ring, vertical strip and, helical strip with different tilt angles of 30°, 45° and 60°. The energy absorption capacity of conventional tubes can be increased up to 46%. This is due to the fact that the material behavior seems to be directed by complicated local strain induced by the metal like-composite tube, where a triaxial strain state has been encouraged

particularly within the tube wall in the case of a 3-helical strip form with tilt angle of 30°.

The objective of this work falls within the concept of non-conventional shock absorber devices. Indeed, the present investigation is an extension of the original idea developed in previous works [START_REF] Baleh | Uniaxial-biaxial transition in quasi-static plastic buckling used as an energy absorber[END_REF][START_REF] Abdul-Latif | Dynamic biaxial plastic buckling of circular shells[END_REF][START_REF] Menouer | New generation of energy dissipating systems based on biaxial buckling[END_REF]. This purely experimental study aims to scrutinize the dynamic plastic buckling of thin-walled square tubes. The adopted concept is based on the patented idea shown in previous works [START_REF] Baleh | Uniaxial-biaxial transition in quasi-static plastic buckling used as an energy absorber[END_REF][START_REF] Abdul-Latif | Dynamic biaxial plastic buckling of circular shells[END_REF][START_REF] Abdul-Latif | Shock-Absorber Device Comprising a Plastically-Deformable Member[END_REF] where a special rig is utilized (called the ACTP). The solution consists of creating a particular complex loading condition, provoking an enhancement in strength properties of the loaded structure due to changes in local deformation mechanisms as demonstrated in [START_REF] Calladine | Strain-rate and inertia effects in the collapse of two types of energy absorbing structures[END_REF]. This rig transforms uniaxial external loading into biaxial loading of combined compression-torsion. Thus, several degrees of loading complexity can be created within the loaded thin-walled tubes. The biaxial plastic buckling, therefore, becomes increasingly complicated once a shear component is added to these sequential phases of compression and bending. It leads consequently to three different strains of compression, bending and shear. Under quasi-static loading, the biaxially crushed aluminum thin-walled square tube has been studied in the work of Menouer et al., [START_REF] Menouer | New generation of energy dissipating systems based on biaxial buckling[END_REF]. That work demonstrated the dependence of plastic buckling on specimen geometry and loading path complexity. The energy absorption capacity of each system under different biaxial loading paths is understood as: The higher the biaxial loading complexity applied, the greater the energy absorbed for a

A C C E P T E D M A N U S C R I P T 8
given structure. A maximum recorded increase in the energy absorbed has been in favor of the Biaxial-60° loading compared to the free-ends case giving an average increase of 47%.

Hence, the same thin-walled square types together with the same loading path complexities are investigated here but under dynamic loading. Note that numerical investigation with optimization will be conducted in future works.

Experimental program

According to a recent development regarding the non-conventional shock absorber proposed in the study of Menouer et al., [START_REF] Menouer | New generation of energy dissipating systems based on biaxial buckling[END_REF], the effect of loading path complexity on the response of thin-walled square aluminum tubes, particularly on the mean collapse load (P m ) and the energy absorbed, has been investigated. It studied several quasi-static combined compressiontorsion biaxial loading paths on the same square aluminum tubes. These loading paths can be decomposed into three groups: (i) uniaxial (free-ends), (ii) moderate biaxial (Biaxial-30°, Biaxial-37° and Biaxial-45°) and (iii) severe biaxial (Biaxial-53° and Biaxial-60°).

In this work, the effect of the dynamic strain rate and its interaction with the loading complexities of the structure has been studied, employing the same tube geometries already tested under quasi-static loading.

For a given tube geometry, the P m , maximum load (P max ) and energy absorbed are examined to evaluate biaxial plastic buckling behavior. The average recorded load-deflection records were calculated and studied.

The tested materials

An annealed commercial aluminum alloy is used because of its ductility, which is considered an important factor for such applications. Its chemical composition is summarized in Table 1 and designated according to the French standard as A50-451 (6060). Its basic mechanical properties are: Young"s modulus of 69 500 MPa, tensile yield strength ( y ) of 200 MPa and ultimate tensile strength ( ult ) of 220 MPa.

The proposed non-convention shock absorber system was designed and prepared using thinwalled square tubes having a cross section of 35mm x 35mm, with 2mm thickness (t). An initial length (L) of 100 mm was utilized and then crushed under dynamic uniaxial compressive load. Hence, this gives two geometrical parameters defined by ζ= c/t having

The ACTP rig

The ACTP device is schematically presented in figure 1. It was made of a tempered steel hollow cylindrical body (1), on which four parallel helicoid grooves were machined [START_REF] Jones | Structural Impact[END_REF]. They are characterized by a defined inclination angle controlling the rate of change of torsional component during collapse. To investigate the effect of this key parameter, five different interchangeable cylindrical bodies (1) were designed having five distinct propeller inclination angles of 30°, 37°, 45°, 53° and 60°. The helicoid grooves are intended to receive a crosspiece [START_REF] Olabi | Metallic tube type energy absorbers: A Synopsis[END_REF] provided with four pivots equipped with bronze rollers (4) to guide it in its descent by imposing a rotational movement. Hence, the two principal parts of this apparatus form a slidehelicoid connection. This mechanism transforms an initial external load of uniaxial nature into a biaxial combined compression-torsion. The bronze rollers (4) were used to minimize friction in the contact zone between the grooves (2) and the crosspiece [START_REF] Olabi | Metallic tube type energy absorbers: A Synopsis[END_REF]. The crushed circular (or square) tube ( 7) is mutually dependent on the crosspiece and cylindrical body by means of a mechanical tube fixation system. The system made principally from a hard steel disc [START_REF] Lu | An experimental study on tearing energy in splitting square metal tubes[END_REF].

Two half-conical shells (8) and a clip [START_REF] Atkins | On the number of cracks in the axial splitting of the ductile metal tubes[END_REF] over which these conical surfaces were machined and assembled in opposition were attached to the disc to maintain the necessary tightening pressure in locking of both crushed tube extremities (figure 1a). For a thin-walled square tube, their extremities fixation system was differently designed with two square clamps (figure 1b) using the same concept as in the circular tubes. Therefore, regardless of the tube type, its deformation is fully conditioned by the crosspiece in its movement of descent and rotation.

It is difficult to evaluate the friction effect on the deformation operation during the tests.

However, an approximate method has been already proposed [START_REF] Baleh | Uniaxial-biaxial transition in quasi-static plastic buckling used as an energy absorber[END_REF][START_REF] Menouer | New generation of energy dissipating systems based on biaxial buckling[END_REF] defining its effect on the total energy absorbed. Therefore, the friction effects and its dependency on the loading complexity have been evaluated for the square tube case in [START_REF] Menouer | New generation of energy dissipating systems based on biaxial buckling[END_REF], as shown in figure 2. In fact, for an axial deflection of 50 mm, it has been recorded that the energy absorbed and the P m vary from 4% in biaxial-30° up to 11% for the Biaxial-60° with a nonlinear evolution depending on the complexity level (i.e., moderate and sever loading).

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A N U S C R I P T

The impact apparatus

All the presented tests were experimentally carried out utilizing a dynamic drop mass bench of a maximum impact velocity of 10 m/s and of a maximum kinetic energy of 2.5 kJ. It is equipped with a dynamic load cell of 20 tons, a 5000 g accelerometer, and a laser beam displacement transducer (series M5L of international Bullier) for a measurement bracket of 100 mm. These instruments are connected to a rapid acquisition chain (2.5 MHz), which ensures the simultaneous recording of these experimental data: force, acceleration and displacement. To assure the synchronization of the acquisition of these essential data, two photocells have been used (figure 3). For the suitably of the collected experimental data, the positioning of these photocells is important and depends on the striker position just before the impact. The tests were conducted under initial impact velocity of about 9 m/s with a mass of 53 kg.

Experimental procedure

Each employed thin-walled square tube should be first mechanically assembled inside the ACTP rig. Then, it was loaded between the platen and the striker of the dynamic drop mass bench under an initial impact velocity of about 9 m/s at room temperature. Having its instrumentation system, the impact apparatus over which the ACTP rig was fixed, connected to a rapid acquisition chain to record the force, displacement and acceleration during collapse process.

As an adopted approach to ensure the experimental results accuracy, each test was repeated, at least six times under the same experimental conditions (applied velocity and room temperature). If the differences in responses of four tests exceed 3%, then other tests should be conducted. Based on the valid tests, mean force-deflection curves were then presented.

This rigorous methodology is expected to improve result reliability and repeatability substantially.

Results and discussion

Preliminary study: validation of the buckling length

In order to totally avoid the Euler buckling mode (EM), which leads to poor energy absorption, different initial lengths were chosen and then tested. This preliminary study which was conducted under uniaxial loading aimed to determine the initial tube length with which

A C C E P T E D M A N U S C R I P T
11 the asymmetric mixed-mode (AMM) can be generated, since the used section has ζ= c/t = 17.5, i.e., its value is in the range of 7.5 ≤ c/t ≤ 40.8 [START_REF] Jones | Structural Impact[END_REF]. Note that the biaxial loading configurations couldn"t be used for currying out this investigation. This is due to the limitation in total allowable length of ACTP rig which is less than 150 mm giving consequently a maximum effective length of 100 mm.

Based on preliminary tests involving five initial lengths [START_REF] Xie | Impact characteristics and crashworthiness of multi-cell, square, thin-walled, structures under axial loads[END_REF]92,138, 184 and 230 mm) for a single square section of (ζ= c/t = 17.5 and = c/L) given in the work of Menouer et al., [START_REF] Menouer | New generation of energy dissipating systems based on biaxial buckling[END_REF],

table 2 summarizes the obtained results. It is obvious that its generated deformation mode is of the AMM type for all the lengths used, except for the length 230 mm, where the Eulerian mode takes over. The length of 100 mm was selected throughout this study. This

Dynamic deformations

Different boundary conditions of uniaxial free-ends and biaxial loadings were applied. Figure [START_REF] Lu | Energy Absorption of Structures and Materials[END_REF] shows two distinct deformation modes of plastic buckling: (i) a classical asymmetric mixed-mode (AMM) and (ii) an axisymmetric mixed-mode with relative warping (referred to as AMMW). It is recognized that the first mode was systematically observed in the free-ends case. This is in accordance with results in the literature [START_REF] Abramowicz | Dynamic axial crushing of square tubes[END_REF][START_REF] Jones | Static and dynamic axial crushing of circular and square tubes[END_REF]. However, under combined biaxial loading paths, the warping level depends exclusively on the inclination angle. In general, these observed deformation modes are similar to those obtained under quasi-static loading [START_REF] Menouer | New generation of energy dissipating systems based on biaxial buckling[END_REF]. Hence, one can conclude that the deformation mode cannot be affected by the applied strain rates. Figure ( 4) illustrates representative specimen photos taken at the end of the crushing process under all used loading paths, showing the top, front and sectioned views.

From the top views (figure 4a), each test has its own warping crown with a given twist angle.

The latter is generated by the torsional component of loading and evidently be determined only via the top view. This angle is measured using one of the sides of the square tube with respect to its initial position as given in figure (4a). These measured twist angle are 7°, 11°, 14°, 27° and 32° for Biaxial-30°, -37°, -45°, -53° and -60°, respectively. In addition, the front views provide another assessment of angles (not twist angles) which were defined differently.

They are generated under the effect of the torsional loading component. Their values are determined through the deviation (or rotation) of the plastic folds from the tube"s initial horizontal cross-sectional plane as shown in figure (4b). These generated angles are 4°, 6°, 10°, 26° and 31° for Biaxial-30°, -37°, -45°, -53° and -60° respectively. Figure (4c) displays the sectioned views targeting the deformed zone for each case giving us useful details about

A C C E P T E D M A N U S C R I P T

the dynamic deformation with respect to the loading configuration. Furthermore, three completed folds in uniaxial and Biaxial-30° have been observed by means of the sectioned views; whereas in Biaxial-37° and -45°, the third semi-completed one was detected, but became incomplete in Biaxial-53° and -60°. This behavior is attributed to the change in tube resistance. As a matter of fact, the resistance is affected by loading complexity, i.e., the higher the inclination angle, the greater the resistance of tubular structures. Consequently, it becomes more resistant to plastic deformation under Biaxial-53° and -60°. This conclusion was also reached in the work of Menouer et al., [START_REF] Menouer | New generation of energy dissipating systems based on biaxial buckling[END_REF]. Hence, one can conclude that the inclination angle (i.e., loading complexity) has an important role on the dynamic deformation.

Once all these angles are determined via the two views, i.e., top and front views, each ratio can be defined by: the measured angle/the inclination angle of the ACTP rig. In fact,  T (and

 F
) is the ratio of the angle induced at the end of crushing process measured by the top view (and the front view) divided by the inclination angle of the ACTP.

Figure [START_REF] Olabi | Metallic tube type energy absorbers: A Synopsis[END_REF] points out the variation of these two ratios:  T and  F . Obviously, three categories of evolution were observed matching directly with the three loading boundary conditions. For the first category, both ratios are equal to zero for the uniaxial loading condition, while for the second one, i.e., the moderate loading complexity (Biaxial-30°, -37° and -45°), their recorded evolutions are relatively slow especially for the top ratio. Then, a jump in their values was seen for the third category, i.e., for Biaxial-53° and -60°. Under moderate loading paths, a distinct difference between  T and  F was clearly observed particularly for Biaxial-30° and -37°. In addition, the  T ratio was always of a higher value than  F . Nonetheless, both ratios

give almost the same ratio in the severe loading zone.

As a key role of the loading complexity, the form of the fold penetration inside the tubes and the induced rotation angles are proportionally affected by the inclination angle, i.e., loading complexity. However, the fold penetration becomes more pronounced at the end of crushing especially in the Biaxial-60° case.

The collapse loads and energy absorbed

Choice of the dynamic effective crushing stroke

In order to suitably identify the influence of dynamic loading condition on specimen behavior, a dynamic effective crushing stroke should be first defined for realistic determination of the dynamic behavior. According to our knowledge, the definition of this stroke is based on: (i) the velocity drop during the crushing process (figure 6) and (ii) the load-deflection behavior

A C C E P T E D M A N U S C R I P T 13
for each applied loading configuration (figure 7). Examination of figure [START_REF] Qiu | Some topics in recent advances and applications of structural impact dynamics[END_REF] shows typical evolutions of impact velocity and their dependence on the loading complexity throughout axial deflection. This stroke should be deduced based on the recorded results of buckling.

Indeed, the entire crushing process can be divided into two distinct buckling regimes. The first is related to  ≤ 20 mm where the dynamic plastic buckling behavior was clearly captured regardless of the loading complexity (figure 7); whereas for  > 20 mm, changes in tube behavior have been observed during the crushing process where quasi-static behaviors were recorded. It was consequently recognized that the crushing stroke of 20 mm gives approximately the limit of dynamic behavior. At this crushing stroke, the variations in impact velocities according to figure [START_REF] Qiu | Some topics in recent advances and applications of structural impact dynamics[END_REF] show a high sensitivity to the loading complexity. Therefore, for uniaxial (Ref), Biaxial-30°, -37°, -45°, -53° and -60°, the corresponding impact velocities the stroke leading therefore to more plastic buckling folds. On the other hand, even with this definition of dynamic crushing stroke (somewhat limited for defining the mean crushing force due to the initial peak load effect), the general trend regarding the gain in P m cannot be changed for more important dynamic crushing stroke.

The collapse loads

The influence of the loading complexity and its rate on the material and structural responses can be demonstrated by the applied collapse load and then the energy absorbed. The crushed structure response can be evaluated by several key parameters, among which is P m . P m is influenced by the strain rate, loading complexity and their interaction (table 3 and figure 7).

As a typical example, the load-deflection curves reveal the loading evolution during the tube collapse (figure 7). It shows the effect of loading boundary conditions on the specimen

A C C E P T E D M A N U S C R I P T 14 response.
For a given dynamic strain rate, the maximum value of P m was recorded in Biaxial-60° of 63.3 kN giving a difference of 32.2 kN compared to the uniaxial free-ends case which gave a maximum global gain of 104% (table 3). Note that this gain can be portioned into two parts (table 4). One is due to the dynamic loading effect (i.e., strain rate sensitivity of the used material). Hence, this effect was assessed by 62% with respect to the same loading complexity under a quasi-static configuration. On the other hand, the gain of 45% was found due to the loading complexity and its interaction with strain rate (table 4). It is important to note that two factors in dynamic uniaxial collapse which should usually be considered are not present in quasi-static collapse. The first is related to the dependence of flow stress on the strain-rate and the second factor is related to the dynamic deformation of structure under dynamic loading. It may be significantly different from the quasi-static mode caused by inertia forces developed within the structure during the impact. Inertia effects depend principally on impact velocity, type and size of structure and the properties of the material [e.g., [START_REF] Calladine | Strain-rate and inertia effects in the collapse of two types of energy absorbing structures[END_REF][START_REF] Tam | Inertia and strain rate effects in a simple plate-structure under impact loading[END_REF][START_REF] Karagiozova | Inertia effects in axisymmetrically deformed cylindrical shells under axial impact[END_REF][START_REF] Hsu | Quasi-static and dynamic axial crushing of thin walled circular stainless steel, mild steel and aluminium alloy tubes[END_REF]. Since we have, in this study, a progressive dynamic buckling type taking place at relatively low impact velocities (lower than 30 m/s) and for a ratio of the mass of the striker and the mass of the specimen higher than 14, therefore the inertia effects can be neglected. In the work, the observed dynamic deformation of the structure under dynamic load is the same as that under quasi-static loading [START_REF] Menouer | New generation of energy dissipating systems based on biaxial buckling[END_REF]. So that, the negligence of those features related to the dynamic behavior of type II subjected to an impact load can be completely acceptable.

Therefore, the proper conclusion to be drawn from this discussion is that only the strain-rate effect is significant in this investigation.

Figure (8a) points out a comparison through the histograms presented for six axial deflections of = 10, 15, 20, 25, 30 and 35 mm. In general, it is recognized that P m decreases with the increase of the axial deflection regardless of the loading path complexity. However, in the case of Biaxial-60° for = 15 mm, P m is greater than that for = 10 mm. Moreover, in Biaxial-30°, -37° and -45° for = 35 mm, P m rise slightly. Nonetheless, P m declines obviously especially in the case of the highest complexity load (i.e., Biaxial-60°) for the same . It can also be concluded that in the range of 25 ≤  ≤ 35 mm, the recorded P m is almost stable for the moderate biaxial loads (i.e., Biaxial-30°, -37° and -45°). This observation could be directly associated to the variation of  T and  F for the same loading configuration (figure 5). The decrease in P m seems to be controlled by the transition from a dynamic configuration to an almost quasi-static one as well as by the attenuation of the peak load effect during the crash, i.e., the greater the axial deflection, the lowest the effect of peak load. The temperature evolution in plastic hinge zone is another parameter to take into account. Actually, its effect increases progressively with tube crushing process due to heat transfer, and becomes more significant in the last part of the crushing particularly with the increase of loading complexity.

It is important to emphasize that it was difficult to evaluate it due to the limited means available to complete this study. In addition, it could be noticed that the greater the loading complexity, the higher is the temperature for a given .

A careful examination of figure (8b) shows that the maximum drop of P m systematically corresponds to Biaxial-45° for  ≤ 25 mm. To interpret this response, a further study should be conducted in the future taking into consideration the plastic behavior of the material at the microscale as in [START_REF] Drusin | Microstructure Analysis of Aluminum Alloy and Copper Alloy Circular Shells after Multiaxial Plastic Buckling[END_REF]. However, this trend changes obviously for more important axial deflections, for example for = 35 mm, where the temperature could play a significant role especially for severe biaxial loads of Biaxial-53° and -60°. Consequently, the drop in P m increases progressively not only with  increasing but also with the increase of the loading complexity. Hence, the highest drop value of 47% was recorded in Biaxial-60° for ≈ 35 mm.

The additional drop in P m may be associated with an increase in temperature in the most deformed area of plastic hinges. However, in the range of  ≤ 20 mm, the drop variation of P m could be interpreted by a competition between the progressive attenuation of the peak load effect by  increasing and the applied loading complexity and its effect of the tube response, where in this range of , the temperature may not reach its effective influence as in the higher values of . Moreover, for 25 ≤  ≤ 35 mm, the variation of drop in P m is limited to moderate loads of Biaxial-30°, -37° and -45°, i.e., their variation is less than 5%. however, under severe biaxial loads (Biaxial-53° and -60°), almost the same drop values were recorded for = 25 mm.

As a conclusion, figure [START_REF] Al-Hassani | Characteristics of inversion tubes under axial loading[END_REF] confirmed the right determination of the dynamic effective crushing stroke is between 20 and 25 mm. For  > 25 mm, P m suffers from a significant decrease due to competition among three principal factors:

(i) transition from a dynamic configuration to a quasi-static configuration;

(ii) progressive attenuation of the peak load effect by the increase of axial deflection and;

(iii) temperature rising in the areas of plastic hinges leading thus to a decrease in the material strength.

Figure [START_REF] Stronge | Long stroke energy dissipation in splitting tubes[END_REF] shows the gain in P m versus the loading path complexity. First, one can find that for the uniaxial dynamic test, 17% of gain in P m was recorded compared to the quasi-static test (table 4). This confirms that specimens are sensitive to the applied strain rate. Likewise, under

A C C E P T E D M

A N U S C R I P T 16 biaxial loading, the global enhancement which rises proportionally with the increase of loading path complexity comes from the strain rate and its interaction with loading complexity. This provokes changes in local deformation mechanisms induced by dislocation mechanisms. Indeed, via the biaxial loading, an additional hardening is notably induced when the material is severely deformed by three different strains: compressive, bending and shear.

In order to understand such sensitivity, this behavior therefore needs to be thoroughly investigated. This observation has also been recorded by Drusin et al., [START_REF] Drusin | Microstructure Analysis of Aluminum Alloy and Copper Alloy Circular Shells after Multiaxial Plastic Buckling[END_REF], where the plastic hinge zones through the aluminium tube that deformed uniaxially have dislocation cells of high-density along with well-defined deformation bands in specific orientations. In the case of a biaxial deformed specimen with 45°, for example, a predominately cellular microstructure with very high-density tangled dislocation network is observed. The presence of high-density tangled sessile dislocations explains the increase of P m and the energy absorbed in this loading case. In work, the inclination angles of 53° and 60° have not yet been studied.

In table [START_REF] Lu | Energy Absorption of Structures and Materials[END_REF], a lowest gain value of 22.5% for the moderate biaxial load (i.e., Biaxial-30°) was observed, then plateau near-maximum was recorded in Biaxial-37° and -45° giving gains of 25.5% and 24.6%, respectively. Under the highest loading complexity of Biaxial-60°, a remarkable increase of 62% was noted.

A simple analytical modeling was developed defining the P m of biaxial plastic buckling and its dependence on the complexity of dynamic loading. In this model, P m depends on: (i) the type of material described by its ultimate tensile strength ( ult ), (ii) the cross-sectional area of structure (A), and evidently (ii) the loading complexity effects defined by the two principal ratios (  and  F ) given above.

Therefore, the mean collapse load ( T m P ) attributable to   is expressed as follows:

T T m A     1 P ult (1)
By analogy, the computational mean collapse load ( F m P ) controlled by  F is assumed to be:

F F m A     1 P ult (2)
For the sake of simplicity, the global mean collapse load of biaxial plastic buckling is considered as a mean value of the above mean collapse loads, we have therefore: Note that in the uniaxial plastic buckling case, the value of both ratios is equal to zero due to the absence of the torsional component, i.e.,   =  F = 0. This leads to the fact that P m =  utl A, i.e., the total dependence of P m on  utl for given material and cross-sectional area.

Table [START_REF]Collapsible impact energy absorbers: an overview[END_REF] and figure [START_REF] Atkins | On the number of cracks in the axial splitting of the ductile metal tubes[END_REF] show a comparison between the experimental values of P m and the computed ones. Hence, these equations can describe the tubes" behavior fairly well under dynamic loading with the maximum difference of 20% recorded in Biaxial-53°. It confirms once again the three categories of loading path complexity (simple uniaxial free-ends, moderate: Biaxial-30°, -37° and -45° and severe: Biaxial-53° and -60°) and the dependence of P m on such categories. It is noted that in the moderate loading configuration, the experimental results of P m vary gradually and almost linearly. However, a sudden increase of P m is noticed as soon as the severe loading complexity is applied. Such an increase would be induced by local changes in mechanical behavior of the material, and this consequently requires further investigation.

The energy absorbed

As far as the energy absorbed is concerned, figure [START_REF] Lu | An experimental study on tearing energy in splitting square metal tubes[END_REF] illustrates a clear improvement in dissipating the kinetic energy during the crushing process. The corresponding curves diverge with the increase in loading path complexity. After = 5 mm, the energy absorbed behaves almost linearly in the case of uniaxial loading; whereas, under sever loading complexities especially for Biaxial-60°, the behavior becomes highly nonlinear. Hence, the energy curves show an evolutionary additional energy absorbed confirmed by figure [START_REF] Derutz | Crushing of a tube between rigid plates[END_REF]. For the chosen dynamic effective crushing stroke of 20 mm, the absorbed energies are: 0.63, 0.77, 0.82, 0.86, 1.01 and 1.29 kJ, respectively. For the uniaxial free-ends, Biaxial-30°, Biaxial-37°, Biaxial-45°, Biaxial-53° and Biaxial-60°, gave respective enhancements of 22.8%, 30.5%, 35.9%, 59.6% and 104%. Such results are induced by the loading complexity, dynamic strain rate and their interaction. Note that in the intermediate loading paths (i.e., Biaxial-30°, -37° and -45°), the energies absorbed differ relatively slightly and are confirmed by their respective evolutions of 0.77, 0.82 and 0.86 kJ. This would be due to a weak interaction between the applied loading complexity and the applied strain rate. Thus, such results require more investigations particularly from the microstructural point of view. However, this material did not demonstrate any sensitivity to quasi-static strain rates as shown in [START_REF] Menouer | New generation of energy dissipating systems based on biaxial buckling[END_REF].

To sum up, the higher the inclination angle, the higher the loading complexity, the greater the interaction between the loading complexity and the strain rate, the greater the P m and the greater the energy absorbed for the dynamic loading rate. As a result, the Biaxial-60° gives the highest P m and energy absorbed which were also confirmed under quasi-static loading [START_REF] Menouer | New generation of energy dissipating systems based on biaxial buckling[END_REF].

Concluding remarks

The objective of this comprehensive experimental study is to develop a new non-conventional energy dissipating system where the strength properties of material and structure can be enhanced under different dynamic loading conditions. Hence, the key point has been to study the effect of the biaxial combined compression-torsion loading condition coupled with the strain rate on the progressive dynamic biaxial buckling response for square aluminum tubular structures. For a given specimen geometry, some parameters control the crushing process in general: loading complexity, its rate and their interaction. By using the well-developed rig (ACTP), a non-conventional biaxial loading path with different complexities is generated from a uniaxial compressive load. Three loading boundary conditions (i.e., simple, intermediate and severe) were investigated under dynamic strain rate studying their effect on the square tubular structures. Five inclination angles of 30°, 37°, 45°, 53° and 60° were used.

Under such loading complexities, three different strains (compression, bending and shear) are simultaneously applied with more complex load/unload conditions inducing local physical phenomena responsible for changes in strength properties. Determination of the dynamic effective crushing stroke was experimentally conducted. With respect to the used dynamic drop mass bench, especially its energetic capacity, it was between 20 and 25 mm. For crushing strokes more than 25 mm, P m suffered from a significant decrease due to competition among three principal factors (i) transition from a dynamic configuration to a quasi-static configuration, (ii) progressive attenuation of the peak load effect by the increase of axial deflection and (iii) temperature rising in the areas of plastic hinges leading thus to a decrease in the material strength. One can conclude that the deformation modes observed in this study were similar to those obtained under quasi-static loadings. In addition, the used aluminum alloy shows an obvious sensitivity to the dynamic loading with respect to the quasi-static one. For the biaxial loading conditions and for  ≤ 25 mm, the higher the inclination angle, the more loading complexity, the more severe is the rates of change of torsional component, the greater the interaction between the loading complexity and the strain rate, and the greater are consequently the values of P m and the corresponding absorbed energy. Therefore, the Biaxial-60°, which is considered as the most complicated loading path, gives the highest P m as well as the energy absorbed. Thus, the enhancement in P m and the energy absorbed compared to the classical uniaxial case are higher than 100% for  = 20 mm. A simple computation of P m which was focused on the biaxial plastic buckling behavior of tubes under dynamic loading can describe the tubes behavior fairly well. Gain due to loading complexity, % Total gain, %

Loading boundary conditions

2

  New experimental investigation of non-conventional dynamic biaxial plastic buckling of square aluminum tubular structures R. Baleh a,c , A. Abdul-Latif a,b , A. Menouer a,d and D. Razafindramary a,b a) Laboratoire Quartz, Supméca, 3, rue Fernand Hainaut, 93407 St Ouen Cedex, France b) Université Paris8, IUT de Tremblay, 93290 Tremblay-en-France, France c) Campus La Salle-Passy Buzenval, 50 avenue Otis Mygatt, 92500 Rueil-Malmaison, France d) Université Mouloud MAMMERI, Laboratoire LMSE, 15000 Tizi-Ouzou, Algérie

A

  C C E P T E D M A N U S C R I P T 9 always a value of 17.5 and = c/L= 0.35. The specimens were subjected to neither heat treatment nor to special machining operation.

  are 6, 5, 4.7, 4.6, 4.5 and 3.4 m/s, respectively. Moreover, this figure points out an upper bound defined by uniaxial loading and a lower one determined by the most complicated loading path (i.e., Biaxial-60°). All other cases which have almost the same values lie between these two bounds. Such behaviors are totally governed by the enhancement of tube resistance increasing by the loading complexity generated by the ACTP rig as shown above. It is worth emphasizing that a dynamic crushing stroke can generally be governed for a given material by (i) the energy capacity provided by the dynamic drop mass bench and (ii) the tube geometry especially its cross-sectional area. In this work, the limitation in the energy capacity of the used drop hammer together with the employed tubes geometry give us a dynamic crushing stroke of 20 mm. It would make sense if the crushing stroke becomes greater than

  this study,  ult = 220 MPa and A: = 136 mm 2 .

24 Figure 2 .Figure 3 .

 2423 Figure 2.Friction effect and its variation with a loading complexity within the ACTP device[START_REF] Menouer | New generation of energy dissipating systems based on biaxial buckling[END_REF]

Figure 4 . 27 Figure 5 . 28 Figure 6 .Figure 7 .Figure 8 . 60 Mean 30 Figure 9 .

 42752867860309 Figure 4. Overview of the observed AMM and AMMW mode and the corresponding dynamic deformations induced during uniaxial and biaxial buckling showing the influence of the loading complexity under dynamic compressive loading, (a) top view, (b) front view, (c) sectioned view

60 Figure 10 .Figure 11 .Figure 12 .

 60101112 Figure 10. Comparison between the experimental and predicted variation of the mean collapse loads versus different loading boundary conditions

(3)