Sinh-Khoa Nguyen

Pierre-Louis Chiambaretto

Miguel Charlotte
email: miguel.charlotte@isae.fr

Philippe Villedieu

Joseph Morlier

Yves Gourinat

Towards an analytical formulation for fluid structure tank vibration analysis: Modal equivalency using granular materials

Keywords:

This paper focuses on a new methodology for substitution of Liquid Hydrogen (LH2) contained in cryogenic tank in vibration analysis by using surrogate granular materials. Our analysis is limited to a 3D tank that is supposed to be fully filled with granular grains and tries to establish a modal equivalence between (tank fully filled with LH2) and (tank fully filled with granular grains) systems. For this, we determine required pre-stresses based on a homogenization technique from properties of grains. After reviewing some important mathematical formulations of vibration and homogenization model, an example of modal equivalence between these two systems is presented. Analytical results are also compared with numerical simulations in order to prove the suitability of the new method.

• H1: Acoustic effects and viscosity are negligible in the fluid domain (LH2). This assumption is notably based both on a Chiambaretto et al.'s estimate of the Helmholtz number [3] that was found to be 10 -2 , and on McCarty's one for the viscosity of LH2 at the storage state [21] that is equal to 120.10 -7 kg/ms.

• H2: Pre-stressed pressure is high enough to prevent relative motions between grains.

Introduction

Liquid Hydrogen (LH2) is largely used in cryogenic engines for launchers due to its high efficiency [START_REF] Sutton | Rocket propulsion elements[END_REF]. However, utilization of this fuel raises some critical technical problems linked not only to explosiveness and fugacity, but also to our understanding of the dynamic behaviour of the cryogenic tanks. LH2 is indeed too dangerous and expensive to be performed in vibration tests in laboratory [START_REF] Ivey | Mated vertical ground vibration test[END_REF]. While for other liquid fuels, surrogate liquids such as water are largely used to study the dynamic behavior of the tank, it is however difficult to apply the same strategy for LH2 due to its very low mass density.

A recent work performed by our research group [START_REF] Chiambaretto | Surrogate granular for modal test of fluid filled tank[END_REF] suggested the idea that pre-stressed grain systems as a surrogate material may possibly help to circumvent the aforementioned experimental issue. While outlining a new methodology for this, these preliminary studies provided indeed good analytical results for two-dimensional structural coupling cases, which tend to support the above hypothesis. Nevertheless, the potential applicability of this approach to more engineering cases of three-dimensional cylindrical cryogenic tanks still remained for investigation. That one constitutes the extension targeted by the present article.

Globally, the approach developed here, and which builds on the one used by Chiambaretto et al. [START_REF] Chiambaretto | Surrogate granular for modal test of fluid filled tank[END_REF], involves two main tasks aimed to define an effective strategy for the substitution of liquid fuel in vibration analysis by using granular materials, as schematized in Fig. 1. In one of the tasks, the vibration of tank-fluid material (TFM) system is considered as one of a tank-solid material (TSM) system while assuming that a modal equivalence between the vibrations of these two different structural/material systems can be found for some frequency bandwidths. In the second task, a modal equivalence between the vibrations of the TSM system and a tank-granular material (TGM) must be found with the help of a homogenization technique. The latter task requires notably determining the suitable pre-pressure that must be applied to an assembly of granular systems -to be modelled as a homogeneous continuum -that achieves the modal frequencies and mode shapes of a cylindrical tank filled with LH2. Besides, as performed before in [START_REF] Chiambaretto | Surrogate granular for modal test of fluid filled tank[END_REF], the aforementioned equivalences are stated by means of a simple and efficient methodology. The strategy and the solutions to the different considered dynamics problems that are notably obtained by Rayleigh-Ritz method rely on the five following hypotheses:

• H3: Granular material can be described as an elastic material with material coefficient depending on pressure.

• H4: Pressure variation during vibration tests is low enough to lin- earize material coefficients.

• H5: The radial displacements of the mode shape of a free circular beam, of a circular membrane on its free edge and of a beam filled by this circular membrane are the same. The appropriateness of this assumption was checked numerically with COMSOL [START_REF] Chiambaretto | Surrogate granular for modal test of fluid filled tank[END_REF], the simulations pointing out that the modal shapes of a circular beam, of a membrane on free edge, and the coupled system, are very similar.

To our knowledge, the foregoing aim and methodology seem to be new in the literature on Tank -Granular material' coupling systems, where researches concentrate mainly on the storage and flow of grains in silos; the main goal of the proposed methodology is indeed to achieve modal equivalency (over some frequency bandwidths) between a tankfluid material (TFM) system and a tank-solid material (TGM) one so that this last material system can be used as a surrogate material in vibratory tests for launcher tanks.

This paper contains three main sections. First, the general strategy, the mathematical formulations for the vibration analysis and the homogenization model are introduced in the 1st section. The 2nd section presents the analytical results for the vibration analyses of an empty tank, a tank fully filled with fluid or else with granular materials. Comparisons between analytical and numerical results are also provided to show the potential of the new methodology. Lastly, in the 3rd section, some important conclusions are drawn and proposals are made in order to find an effective strategy for a surrogate granular material in the future.

Mathematical formulations

Methodology for substitution of Liquid Hydrogen in vibration analysis

As already stated, the final purpose of the ongoing project is to find a surrogate granular material to substitute the liquid fuel in vibration tests, the concerned liquid fuel being LH2 in our targeted application case. The surrogate material must satisfy the condition that the most representative mode shapes and eigenfrequencies of tank filled with grains are the same as the ones of tank filled with liquid. For our study, the most representative mode shapes and eigenfrequencies of interest are related to the lowest frequencies of vibration, but the criterion of equivalence may be adjusted to other frequency ranges.

From a mathematical point of view, our criterion of equivalence is said to be fulfilled if we can find a granular material solving the following equation

+ + = + K K M M K M M T G T G T T F (1)
with -K T , M T : respectively the modal stiffness and the modal mass of the tank. -K G , M G : respectively the modal stiffness, the modal mass of the granular assembly. -M F : modal mass added by the fluid. -Based on this idea, the following scheme is proposed in order to find the required pre -stress value for each vibration mode.

Fig. 1. Modal equivalence between fluid filled beam and grains filled beam using a homogeneous model for granular material [START_REF] Chiambaretto | Surrogate granular for modal test of fluid filled tank[END_REF] Details in this scheme will be demonstrated by an example in Section 3.

Radial vibrations of empty tanks, tanks filled with LH2, and tanks filled with grains

Vibration of empty tank

Within our dynamic modelling, the cryogenic tank is considered as a thin shell structure. Donnell -Mushtari's thin shell theory is used in all our analyses in order to determine modal shapes and eigenfrequencies of lobed modes. For illustration, Fig. 2 shows a structural instance of circular cylindrical shells with its coordinate system. Here R is the radius of the circular cylindrical shell, h is the thickness, L is the length, θ is the angle with respect to the vertical axis, z is the axis along its length, and finally u, v, and w are the displacements in the z, θ, and r directions, respectively, r is the distance from the middle surface to an arbitrary point on the shell.

According to Leissa [4], the generalized modal mass and stiffness of such a cylindrical shell can be expressed respectively as kinetic and potential energies:

= + + M u v w Rd dzdr ( ) ij T L h ij ij ij 0 0 2 2 2 2 2 h 2
(2)

= + + K Rd dzdr ( ) ij T L h z z z z 0 0 2 2 h 2 (3)
Three components of displacement field are defined as follows:

= u A cos i j z L ( )cos( ) ij ij (4) = v B sin i j z L ( )sin( ) ij ij (5) = w C cos i j z L ( )sin( ) ij ij (6)
Here, i denotes the lobes following the circumference and j denotes the lobes in longitudinal direction. Parameters A, B and C are identified from equations detailed in the Appendix A.

The strain tensor is defined following Donnell-Mushtari's shell theory [START_REF] Leissa | Vibration of shells[END_REF]

like = u z r w z z 2 2 (7) = + R v w R r R w 1 2 2 2 (8) = + v z R u r R w z 1 2 z 2 (9)
and the stress tensor is determined with the help of Hooke's linear elastic law like

= + E 1 ( ) z z 2 (10) = + E 1 ( ) z 2 (11) = + E 2(1 ) . z z (12)

Fluid -structure interaction

In this work, the tank is assumed fully filled so that sloshing effects can be neglected. In 2D case, each slice of tank can be considered as a circular beam. With the presence of a non-compressible fluid, the mode shapes for each mode remain the same as for the beam without fluid; however the frequency decreases due to the inertia added by fluid [START_REF] Axisa | Modelling of mechanical systems: fluid-structure interaction[END_REF][START_REF] Paidoussis | Fluid Structure interactions: slender structures and axial flow[END_REF][START_REF] Paidoussis | Fluid Structure interactions: slender structures and axial flow[END_REF]. This phenomenon can be decreased by an added mass due to the fluid on each mode. In the case of lobed modes, this added mass by unit of length is defined by:

= M F z R i ( ( )) a D F ij w F (2 ) 2 2 (13)
Here, = ( )

F z C ( ) . sin ij w j z L
denotes the amplitude of radial motion which depends on longitudinal position of 2D tank's slice. It should be noted that F z ( ) ij w has already been described in the previous section as the radial displacement of tank.

The added mass by fluid in 3D case can be considered as the integration of 2D slice along its length due to the absence of viscosity

= = M M F z dz C j z L R i dz . ( ( ) . sin( ) . ij F L a D F ij w L F 0 (2 ) 2 0 2 2 (14)
Each eigenfrequency of whole TFM system can be found then according to the following equation

= + f K M M 1 2 ij TF ij T ij T ij F ( 15
)
where

- + M M ij T ij
F : effective modal mass for the mode ij of the TFM system.

-K ij T : effective modal stiffness for that mode ij.

Interaction between granular solid and tank in the vibration analysis

We now consider the granular assembly as a continuum solid whose important mechanical properties such as Young's modulus, Poisson's ratio and mass density are determined with the help of a homogenization model.

Following the previous research by Chiambaretto et al. [START_REF] Chiambaretto | Surrogate granular for modal test of fluid filled tank[END_REF], in 2D case, the vibration of homogenized granular material surrounded by a circular beam is considered as in-plane vibration of a membrane. Important mechanical properties of a "granular-membrane" are the Young's modulus, the Poisson's ratio and the mass density denoted respectively as (E , , G G G ). The modal mass and the modal stiffness of the "granular-

= + M R U U d ( ( ) ( )) i G D M G r (2 ) 0 1 ( ) 2 2 2
(16)

= + + + + + + K E U U U i U U i U i U U U d 1 2 1 2 i G D r r r r r t t (2 ) 0 1 2 2 2 2 (17)
with U r , U , U r , U are defined in [START_REF] Chiambaretto | Surrogate granular for modal test of fluid filled tank[END_REF]. In 3D, mechanical properties in vibration are supposed to be considered as integration of 2D slice along the length of tank with a sinus function.

The modal mass and modal stiffness of solids inside the tank are determined by:

= + M R U z U z d dz ( ( , ) ( , )) ij G D L M G r (3 ) 0 0 1 ( ) 2 2 2
(18)

= + + + + + + K E U U U i U U i U i U U U d dz 1 2 1 2 ij G D L r r r r r t t (3 ) 0 0 1 2 2 2 2 (19)
with

- = ( ) U r z U r sin ( , ) ( ) r r j z L - = ( ) U r z U r sin ( , ) ( ) r j z L - = ( ) U r z U r sin ( , ) ( ) r r j z L - = ( ) U r z U r sin ( , ) ( ) j z L -= r R .
The 2D-case displacement field is detailed in Appendix B.

In those formulas, the superscripted character « » denotes the differential of the displacements with respect to radial direction of membrane.

Finally, the first frequencies of the TGM system are determined as follows:

= + + f K K M M 1 2 ij TG ij T ij G ij T ij G ( 20
)
where

- + M M ij T ij
G : effective modal mass for the mode ij; of the TGM system

- + K K ij T ij
G : effective modal stiffness for that mode ij.

Vibration of tank in flexion mode

To characterize the effective flexural modes of the empty tank as well as the TGM and TFM systems, the flexural modes are modeled as a Timoshenko's beam. Each system is modeled as a clamped -free beam which carries a finite mass at the free end as illustrated on Fig. 3.

In general, the eigenfrequencies and eigen-mode shapes along the longitudinal direction of the beam are determined from differential equations proposed by Brutch and Mitchell [START_REF] Leissa | Vibration of shells[END_REF]

+ = EI z KAG y z I t 0 2 2 2 2 (21) = A y t KAG y z z 0 2 2 2 2 (22)
and boundary conditions that are specified in Appendix C. These equations involve notably the following material and geometrical constants:

-A: cross sectional area of beam (m 2 ) -E: Young's modulus of beam (Pa) -G: shear modulus (Pa). 1) with as the Poisson's ratio.

= + G E 2(
-I: moment of inertia (m 4 ).

-L: length of beam (m).

-Y: deflection of beam's central line -: slope due to bending.

-K: factor depending on shape of cross -section.

The solutions to the differential equations given in Eqs. ( 24) and (25) are found to be like

= y Y z t ( )cos( ) (23) = z t ( )cos( ) (24)
with the modal shapes

= + + + Y z C b z L C b z L C b z L C b z L ( ) cosh sinh cos sin( ) 1 2 3 4
(25)

= + + + z C b z L C b z L C b z L C b z L ( ) ' sinh ' cosh ' sin ' cos( ) 1 2 3 4 (26)
Here, the following non -dimensional variable is introduced

= b L EI A 1 2 (27)
The related natural frequencies of beam are found by solving the ensuing equation

+ + + + = s R R R R R R R R s R R R R R R R R ( ' ' ' ) ( ' ' ' ) 0 2 2 3 4 3 4 4 ' 1 1 4 2 2 2 3 2 3 1 ' 2 2 1 (28) the parameters , , R 1 , R 2 , R 3 , R 4 , R' 1 , R' 2 , R' 3 , R' 4 being also specified in the Appendix (A3).
In the application for the TFM (i.e. tank-fluid) and TGM (i.e. tankgranular) systems, the Eqs. ( 22) and (23) mainly require explicit values for the productsEI , A, KAG and I but not necessary for E I , , A , , K G , .

In the case of the tank fully filled with fluids, we suppose that the embedded fluid does not contribute to the global stiffness of the TFM system but to the global inertia. Thus, the following mechanical properties are calculated

= EI E R R . 4 ( ) T ext in 4 4
(29)

= + A R R R . ( ) . ( ) T ext in f in 2 2 2 (30) = M L A TF (31)
Fig. 3. Clamped -free beam with a mass at the end.

In case of the tank filled with grains, we suppose that the embedded grains contributes to both the global stiffness and inertia of the TGM, what implies the following mechanical properties

= + EI E R R E R . 4 ( ) . 4 ( ) T ext in G in 4 4 4 (32) = + A R R R . 2 ( ) . 2 ( ) T ext in G in 2 2 2 (33) = M L A TG (34)
Besides, the term KAG in Eq. ( 22) is substituted by:

= + KAG K A G K A G T T T G G G (35)
with:

K T :shear factor for the section of tank (in form of a hollow circle) [START_REF] Han | Dynamics of transversely vibrating beams using four engineering theory[END_REF] =

+ + + + + + K m m m 6(1 )(1 ) (7 6 )(1 ) ( 20 12
)

T T T T 2 2 2 2 2 (36) where = m R R in ext
and T is the Poisson's ratio of the tank material K G : shear factor for the section de granular solid (in form of a circle) [START_REF] Han | Dynamics of transversely vibrating beams using four engineering theory[END_REF]:

= + + K 6(1 ) 7 G G G (37)
where G is the Poisson's ratio of the homogenized granular material -A T : cross sectional area of the tank -A G : cross sectional area of granular solid inside the tank -G T : shear modulus of the tank's constitutive material -G G : shear modulus of the homogenized granular material.

Those formulas will be used in Section 3.3 to characterize the eigenfrequencies that are compared with numerical simulations of the tank filled with different liquids, including LH2, or grains.

Generality on the homogenization model

As we can infer from the mathematical formulations mentioned in Sections 2.2 and 2.3, the eigenfrequencies of TGM system depend on both the overall stiffness (ie. the global Young's modulus and Poisson's ratio) and the mass density of the homogenized granular assembly. Thus, it is important to select a suitable homogenization process that relates the micro-mechanical properties of the grains to the required properties.

In general, a sample of granular materials contains a lot of grains with different sizes, morphologies and stiffnesses. The global behavior of the granular sample depends on the properties and arrangement of these grains in the system. The interaction between two grains of the granular system is defined by contact laws. In the previous works by [START_REF] Chiambaretto | Surrogate granular for modal test of fluid filled tank[END_REF][START_REF] Grippon | Guideline of a metamaterial for launcher tanks dynamic experiments[END_REF], the contact law proposed by Hertz and Mindlin [START_REF] Hertz | Uber die berührung fester elastischekorper (On Contact Between Elastic Bodies)[END_REF][START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF] was used in a DEM simulation in order to model a laboratory test, whilst, in our modeling, global mechanical properties of the granular system are required. In literature, there are several homogenization approaches in order to estimate the macroscopic elastic mechanical properties of granular assembly. In this paper, we use an approach proposed by Emeriault and Cambou [START_REF] Emeriault | Micromechanical modeling of anisotropic non-linear elastic of Granular medium[END_REF].Their formulas for the global Poisson's ratio and Young's modulus are determined from the shear modulus and Poisson's ratio of the grains like (38) -D ¯is the average diameter of grains -N is the average number of contacts in one unity of volume, this operator is defined by the formula

= + + µ µ µ µ 2(1 )( 5
= + E E µ µ 1 2(1 )(1 ) ( 5 4
= N Z n V . 2
with n is the total number of grains, V is the volume of granular assembly, Z is the coordination number defined as the average number of contact per particle. This parameter characterizes the local arrangement of grains in the system. Many studies in literature show that there is a relation between the coordination number Z and the void ratio e of a given granular materials [START_REF] Chang | Initial moduli of particulated mass with frictional contacts[END_REF]. Some empirical models were deduced to relate these two important parameters [START_REF] Smith | Packing of homogeneous spheres[END_REF][START_REF] Yanagisawa | Influence of void ratio and stress condition on the dynamic shear modulus of granular media[END_REF], In this paper, we attempt to use the model proposed by Field [START_REF] Field | Towards the statistical definition of a granular mass[END_REF]:

= + Z e 12 1 (41)
-µrepresents the fraction of the deviatoric stress tensor supported by the normal components of contact forces. This parameter may vary from 0 to 1 and is identified by curve-fitting technique with an oedometric test.

Analytical results and comparison with numerical simulations

Modeling the oedometric test for the evolution of Young's modulus and mass density

The TGM coupling needs to specify the effective mechanical properties of granular materials. These properties are identified from an oedometric test. In this section, we attempt to model the oedometric test performed by [START_REF] Chiambaretto | Surrogate granular for modal test of fluid filled tank[END_REF] with the homogenization formulas proposed in Section 2.4. The dimension and mechanical properties of grains related to this test are provided in the Appendix D. By supposing that the Poisson's ratio of homogenized media is equal to 0.275 we found by curve-fitting technique that = µ 0.3. The modeling and experimental results for the behavior of granular materials in terms of axial stress and strain are presented in Fig. 4.

During the oedometric loading, the mass density of granular material increases, given that the total mass of solid grains is constant and the total volume of whole sample decreases. The Young's modulus of granular media depends not only on grain stiffness but also on mean pressure applied on material; this relation is described with formulas (39)-(41). Thus, during a compression test, Young's modulus of material increases. These evolutions of mass density and Young's modulus in terms of axial stress are illustrated in Fig. 5a andb.

By observing Fig. 5b, we could also deduce that Young' modulus of the whole granular assembly evolves enormously during compression. In detail, this mechanical property increases from 0 to 10 Mpa when the axial stress is increased progressively from 0 to 25 kPa. By contrast, we observe on Fig. 5a that the mass density of material remains almost constant during the oedometric test since this property increases only 0.2% from its initial value (from 351.5 kg/m 3 to 352.5 kg/m 3 ). By virtue of these facts, we can conclude that the eigenfrequency for each mode of vibration of the TGM system increases during the compression. This idea is essential for a new methodology to substitute LH2 in vibration analyses which will be presented in the forthcoming sections.

Analytical results and comparison with numerical simulations for the radial vibrations

This section presents the analytical results and comparison with numerical simulations done with the help of COMSOL for the different couplings, ie. TFM and TGM systems, and for the different lobed-and flexural-modes.

Analytical results of the vibration analysis for the TFM system

In this section, we attempt to find the eigenfrequencies of tank in radial vibration (lobed modes). The considered tank is a polycarbonate cylinder with an external radius of 100 mm, a thickness of 3 mm and 700 mm of length. Polycarbonate was chosen for its transparency allowing an external determination of the packing fraction. Geometry and mechanical properties of tanks are described in Table 1. The upper bound of tank is covered with an aluminium disk with a thickness of 1 cm, total weight of the disk is 0.8482 kg.

All our forthcoming analyses suppose that the elastic limit of considering granular material is found at 150 kPa. In reality, compression tests need to be realized in order to find this yield.

The tank is assumed to be fully filled with different fluids including LH2.

The analytical results were found and compared with numerical results done with the help of COMSOL [START_REF]Comsol user's manual[END_REF]. Two finite element method (FEM) models have been realized using 2nd order tetrahedral elements. The first one models the empty tank and the second one models the tank fully filled with fluid system. The tank is clamped at the bottom. Geometry and meshing for TFM are shown in the Fig. 6.

In the fluid domain the variable used is the pressure. The three equations govern the behaviour and the coupling [START_REF] Axisa | Modelling of mechanical systems: fluid-structure interaction[END_REF][START_REF] Paidoussis | Fluid Structure interactions: slender structures and axial flow[END_REF][START_REF] Paidoussis | Fluid Structure interactions: slender structures and axial flow[END_REF][START_REF] Morand | Fluid -structure interaction[END_REF]. .

F 2
In the numerical model of the coupling system, the number of degrees of freedom is 137316. The eigenfrequency values of this system are archived by natural frequency analysis in COMSOL. Monolithic method is used to solve the fluid -structure interaction problem. A good agreement between analytical and numerical results was found. It shows the suitability of the analytical tool for the shell theory used and hypothesis for interaction between fluid and structure. The observed difference between analytical and numerical results comes from the shell theory (in this case, it is Donnell-Mushtari's theory), however, more complex theories may give a better agreement between two approaches.

Analytical results of vibration analysis and comparison to numerical simulation for TGM system

By applying the proposed methodology in Section 2.1 and analytical formulations in Sections 2.2 and 2.4, we can determine the required pre -stresses applied to granular assembly in order to substitute LH2 in vibration analysis. The required axial pre-stress for each mode is determined as intersection of the frequency -axial stress curve of the TGM system and the expected frequency of the TFM system. Here are examples for the modes (i, j) = (2, 1), (2, 2) and [START_REF] Ivey | Mated vertical ground vibration test[END_REF][START_REF] Chiambaretto | Surrogate granular for modal test of fluid filled tank[END_REF].

By observing Fig. 7, we could deduce that we need different values of pre -stress in order to substitute a same fluid (LH2 in this case) for different vibration lobed modes.

Once axial stress is found, Young's modulus and mass density of assembly are easily determined from mechanical characteristic curves presented in Fig. 5.

The required axial pre-stress and the mechanical properties for TGM system to substitute the LH2 are shown in Table 3. The analytical results are also compared with numerical results with the help of COMSOL. In FEM simulation, a same model as the one in Section 3.2.1 is built with one difference, the homogenized granular solid is filled in the tank instead of fluids. Both the tank and the granular media are meshed with 2nd order tetrahedral elements using displacements as variables. Only the material properties are different.

The results in Table 3 show the suitability and potential of the proposed methodology for lobed modes. Indeed, we obtain good agreement between the numerical and analytical approaches. More specifically, the numerical results for the TFM and TGM systems agree with the related analytical results. Moreover, by using the Young's modulus (E G ) and mass density ( G ) of homogenized solid from Table 3, we found the good agreement between TFM and TGM systems for different lobed modes in the numerical simulations.

Analytical results and comparison with numerical simulation for the vibrations in longitudinal direction

As discussed in Section 2.4, Young's modulus of granular media evolves enormously during a compression test, hence, the eigenfrequency is increased for each vibration mode. Based on this idea, a methodology similar to the one mentioned in Section 2.1 was proposed to find the required pre-stress that allows substituting the LH2 in vibration analyses.

By applying this strategy, we found that the testing material is too heavy to be able to substitute the LH2 in such a vibration analysis. For example, the eigenfrequency of a tank filled with LH2 is 58.79 Hz for flexural mode 1, however, the maximal frequency of a tank filled with the grains used to run the in-lab experiment is only 43.5 Hz at 90 kPa (See Fig. 8).

This result shows evidence that a lighter material is necessary for flexural modes analysis. In order to analyze and verify the proposed methodology, a "new pseudo-material" with density of 150 kg/m 3 is supposed to take place in the tank. As we can observe on Fig. 9, different values of pre stress needed to be applied on granular materials in order to substitute LH2 for different flexural modes in vibration analysis. For the mode 1, an axial stress of 59,000 Pa is required, however, a smaller value (equal to 23,000 Pa) is necessary for the mode 2.

In Table 4, analytical and numerical results were compared for the first 4 vibration modes. One observes a good agreement between analytical and numerical results illustrating the suitability of the proposed methodology. It is important to notice that the axial stress applied on granular assembly increases enormously from 0 to 90 kPa, however, frequency of the TGM system evolves very slightly. For example, for mode 1, frequency increases 2.6% (from 57.5 Hz to 59 Hz), for mode 2, the frequency increases 1% (from 295 Hz to 298 Hz). This result suggests that, for flexion modes, it is possible to use mechanical properties of one particular mode to study other modes. In other words, this highlights the fact that only one value of pre stress is necessary for vibration analysis. This idea is demonstrated with the following example where the axial stress, Young's modulus and mass density necessary for mode 1 were used to determine the eigenfrequencies of the modes 2, 3 and 4. By comparing analytical results with the COMSOL numerical simulation in Table 5, we could conclude that only one pre axial stress is required to analyze all flexural vibration modes of cryogenic tank filled with the liquid fuel. This result could explain that, for flexural modes, the mass density of granular assembly contributes a very important role to the global mass of the whole system; on the other hand, the stiffness of grains plays a less important role in the global stiffness of the analyzed system.

We define the contribution of the granular solid system to vibration system in terms of stiffness and mass density as follows:

= + P E I E I E I (%) G E G G G G T T (42) = + P A A A (%) G G G G G T T (43) with:
-E I T T , , A , T T are respectively Young's modulus, inertia moment, mass density and cross section of tank.

-E I G G , ,
A , G G are respectively Young's modulus, inertia moment, mass density and cross section of homogenized granular solid inside the tank.

Table 3

Comparison between analytical and numerical results for tank filled with granular material for lobed modes. Fig. 10a andb show the evolution of the contribution of the granular solid to the global stiffness and mass. As we can observe, during a compression to 90 kPa, the granular assembly contributes only up to 3.5% of the global stiffness; however, its contribution to the global mass density (per unit length) is always around 54%. This explains why in flexural vibration analysis, we should focus on the mass density of the grains rather than their stiffness.

Conclusion and perspectives

In this paper, we investigated the potential applicability of a new methodology in aerospace engineering, which may allow the substitution of Liquid Hydrogen in vibration analyses by surrogate granular materials. In general, the mass density and Young's modulus of granular materials evolve during compression, especially for the latter one. For our purpose, mechanical properties of granular materials were determined in our work by a homogenization process. Then, analyses on the vibrations of an empty tank, a tank filled with LH2 and a tank filled with a granular-based material solid were realized.

For the vibration in the tank radial direction (lobed modes), Donnell-Mushtari's shell theory was used in order to find the eigenfrequencies and mode shapes of the vibrating tank. With the presence of a non -compressible fluid such as LH2, the mode shapes remain the same. However, the eigenfrequency of each vibration mode decreases due to the increase in inertia. The vibrations of the homogenized granular material were modeled as the integration of 2D membrane along its length with a shape function. Based on the enormous evolution of the frequencies of the TGM system, we were able to propose an original methodology to substitute fuel liquid in vibration analysis. In this new method, it is necessary to compute the evolution of the eigenfrequency for each mode during compression and find the required pre axial stress which gives the intersection between frequency -stress curve given by TGM system and the expected frequency given by the TFM system. To summarize, for lobed modes, we need different pre stresses for different modes in order to substitute a fluid in vibration analysis. For the flexural modes, Timoshenko's beam theory was used in order to determine the eigenfrequency of each vibration mode. The same methodology as the previous one was proposed to find the required pre stress to substitute the LH2 in vibration analyses. Contrary to the lobed mode case, we only need one value of pre stress in order to all flexural modes. This result was obtained due to the important contribution of granular mass density to global modal mass of the whole system. Analytical results were compared to numerical results simulated with the help of COMSOL. Good agreements between two approaches were found and showed the suitability and potential of the proposed methodology.

This paper also gives an idea of how to choose a surrogate granular material to substitute LH2 in vibration analysis. Based on results presented in Section 3, a material with low mass density should be chosen in order to achieve the flexural modes because eigenfrequencies of these modes depend mostly on mass density of the surrogate materials. Then, the lobed modes can be found by increasing the pre -stresses applied on granular materials.

In the future, it is important to conduct two types of experiment.

The first one focuses on static compression test in order to determine the elastic limit of one particular granular material. The second one focuses on vibration test with tank filled with granular grains. This experiment is important to show agreement between reality, and the analytical and numerical methods.

B. Displacement field for 2D membrane in "in-plane" vibration

The components of displacement field and their derivation with respect to the radial direction for a 2D membrane in "in-plane" vibration (Section 2.2.3) were proposed by Kim et al [START_REF] Kim | Exact solutions of in-plane natural vibration of a circular plate with outer edge restrained elastically[END_REF].

= + + + + U A J J A J c J c 1 2 { ( ( ) ( )) ( ( ) ( ))} r i i i i 1 1 1 2 1 1
(B1)

= + + + + U A J J A J c J c 1 2 { ( ( ) ( )) ( ( ) ( ))} i i i i 1 1 1 2 1 1 (B2) = + + + U A J J J c A J c J c 1 4 { [2 ( ) ( ) ( )] [ ( ) ( )]} r i i i i i 1 2 2 2 2 2
(B3)

= + + + U A J J c A J c J c J c 1 4 { [ ( ) ( )] [ 2 ( ) ( ) ( )
]}.

r i i i i i 1 2 2 2 2 2
(B4)

Here A 1 and A 2 are two variables that are linked by the equation: (B9)

= D D A A D D
In those equations, c is the variable which depends on the Poisson's ratio of the homogenized granular material by the following equation:

= c 2 1 G (B10)
and is the variable which depends on the mechanical properties and the vibration frequency of the homogenized granular material by following equation

= f E 2 (1 ) G G 2 (B11)
The vibration frequency of each mode is found by solving the characteristic equation:

+ + = + + + J J c J c J c J J c J c 2 ( ) ( ) 2 ( ) ( ) 2( 1) ( ){ ( ) ( )} 0 i i i i i i i 1 2 2 2 2 2 2 (B12)

Fig. 2 .

 2 Fig. 2. Geometry of a 3D shell.

  -m , G m are respectively Poisson's ratio and shear modulus of the grains -M is the mean stress defined by = tr ( )

Fig. 4 .

 4 Fig. 4. Experimental and analytical results modeled by proposed homogenization law.

  the fluid domain on the boundary with the structure 0 .

Fig. 5 .

 5 Fig. 5. Evolutions of (a) the mass density and (b) the Young's modulus of granular assembly during the compression test.

Fig. 6 .

 6 Fig. 6. Numerical model in COMSOL for tank filled with fluid (TFM) system.

Fig. 9a and

  b show two other examples for the flexion modes 1 and 2.

Fig. 7 .

 7 Fig. 7. Evolution of eigen frequencies with respect to axial stress -determination of required axial pre-stress for lobed modes, with:(a) Mode (i, j) = (2,1) (b) Mode (i, j) = (2,2) (c) Mode(i, j) = (2,3).

Fig. 9 .

 9 Fig. 9. Evolution the eigen frequencies with respect to axial stress -determination of required axial pre-stress for flexural modes, with: (a) Mode 1 and (b) Mode 2.

Fig. 8 .

 8 Fig.8. Evolution of the eigenfrequency with respect to axial stress for the flexural mode 1, using the testing material.

1

  for all (i, j), meanwhile the value of A/C varies with different mode shapes. It results finally then that the displacement field can be defined as: the parameter A ij being determined for each mode and takes the values presented in TableA.

Fig. 10 .

 10 Fig. 10. Contribution of the granular material to (a) the global stiffness and (b) the global mass during compression.

Table 1

 1 Dimension and mechanical properties of tank.

	External radius	0.1 m
	Internal radius	0.097 m
	Thickness	0.003 m
	Young's modulus	3.3 Gpa
	Poisson's ratio	0.45
	Mass density	1190 kg m -3

Table 2

 2 Comparison between analytical and numerical results for lobed modes (a) of the empty tank and the tank filled with LH2, and (b) the tank filled with other liquids.

			Empty tank			F = 72 (kg m -3 )
			Numerical (Hz)	Analytical (Hz)	(%)	Numerical (Hz)	Analytical (Hz)	(%)
	(a)
	j = 1	i = 2	219	236.98	8.21	165.1	178.10	7.87
		i = 3	232.7	242.9	4.38	184.4	193.40	4.88
		i = 4	393	417.55	6.25	322.7	346.40	7.34
	j = 2	i = 2	475.8	491.01	3.2	362.4	369.23	1.88
		i = 3	348.1	334.66	3.86	278.9	266.57	7.41
		i = 4	436.5	450.45	3.2	360.5	373.77	3.68
	j = 3	i = 2	781.2	792.99	1.51	600.6	596.44	1.03
		i = 3	530.8	510.25	3.87	429.1	406.55	5.26
		i = 4	525.8	530.25	0.85	437.9	440.87	0.68
			F ==2000(kg m -3 )			F =5000(kg m -3 )
			Numerical (Hz)	Analytical (Hz)	(%)	Numerical (Hz)	Analytical (Hz)	(%)
	(b)
	j = 1	i = 2	46.32	50.09	8.14	29.69	32.11	8.16
		i = 3	55.8	58.85	5.47	35.32	37.89	7.29
		i = 4	104.8	113.29	8.1	67.78	73.29	8.61
	j = 2	i = 2	102.9	103.87	0.94	66.06	66.59	0.9
		i = 3	85.05	81.16	4.57	54.77	52.26	4.58
		i = 4	117.7	122.30	3.91	76.11	79.12	3.95
	j = 3	i = 2	172.6	167.83	2.76	110.7	107.60	2.8
		i = 3	132.2	123.83	6.3	85.15	79.74	6.35
		i = 4	144.0	144.06	0.4	93.18	93.20	0.2

Table 4

 4 Comparison between analytical and numerical results for flexural modes, using different pre -stresses.

		Axial stress (Pa)	EG(Pa)	G (kg m -3 )	Numerical TFM f (Hz)	Numerical TGM f (Hz)	Analytical TFM TGM f (Hz)
	Mode 1	5.86 × 10 4	1.341 × 10 7	87.90	59.7	60.00	58.68
	Mode 2	2.3 × 10 4	9.814 × 10 6	87.68	300	288.40	289.25
	Mode 3	2.17 × 10 4	9.581 × 10 6	87.67	664	614.20	639.01
	Mode 4	900	3.31 × 10 6	87.46	993	942.6 0	971.89

Table 5

 5 Comparison between analytical and numerical results for flexural modes -Using one pre-stress.

		Axial stress (Pa)	EG(Pa)	G (kg m -3 )	Numerical TFM f (Hz)	Numerical TGM f (Hz)	Analytical TGM TFM f (Hz)
	Mode 1	5.86 × 10 4	1.341 × 10 7	87.9	59.7		58.67
	Mode 2	5.86 × 10 4	1.341 × 10 7	87.9	300	293.4	291.91
	Mode 3	5.86 × 10 4	1.341 × 10 7	87.9	664	631.6	646.16
	Mode 4	5.86 × 10 4	1.341 × 10 7	87.9	993	922.7	1005.1

Table D

 D Geometric and mechanical mean properties of the grains.

	Number of grains	35 000
	Radius	1.665 mm
	Specific weight	6.03.10 -4 g mm -3
	Em	600 MPa

m 0.015

Acknowledgement

This research is funded by the Région Midi -Pyrénées, France (research grant FEDER 2014 -2020). The authors acknowledge the Région Midi-Pyrénées of France for the permission to publish this work. We gratefully thank also the anonymous reviewers for their valuable suggestions and comments which have help us improving this paper.

Appendix A. Identification of parameters A, B, C for Donnell-Mushtafi's shell theory

Three parameters of Donnell-Mushtari's theory [START_REF] Leissa | Vibration of shells[END_REF] in the Section 2.1.1 are determined from a system of equations mentioned below. By solving these equations, the ratios between the three parameters (A, B, C) can be found from

which relies on the following constant parameters

Appendix C. Variables of Timoshenko's beam theory

The parameters , , R

3 are calculated with the following expressions [START_REF] Bruch | Vibrations of a mass -loaded clamped-free Timoshenko beam[END_REF]:

Table A

Values of parameter A for different lobed modes.

The initial boundary value problems for a homogeneous and linear elastic beam that is clamped at its left extremity and loaded by a mass M at its right extremity is:

2 (k: gyration radius).

Appendix D. Dimension and mechanical properties of granular grains

The sample in the Section 3.1 contains 35 000 grains made of epoxy with mineral charges. The radius of all grains is around 1.665 mm, the specific weight of each grain is 6.03.10 -4 g/mm 3 . The Young's modulus (E m ) and Poisson's ratio ( m ) are respectively equal to 600 Mpa and 0.015. The method used to identify the mechanical properties of each grain is detailed in [START_REF] Grippon | Guideline of a metamaterial for launcher tanks dynamic experiments[END_REF]. Details of grain properties are shown in the Table D.


RM, MPH a , Chloé Barasinski
email: cbarasinski@chu-clermontferrand.fr

MD, MPH a,b , Didier Anne Debost-Legrand

MD Lemery

MD, PhD Françoise Vendittelli

Practices during the active second stage of labor: a survey of French midwives

Keywords: delivery position, manual perineal support, midwifery practice, perineal massage, pushing second stage, second stage of labor

Objective:

The principal objective of our study was to describe the practices reported by French midwives during the active second stage of labor (expulsion phase).

Design: This cross-sectional Internet survey questioned French midwives who attended at least one delivery in 2013.

Setting: This open survey was posted on a website from June 15 through December 1, 2014.

Participants: 1496 midwives from 377 maternity units participated in the study.

Measurements and findings:

The midwives most often reported suggesting horizontal positions during the active second stage (supine with footholds, lithotomy, lithotomy with knees turned in, or lateral positions). Non-horizontal positions were more often proposed by midwives in level I units (p<0.0001). Almost half the midwives responding (46.4%), especially those working in level III units (51.1%, p = 0.006), advised Valsalva pushing. The mean maximum pushing time was 35.3 min ± 12.8 min. Nearly all the midwives favored the "hands on" technique at delivery (91.4%), and 24% reported using warm compresses on the perineum at delivery.

Key conclusion: Most midwives advised horizontal positions for delivery. The practices of

French midwives differed as a function of where they worked. The midwives, especially those in level III facilities, reported that they cannot always ensure "physiological childbirth".

Implications for practice:

The practices of French midwives must become more evidencebased. The development of professional guidelines for midwives in France appears essential.

This study also helps to prioritize national training for midwives.

Introduction

France has the highest birthrate in Europe, with around 800,000 babies born each year [START_REF] Bellamy | Bilan démographique 2015 : le nombre de décès au plus haut depuis l'après-guerre[END_REF]. Childbirth is a frequent event and nearly all (99.1%) of deliveries occur in hospitals (INSEE, 2014). In 2013, there were 519 maternity units nationwide (DRESS). Around half of the births (47%) took place in maternity units with 500 to 1999 deliveries a year, 41% in those with at least 2000 deliveries a year and 2.5% in facilities with fewer than 500 deliveries per year (Court of Auditors : Cour des Comptes, 2014). France is one of the European countries where women use epidural analgesia most often. In 2012, for example, 75.0% of women in spontaneous labor had epidurals (Audipog).

A midwife is routinely assigned to support each parturient, in all hospitals, public or private.

Most births result from non-operative vaginal deliveries (66.9%), accompanied most often by a midwife (79.7%) [START_REF] Blondel | Les naissances en 2010 et leur évolution depuis 2003[END_REF].

During the expulsive phase, midwives can advise one or several positions, choose to direct (or coach) the pushing or not, and use different techniques for perineal protection. The positions used during delivery remain controversial [START_REF] Gupta | Position in the second stage of labour for women without epidural anaesthesia[END_REF][START_REF] Kibuka | Position in the second stage of labour for women with epidural anaesthesia[END_REF]. Vertical positions, for example, are often said to reduce the risk of aortocaval compression, improve acid-base outcomes in newborns, enable stronger and more efficient uterine contractions, and facilitate fetal progression through the effects of gravity and improved pelvic alignment [START_REF] Gupta | Position in the second stage of labour for women without epidural anaesthesia[END_REF]. Nonetheless, some vertical positions may create an excess risk of severe perineal lacerations [START_REF] Elvander | Birth position and obstetric anal sphincter injury: a population-based study of 113 000 spontaneous births[END_REF], which increase the volume of postpartum blood loss and thus the risk of an immediate postpartum hemorrhage [START_REF] De Jonge | Increased blood loss in upright birthing positions originates from perineal damage[END_REF][START_REF] Vendittelli | Le type de poussée et la position maternelle pendant le 2e stade du travail ont-ils un impact sur les issues obstétricales ou néonatales?[END_REF]. An Australian study compared 9 maternal positions for delivery and found the semi-recumbent position most closely associated with perineal tears or lacerations (OR 1.16, 95% CI 1.01-1.33) [START_REF] Soong | Maternal Position at Midwife-Attended Birth and Perineal Trauma: Is There an Association?[END_REF]. In another study, the lithotomy, compared with the sitting, position was found to be associated more often than other positions with obstetric anal sphincter injuries (OASIS) in nulliparous (aRR 1.17, 95% CI 1.06-1.29) and parous women (aRR 1.66, 95% CI 1.35-2.05) [START_REF] Elvander | Birth position and obstetric anal sphincter injury: a population-based study of 113 000 spontaneous births[END_REF]. At the same time, there are different techniques that midwives can use to reduce these tears and lacerations. The metaanalysis by [START_REF] Aasheim | Perineal techniques during the second stage of labour for reducing perineal trauma[END_REF] concluded that the use of warm compresses during expulsion appear to reduce severe perineal lacerations significantly as does perineal massage during delivery, compared with the "hands-off" technique. Moreover, although hands-off appears to reduce episiotomy rates [START_REF] Aasheim | Perineal techniques during the second stage of labour for reducing perineal trauma[END_REF], the development of a "hands-on" policy in several departments reduced the number of OASIS [START_REF] Hals | A multicenter interventional program to reduce the incidence of anal sphincter tears[END_REF][START_REF] Laine | Incidence of obstetric anal sphincter injuries after training to protect the perineum: cohort study[END_REF], and a meta-analysis of cohort studies also found that hands-on had a strong protective effect against OASIS [START_REF] Bulchandani | Manual perineal support at the time of childbirth: a systematic review and meta-analysis[END_REF]. Thus numerous experts strongly recommend the use of perineal support/hands-on at delivery [START_REF] Ismail | Perineal support and risk of obstetric anal sphincter injuries: a Delphi survey[END_REF]. The type of pushing used at delivery may also affect both the maternal perineum and fetal well-being (Barasinski et al., 2016). The meta-analysis by [START_REF] Lemos | Pushing/bearing down methods for the second stage of labour[END_REF] confirmed the absence of scientific evidence for preferring one pushing technique over another.

There are no French guidelines for physiological childbirth, except for the first set of guidelines issued by the French National College of midwives (CNSF) about oxytocin augmentation [START_REF] Dupont | Oxytocin administration during spontaneous labor: Guidelines for clinical practice. Guidelines short text[END_REF]. Moreover, although studies have examined midwives practices during the intrapartum period [START_REF] Hanson | PRACTICES: Part 1: Position Use and Preferences[END_REF]RCM, 2010), no nationwide survey in France has reported these practices here. Our research hypothesis is that the practices of French midwives differ in some ways from those of their colleagues in other developed countries.

The principal objective of our study was to describe the practices reported by French midwives during the active second stage of labor. Our secondary objective was to assess whether these practices differ according to midwives' experience or the level of the maternity ward where they work (as defined below).

Methods

Design

We conducted a cross-sectional Internet survey of French midwives, both hospital-based and in private practice, who perform deliveries in equipped facilities (i.e., not home births). In our convenience sample, only midwives who attended at least one delivery in 2013 were eligible.

The midwives participating in this study were informed on the survey home page of the time required to complete the survey, their right to withdraw from the study (via a personal number obtained at the end of the study), and the study's objectives. No identifying information was collected. The relevant ethics committee approved this study on March 31, 2014 (CECIC Rhône-Alpes-Auvergne, Grenoble, IRB 00005921).

Development and pretesting of the questionnaire (Supplementary file 1).

The questionnaire covered the different practices known to be used in France, during all three stages of delivery. It was pilot tested by 10 midwives practicing in maternity wards of different levels (I, II, and III), different sizes (<500, ≥500 -≤1500, and >1500 births/year), and different statuses (public or private). Level I units have no neonatology department, while level II units have a department of neonatology in the same building as or in immediate proximity to the delivery suite, and level III units have, in addition to a neonatology unit, a neonatal intensive care in the same building as or in immediate proximity to the delivery room. Accordingly, the higher the hospital level, the more likely it is to deal with high-risk and complicated pregnancies. The online version was tested on different brands of computers and different versions of browser software.

Recruitment process

This open survey was posted on a website from June 15 through December 1, 2014. Midwives were recruited for the survey through the e-mail lists and websites of the national council of midwives and the national college of French midwives. A single reminder e-mail was sent.

Survey administration

The website was exclusively devoted to the survey, and the midwives had direct access to its information page. No incentives were offered for participation in this voluntary survey. Once the midwife agreed to complete the questionnaire, the first question was whether she had attended a delivery in 2013 (the inclusion criterion). If the response was negative, the survey ended.

The survey comprised 46 closed questions, 35 of which are used in this study (Supplementary File 1). Six questions specifically concerned position, six pushing methods, 8 perineal protection and support techniques, and 3 placental delivery. For the part about birthing positions, 8 different positions were proposed to the midwives, each illustrated by a photograph. We chose these positions from the earlier practice surveys [START_REF] Hanson | PRACTICES: Part 1: Position Use and Preferences[END_REF]RCM, 2010) and included several versions of the supine position. The questions were organized according to the chronology of the delivery, with no randomization of item order. We used some adaptive questions to streamline the questionnaire when possible. The items about practices during expulsion were distributed onto 4 separate screens (positions, pushing methods, perineal protection, and perineal support techniques). Practices for the third stage of labor were on another page. The questionnaire ended with five questions to assess the midwives' feelings about facilitating physiological childbirth. All items were mandatory and if one was not completed, the respondent could not go on the next screen. Participants could change their answers throughout the survey via the table of contents to the left of the questionnaire. Because many midwives responded from their workplace, and therefore several midwives used the same computer, software controls to prevent multiple entries from the same individuals could not be used.

Analysis

Only completed questionnaires were analyzed; there was no time limit for their completion.

Our principal outcome was the prevalence of the practices midwives reported. The responses are reported as percentages of all respondents. They were subsequently compared according to maternity unit level and according to the midwives' experience (years of practice) (≤ 5; 6-15; > 15). The Chi2 test (or Fisher's exact test, as appropriate) was used to compare the qualitative variables, and Student's T test for the quantitative variables. The threshold of significance was set at p<0.05. The statistical analyses were performed with SAS software (version 9.4, SAS Institute, Inc, Cary, NC, USA, 2002USA, -2012)).

Findings

Overall, 1550 midwives responded to the questionnaire, but 54 were ineligible for the study (27 midwives in private practice without access to an equipped facility and 27 midwives who qualified only in 2014). These 1496 midwives practiced in 377 maternity units. The participation rate in this study was 30.5% (denominator calculated from the annual national statistics for French healthcare facilities: SAE-https://www.sae-diffusion.sante.gouv.fr/). Table 1 summarizes the respondents' characteristics. They had been in practice for a mean of 9.7 ± 8.1 years: 38.9% for 5 years or less, 40.7% for 6 to 15 years, and 20.4% for more than 15 (p=0.04). The largest percentage of respondents worked in level II maternity wards (44.5% vs. 31% level III and 24.5% level I, p= 0.04).

Birth position

Most midwives reported that they proposed horizontal positions during the active pushing phase: supine with footholds (79.8%), lithotomy (80.0%), lithotomy with knees turned in (86.7%), and lateral one (90%) (Table 2). Positions other than supine were proposed more often by midwives in level I units than those in level II or III facilities (p<0.0001) (Table 2).

Midwives with ≤5 years of experience proposed the lithotomy position and the supine position with footholds more often than the other midwives (respectively, p <0.0001 and p=0.0002) (Table 2). Those with more than 15 years of experience, on the other hand, proposed the lateral position significantly less often than the others (p<0.009). Midwives reported that their preferred positions were lithotomy with knees turned in (35.2%), then lithotomy (28.6%), and next supine with footholds (23.8%). Only 16.2% of the midwives always let the woman choose her birthing position: 21% of those in level I, compared with 16.7% in level II and 11.9% in III units (p=0.005) (Table 3). We note that a quarter of the midwives reported that they never or rarely left the choice of birthing position to the woman (Table 3). Most midwives reported that they most often had women use footholds (56.8%) or stirrups (66.1%) (Table 3). Stirrup use was most frequently reported by level III midwives (p<0.0001) (Table 3). Few midwives used the support bar to position women at delivery (78.3% never or rarely).

Pushing methods

Around a third of the midwives always let women choose the type of pushing, especially those with 6 to 15 years of experience and those with more than 15 years, compared with those with ≤ 5 years (respectively, 31.5% and 29.8% vs. 21%, p <0.0001) (Table 4). Valsalva pushing was advised most often by midwives (46.4%), especially those with ≤ 5 years of experience (59.1%) and those in level III units (51.1%) (Table 4). Nearly all midwives encouraged women during their efforts (94.7%), especially those with ≤ 5 years of experience (96.9%, p = 0.0006) (Table 4). We found a mean maximum pushing time of 35.3 min (± 12.8 min), longer for level-I midwives (36.9 min ± 12.5, p<0.0001) (Figure 1). Midwives with ≤ 5 years of experience reported a mean duration of active pushing shorter than the others (34.2 min ± 9.2, p=0.02) (Figure 2).

Perineal techniques

Half of the midwives reported they did not provide perineal massage during delivery (53.5%), while one third (31.8%) of those with ≤ 5 years of experience did, on both the cutaneous and vaginal sides (p=0.0005) (Table 5). One quarter of the midwives reported using warm compresses on the perineum at delivery; this was significantly more frequent among those in level I units (33.6%, p<0.0001) (Table 5). Most midwives preferred the hands-on technique at delivery (91.4%), especially those in level III units (95.3%, p = 0.0006) and those with ≤ 5 years of experience (95.5%, p<0.0001) (Table 5). Most midwives used Ritgen's maneuver rarely or never (78.3%) (Table 5). It was performed often or most of the time by 25.7% of level-III midwives (p=0.01) and 25.9% of the midwives with more than 15 years of experience (p=0.04) (Table 5). At restitution, the most frequent practice was external rotation of the fetal head (54.4%); exaggerated restitution was reported significantly more often by level-III midwives (p= 0.001) and those with ≤ 5 years of experience (p = 0.002) (Table 5). a Level I = maternity ward without a neonatology department; Level II = maternity ward with a neonatology department; Level III = maternity unit with a neonatology department and a neonatal intensive care unit (NICU).

Third stage of labor

For placental delivery, 75.9% of the midwives said that they changed the woman's initial position; this was more frequent among the youngest midwives (≤ 5 years of experience and between 6 and 15; respectively, 80.9% and 76.8%) than those with more than 15 years of experience (64.3%, p<0.0001). This practice did not differ by maternity ward level. Among those who reported a change in the mother's position, 97.1% recommended a supine position for this phase, especially those in level III units (98.6%, p=0.03).

Facilitating physiological childbirth

Among the respondents, 81.9% of the midwives thought that their labor management often or always ensured physiological childbirth. Three quarters of the midwives (76.7%) working in level III units thought so, versus 82.8% of those in level II units and 86.9% of those in level I (p=0.0006). Overall, only 38.2% of midwives were comfortable with all the maternal positions for fetal expulsion: 48.4% in level I units compared with 38.0% in level II and 30.4% in level III units (p<0.0001). It was also the case for 43% of the midwives with more than 15 years of experience versus 40.4% of those with 6 to 15 years of experience and 33.3% of those with ≤ 5 years (p=0.007).

Discussion

This large cross-sectional study enables us to describe the practices of French midwives, until now little documented, especially for physiological births. Although epidemiologic studies describe these deliveries, their outcomes, and the women's characteristics [START_REF] Blondel | Les naissances en 2010 et leur évolution depuis 2003[END_REF]Audipog), few French studies have focused on midwives' practices. A recent French study conducted in different maternity units and covering only 551 women with spontaneous vaginal deliveries found practices similar to ours, with 86.1% of deliveries in supine (or lithotomy) positions, 3.8% in lateral positions, 0.9% on hands and knees, and 1% upright positions [START_REF] Desseauve | Dans quelle position les femmes accouchent-elles en 2015 ? Résultats d'une étude prospective régionale multicentrique[END_REF]. In our study, 87.6% of midwives reported that they prefer dorsal positions, whether supine with footholds or lithotomy with knees turned in or not. Regular use of stirrups was also reported by 66% of midwives. A Canadian survey in 2006 also found stirrup use in 57% of deliveries [START_REF] Chalmers | Rates of Interventions in Labor and Birth across Canada: Findings of the Canadian Maternity Experiences Survey[END_REF]. Although stirrup use does not appear to be associated with an increase in perineal tears or lacerations [START_REF] Corton | A randomized trial of birthing with and without stirrups[END_REF], other studies have found that the lithotomy position is a risk factor for OASIS, especially in parous women [START_REF] Elvander | Birth position and obstetric anal sphincter injury: a population-based study of 113 000 spontaneous births[END_REF]. Moreover, stirrup use constrains the woman to maintain the same supine position through her active pushing, which can have a harmful effect on fetal oxygenation [START_REF] Carbonne | Maternal position during labor: effects on fetal oxygen saturation measured by pulse oximetry[END_REF]. The National Institute for Health and Care Excellence (NICE) recommends that British physicians discourage women woman from lying in supine or semi-supine positions during the second stage of labor (NICE, 2017).

Nonetheless, the study on birthing positions published by the Royal College of Midwives found that 51% of the women who gave birth normally used a semi-supine position, and only 3% the lithotomy position (RCM, 2010). A Swedish study found a lower percentage of lithotomy positions at delivery, with the rate varying by parity (nulliparous ~25%, parous ~ 10%) [START_REF] Elvander | Birth position and obstetric anal sphincter injury: a population-based study of 113 000 spontaneous births[END_REF]. In the same study, sitting was the position used most often by both nulliparous (38.5%) and parous (41.3%) women [START_REF] Elvander | Birth position and obstetric anal sphincter injury: a population-based study of 113 000 spontaneous births[END_REF]. A Dutch study, on the other hand, found supine positions in 89.7% of deliveries in 2005, both in hospital and at home [START_REF] De Jonge | Are there inequalities in choice of birthing position?[END_REF]. These authors observed different social and demographic factors associated with the use of a non-supine position, including the mother's age (≥ 36 years), a higher education level, and homebirth.

In France, during the second stage, "delayed pushing" is the standard clinical practice. The French National College of Gynecologists-Obstetricians (CNGOF) recommends delayed pushing (2 h after dilation) for women with epidural analgesia, to reduce the number of difficult operative vaginal deliveries (Vayssière et al., 2011). There are, however, currently no guidelines about the type of pushing to be used. In our study, 46.4% of midwives advised closed-glottis pushing, also known as Valsalva pushing (Barasinski et al., 2016). We found very few studies describing techniques used by midwives across the globe. A cross-sectional Japanese study found that Valsalva pushing was used in all cases by 37.9% of hospitals, 50% of clinics, and only 10% of midwifery birth centers [START_REF] Baba | A cross-sectional survey of policies guiding second stage labor in urban Japanese hospitals, clinics and midwifery birth centers[END_REF]). An

American study found that 25.7% of women having a first vaginal birth and 16.3% of parous women who participated in a randomized clinical trial of perineal management techniques used Valsalva pushing [START_REF] Albers | Factors Related to Genital Tract Trauma in Normal Spontaneous Vaginal Births[END_REF]. Another American study found more directive approaches when women have an epidural [START_REF] Osborne | Directive Versus Supportive Approaches Used by Midwives When Providing Care During the Second Stage of Labor[END_REF]. According to the Royal College of Midwives, there is no good evidence to justify the use of directed pushing with Valsalva breathing, and indeed the practice of sustained breath holding in directed pushing may be harmful (RCM, 2012).

The mean maximum duration of pushing in our study was 35.3 minutes. In France, the CNGOF guidelines recommend operative vaginal delivery when the pushing stage exceeds 30 minutes, subject to no anomalies in the fetal heart rate [START_REF] Dupuis | La surveillance foetale pendant l'expulsion[END_REF].

Nonetheless, this policy is not followed internationally (Le [START_REF] Ray | Durée des efforts expulsifs : données de la littérature[END_REF], and durations of active pushing can exceed 4 hours [START_REF] Grobman | Eunice Kennedy Shriver National Institute of Child Health and Human Development (NICHD) Maternal-Fetal Medicine Units (MFMU) Network[END_REF].

Looking at perineal protection techniques, the US study by [START_REF] Albers | Factors Related to Genital Tract Trauma in Normal Spontaneous Vaginal Births[END_REF] found that 34.5% of women with first vaginal births and 31.9% of parous women had warm compresses at delivery, and an Australian survey found that 44.9% of the midwives questioned used these compresses [START_REF] East | Midwives' and doctors' perceptions of their preparation for and practice in managing the perineum in the second stage of labour: a cross-sectional survey[END_REF]. Only 24% of the midwives in our study reported using them at delivery. Our survey showed widespread use of the hands-on technique (91.4% of respondents), although it too appears to be used much less frequently internationally: in Australia, between 37% and 60.9% of midwives use it, depending on the study [START_REF] Ampt | Perineal management techniques among midwives at five hospitals in New South Wales -A cross-sectional survey[END_REF][START_REF] East | Midwives' and doctors' perceptions of their preparation for and practice in managing the perineum in the second stage of labour: a cross-sectional survey[END_REF] and in England, 48.6% [START_REF] Trochez | Hands on or hands off the perineum: a survey of care of the perineum in labour (HOOPS)[END_REF]. The contradictory data in the literature about manual perineal support et al., 2015) is mirrored by the different attitudes professionals have towards this practice. Nonetheless, this common practice in France is associated with a lower rate of severe perineal lacerations (3 rd and 4 rd degree) (0.54% in 2012, Audipog) than in other countries not using this practice (Sweden between 4.2% and 5.7%, Denmark 3.6%, Norway 4.1%, England 3.1%) [START_REF] Ekéus | Increasing incidence of anal sphincter tears among primiparas in Sweden: A population-based register study[END_REF]Birthplace in England Collaborative Group, 2011;[START_REF] Laine | Changing incidence of anal sphincter tears in four Nordic countries through the last decades[END_REF].

Our study shows that practices differ among midwives as a function of their experience but also of their place of work, as previously reported in the literature [START_REF] Wiklund | Swedish midwives' rating of risks during labour progress and their attitudes toward performing intrapartum interventions: a web-based survey[END_REF][START_REF] Zinsser | Midwives' attitudes towards supporting normal labour and birth -A cross-sectional study in South Germany[END_REF]. Different authors have observed more physiological practices in out-ofhospital birth centers [START_REF] Zinsser | Midwives' attitudes towards supporting normal labour and birth -A cross-sectional study in South Germany[END_REF][START_REF] Baba | A cross-sectional survey of policies guiding second stage labor in urban Japanese hospitals, clinics and midwifery birth centers[END_REF]. The Birthplace study found that its large cohort of women planning birth in a midwifery unit and multiparous women planning birth at home experienced fewer interventions than those planning birth in an obstetric unit (Birthplace in England Collaborative Group, 2011). French midwives working in level I units, where only women with low-risk pregnancies can give birth, considered that physiological childbirth took place more often than those in level II and III facilities. We found that these midwives used more varied practices with more non-recumbent birth positions, more openglottis pushing, more spontaneous pushing, and more perineal application of warm compresses. Nonetheless, another study failed to find any difference in the observed rate of interventions (induction of labor, cesarean section, operative vaginal delivery, and episiotomy) in France according to hospital level. Instead, it found that intervention rates appear to depend on maternity unit status -public or private [START_REF] Coulm | Obstetric Interventions for Low-Risk Pregnant Women in France: Do Maternity Unit Characteristics Make a Difference[END_REF].

Nonetheless, the practices they examined were not, as those in our study were, specific to midwifery; indeed, theirs were more specific to obstetricians. Some countries have analyzed their optimality of their practices according to current evidence to identify the optimal processes of care and the clinical outcomes achieved in relation to background risk factors [START_REF] Sheridan | Measuring the best outcome for the least intervention: can the Optimality Index-US be applied in the UK?[END_REF]. Developed by [START_REF] Wiegers | An approach to measuring quality of midwifery care[END_REF], the "Optimality Index" seems to be a useful approach to studies involving midwifery care (American College of Nurse-Midwives, 2016). It would be interesting to use this process in France by adapting this tool, or to develop an international tool for the comparison of background risk and outcomes across a range of care settings [START_REF] Sheridan | Measuring the best outcome for the least intervention: can the Optimality Index-US be applied in the UK?[END_REF].

Strengths and Limitations

Our low participation rate (30.5%) [START_REF] Eysenbach | Improving the Quality of Web Surveys: The Checklist for Reporting Results of Internet E-Surveys (CHERRIES)[END_REF] was nonetheless an underestimate because the exact number of midwives participating in deliveries in each maternity ward in France is unavailable. We used the national database of healthcare facility statistics (SAE), which reports the number of midwives practicing in maternity units, regardless of their actual work in those units. Some midwives, however, no longer work in the delivery room (eg, they may do prenatal and postnatal care, or administration), which artificially diminishes our participation rate. Our participation rate is nonetheless good for a survey of practices among the general population of midwives, since response rates in other general population-based studies have been less than 15% [START_REF] Arrish | Australian midwives and provision of nutrition education during pregnancy: A cross sectional survey of nutrition knowledge, attitudes, and confidence[END_REF][START_REF] George | Measuring oral health during pregnancy: sensitivity and specificity of a maternal oral screening (MOS) tool[END_REF]RCM, 2010). We could not define a target sample to determine a representative sample of the population of French midwives involved in the delivery of babies because there is no national register of all

French midwives that includes their professional characteristics. Moreover, French midwives are not required to join the French National College of French Midwives (CNSF). Another weakness of our study is its reliance on midwives' self-report. That is, professionals do not always do what they say they do. In 2013 France had 519 maternity units (DRESS); accordingly, respondents reported practices in 72.7% of French public and private maternity units.

In conclusion, the practices reported by French midwives are not always consistent with the scientific literature or with a physiological approach to birth. These practices vary as a function of the midwife's experience and the level of the maternity ward where she works.

The absence of professional guidelines for midwives in France surely plays a role in this finding, especially since the scientific literature is written almost exclusively in English which is not read by most of the French midwives. Moreover, although research in midwifery is highly developed in some countries [START_REF] Luyben | The current state of midwifery and development of midwifery research in four European countries[END_REF], it is only just beginning in France.

Thus, in 2010, only 10 midwives in France had a PhD. Studies of midwifery care have recently begun to appear (Le [START_REF] Ray | Lateral asymmetric decubitus position for the rotation of occipito-posterior positions: multicenter randomized controlled trial EVADELA[END_REF]Barasinski and Vendittelli, 2016), however, and are the beginning of the development of a culture of research and the appropriation of evidence-based midwifery by French midwives. A better description of midwives' practices in all countries will help us to understand how they affect the health of women and newborns by comparing national maternal and neonatal indicators. This study is useful to the development of a qualitative research project that would allow us to identify the obstacles to and facilitating factors for the appropriation of international evidence-based practices by French midwives.

Table 1 -

 1 Characteristics of responding midwives, globally, by maternity unit levels, and by midwives' experience

		Global
	Characteristics		Results by maternity unit level		Results by experience
		results
		Total	Level I a	Level II a	Level III a		≤ 5 years	6-15 years	> 15 years
		N= 1496	N= 366	N= 666	N= 464		N= 582	N= 609	N= 305
						p				p
		%	%	%	%		%	%	%
		[Mean ± SD]	[Mean±SD]	[Mean±SD]	[Mean±SD]		[Mean±SD]	[Mean±SD]	[Mean±SD]
	Age	[33.4 ± 7.9] [34.4 ± 8.5] [33.3 ± 8.0] [32.7 ± 7.3]	0.009	[26.8 ± 2.1] [33.1 ± 3.2] [46.3 ± 5.6] <0.0001
	Sex (women)	95.7	95.9	95.4	95.9	0.87	93.3	96.1	99.3	<0.0001
	Type of practice
	Hospital	92.9	22.3	45.1	32.6	<0.0001	39.3	41.3	19.4	0.007
	Mixed	5.9	52.3	36.4	11.4		34.1	35.2	30.7

a Level I = maternity ward without a neonatology department; Level II = maternity ward with a neonatology department; Level III = maternity unit with a neonatology department and a neonatal intensive care unit (NICU).

Table 2 -

 2 Positions proposed by midwives during the active second stage, globally, by maternity unit level and by midwives' experience.

		Global
	Practices proposed		Results by maternity unit level		Results by experience
		results
		Total	Level I a	Level II a	Level III a		≤ 5	6-15	>15
							years	years	years
		n=1496	N= 366	N= 666	N= 464	p	n=582	n=609	n=305	p
		%	%	%	%		%	%	%
	Supine with footholds	79.8	79.0	80.2	79.7	0.90	83.5	80.1	71.8	0.0002
	Knees-elbows	31.7	39.1	32.3	25.0	<0.0001	30.4	32.5	32.5	0.70
	Sitting position	39.5	48.4	38.9	33.4	<0.0001	36.3	40.1	44.6	0.05
	Kneeling position	28.8	39.3	28.1	21.6	<0.0001	26.5	32.0	26.9	0.07

a Level I = maternity ward without a neonatology department; Level II = maternity ward with a neonatology department; Level III = maternity unit with a neonatology department and a neonatal intensive care unit (NICU).

Table 3 -

 3 Midwives' practices for positioning women at delivery, globally, by maternity unit level and by midwives' experience.

		Global
	Practices proposed		Results by maternity unit level			Results by experience
		results
		Total	Level I a	Level II a	Level III a		≤ 5	6-15	>15
							years	years	years
		n=1496	N= 366	N= 666	N= 464	p	n=582	n=609	n=305	p
		%	%	%	%		%	%	%
	Choice of position left to the
	woman
	Never or rarely	25.1	21.6	24.9	28.0	0.005	28.0	21.5	26.6	0.11
	Often or most of the time	58.7	57.4	58.4	60.1		57.0	61.3	56.7
	Always	16.2	21.0	16.7	11.9		15.0	17.2	16.7

Table 4 -

 4 Practices proposed by midwives for pushing, globally, by maternity unit level and by midwives' experience.

		Global
	Practices proposed		Results by maternity unit level			Results by experience
		results
		Total	Level I a	Level II a	Level III a		≤ 5	6-15	>15
							years	years	years
		n= 1496	N= 366	N= 666	N= 464	p	n= 582	N=609	N=305	p
		%	%	%	%		%	%	%
	Choice about type of pushing
	left to woman
	Never or rarely	14.6	13.1	15.3	14.9	0.34	18.2	13.6	9.8	<0.0001
	Often or most of the time	58.3	56.3	57.7	60.8		60.8	54.8	60.3
	Always	27.1	30.6	27.0	24.4		21.0	31.5	29.8

a Level I = maternity ward without a neonatology department; Level II = maternity ward with a neonatology department; Level III = maternity unit with a neonatology department and a neonatal intensive care unit (NICU).

b closed-glottis pushing

Table 5 -

 5 Midwives' practices for perineal protection and techniques of perineal support, globally, by maternity unit level and by midwives' experience.

		Global
	Practices proposed		Results by maternity unit level			Results by experience
		results
		Total	Level I a	Level II a	Level III a		≤ 5	6-15	>15
							years	years	years
		n=1496	N= 366	N= 666	N= 464	p	n=582	n=609	n=305	p
		%	%	%	%		%	%	%
	Perineal massage
	No	53.5	51.9	52.3	56.7	0.13	48.8	58.6	52.4	0.0005
	Yes, cutaneous side	7.1	8.2	8.0	5.0		8.1	6.1	7.2
	Yes, vaginal side	7.3	5.7	8.4	6.9		6.0	6.6	11.2

DRESS: Bureau of research, studies, evaluation and statistics (Direction de la Recherche,