Denis Duhamel
email: denis.duhamel@enpc.fr

Computation of the dynamic response of finite two-dimensional periodic structures

The study of periodic media is mainly focused on one-dimensional periodic structures (periodic along one direction), to determine the dispersion curves or for the calculation of the response to an external excitation. Effective methods such as the Wave Finite Element (WFE) have been obtained for such computations. Two-dimensional periodic media are more complex to analyse but dispersion curves can be obtained rather easily. Obtaining their response to an excitation is much more difficult and the results mainly concern infinite media while for finite media, few results are available. In this communication, the response of finite two-dimensional periodic structures to an excitation is studied by limiting oneself to structures described by a scalar variable (acoustic, thermal, membrane behaviour) and having symmetries. Using the WFE for a rectangular substructure and imposing the wavenumber in one direction, we can calculate the wavenumbers and mode shapes associated with propagation in the perpendicular direction. By building solutions with null forces on parallel boundaries, we can decouple the waves in the two directions parallel to the sides of the rectangle and solve each case by a FFT. Summing the contributions of all these waves gives the global solution with a low computing time even for a large number of substructures. Examples are given for the case of a two-dimensional membrane.

Introduction

Authors interested in two-dimensional periodic media first aimed to determine the dispersion curves like [START_REF] Abdel-Rahman | Matrix analysis of wave propagation in periodic systems[END_REF] who considered wave propagation in one, two and three-dimensional periodic structures using finite element models or [START_REF] Phani | [END_REF] who considered two-dimensional wave propagation in periodic lattices made of beams. More complex structures were considered by [START_REF] Manconi | Modelling wave propagation in two-dimensional structures using a wave/finite element technique[END_REF][START_REF] Zhou | International Congress on Sound and Vibration[END_REF][START_REF] Zhou | [END_REF]6] who computed dispersion relations for wave propagation in two-dimensional periodic structures by the Wave Finite Element method (WFE). Most of these authors considered a reduced model obtained by the Craig-Bampton method while in [7,8,9,[START_REF] Van Belle | International Conference on Noise and Vibration Engineering[END_REF] the reduction of the boundary degrees of freedom was considered in addition and [START_REF] Hussein | [END_REF] projected the mass and stiffness matrices on a reduced set of Bloch modes.

One can also be interested in the response of a periodic medium to an external excitation. For instance, the computation of infinite doubly periodic structures for harmonic electromagnetic fields was considered by [12] using hybrid Finite Element/Boundary Integral and periodic Green's functions. The forced response of infinite two-dimensional periodic media using WFE was computed by [13] while [14] focused on homogeneous media still using WFE but with a contour integration to improve the computation of some integrals giving the force response.

The responses of two-dimensional finite and infinite periodic structures to point harmonic and impulsive forces were also computed by [15,16]. For finite structures, he found the solution by a modal summation of modes with periodic boundary conditions and then extended the solution to infinite structures for points far enough from the load. Experiments were also done by [17] for the seismic isolation by two-dimensional finite periodic foundations and they found interesting isolation effects for frequencies band gaps even for a small number of substructures for which comparisons with full Abaqus computations were done. In another domain, [START_REF] Mencik | International Conference on Noise and Vibration Engineering (ISMA 2022[END_REF] computed the acoustic radiation of two-dimensional nearly periodic metamaterial plates with 6 × 6 substructures by the finite element method with the Craig-Bampton reduction and interpolation strategies to reduce the computational cost. In [START_REF] Van Belle | Proceedings of Euronoise[END_REF] two-dimensional periodic metamaterial structures were computed using interior dofs reduction by the Craig-Bampton method and an interface dofs reduction while [START_REF] Qu | International Conference on Noise and Vibration Engineering (ISMA 2022)[END_REF] used an approximate solution built from a linear superposition of waves and computed finite two-dimensional periodic structures up to 30 × 30 substructures. In all cases, the number of substructures was very limited of the order of a few hundred to the maximum and the main goal was to develop reduction methods for substructures to accelerate calculations.

In this paper, we consider the calculation of a finite two-dimensional periodic medium made of a large number of substructures under external excitations. We will limit ourselves to media described by a scalar equation such as during the propagation of acoustic waves or vibrations of membranes and having symmetries with respect to two orthogonal planes parallel to the edges of a substructure. One bases on the WFE by modeling a rectangular substructure by a finite element model (FEM). By imposing the wavenumber in one direction, we can numerically calculate the wavenumbers and mode shapes associated with propagation in the perpendicular direction. By taking appropriately chosen solutions, we can decouple the waves in the two directions parallel to the sides of the rectangle. The solution of each of these two problems is obtained by a fast Fourier transformation which gives the amplitudes associated with the waves. We thus obtain the global solution for a two-dimensional periodic medium with a large number of substructures with a low computing time.

The paper is organised as follows. In section 2, wave modes for two-dimensional periodic media are computed. Special symmetric solutions are computed in section 3 and then the equations of the global problem are set and solved in section 4. Section 5 presents some numerical results for the case of a two-dimensional membrane with many substructures before the conclusion.

Two-dimensional wave modes 2.1. Relations on boundary variables

The discrete dynamic equation of a substructure obtained from a FE model at a circular frequency ω and for the time dependence e iωt is given by:

(K -ω 2 M)q = f (1)
where K and M are the stiffness and mass matrices, respectively. f is the loading vector and q the vector of the degrees of freedom (dofs). Introducing the dynamic stiffness matrix D = K-ω 2 M, decomposing the dofs into boundary (b) and interior (i) dofs, and assuming that there are no external forces on the interior nodes, result in the following equation:

D bb D bi D ib D ii q b q i = f b 0 (2)
The interior dofs can be eliminated using the second row of equation ( 2), which results in The subscript b will be dropped in the following and only boundary dofs will be considered.

q i = -D -1 ii D ib q b , f b = D bb -D bi D -1 ii D ib q b = D b q b (3)

Condensation of transverse degrees of freedom

The substructure is meshed with an equal number of nodes on their opposite sides. The boundary dofs are decomposed into left (L), right (R), bottom (B), top (T ) dofs and associated corners (LB), (RB), (LT ) and (RT ) as shown in figure 1. The global vector of boundary dofs is defined as q = t t q LB t q L t q LT t q RB t q R t q RT t q B t q T (4)

Consider waves with a given propagation constant λ y along y. The longitudinal dofs vector is defined as q l = t t q LB t q L t q LT t q RB t q R t q RT . Thus, equation ( 3) is rewritten as

  D ll D lB D lT D Bl D BB D BT D T l D T B D T T     q l q B q T   =   f l f B f T   (5)
Using pseudo periodic conditions with the given propagation constant λ y and the effort equilibrium at the bottom side of the substructure, relations between the transverse dofs are given by

q T = λ y q B , f B + 1 λ y f T = 0 (6)
Multiplying the third row of equation ( 5) with 1 λy , taking the sum of the second and third rows of equation ( 5), using conditions (6), lead to

q B = -D BB + D T T + 1 λ y D T B + λ y D BT -1 D Bl + 1 λ y D T l q l (7)
Using ( 6) and ( 7), the first row of equation ( 5) becomes

f l = D ll -(D lB + λ y D lT ) D BB + D T T + 1 λ y D T B + λ y D BT -1 D Bl + 1 λ y D T l q l = D l q l (8)
which defines the dynamic stiffness matrix D l for longitudinal dofs.

One-dimensional eigenvalue problem

Using the pseudo periodic conditions also leads to relations between longitudinal dofs q R = λ x q L q RB = λ x q LB q RT = λ x λ y q LB q LT = λ y q LB (9) From the pseudo periodic conditions (9), it can be seen that all components of the vector q l depend on the reduced set of dofs defined by q r = t t q LB t q L . This can be expressed as

q l = (W 0 (λ y ) + λ x W 1 (λ y )) q r ( 10
)
where the matrices W 0 (λ y ) and W 1 (λ y ) depend on the propagation constant λ y and are given by

W 0 (λ y ) =         I O O I λ y I O O O O O O O         W 1 (λ y ) =         O O O O O O I O O I λ y I O         (11)
The equilibrium conditions between adjacent substructures can be written as

λ x W T 0 ( 1 λ y ) + W T 1 ( 1 λ y ) f l = 0 (12)
Combining ( 8), ( 10) and ( 12), lead to

λ x W T 0 ( 1 λ y ) + W T 1 ( 1 λ y ) D l (λ y ) (W 0 (λ y ) + λ x W 1 (λ y )) q r = 0 (13)
A small damping is introduced by taking (1+ξi)ω instead of ω with ξ = 0.0001. So we always have |λ x | = 1 and the right-going solutions must have a decreasing amplitude as we move on the right, meaning that

|λ x | < 1, |λ x | > 1 (14)
This allows to define the two sets of eigensolutions

Λ(λ y ) = [λ x1 (λ y ) • • • λ xn (λ y )] , Λ (λ y ) = [λ x1 (λ y ) • • • λ xn (λ y )] Φ q (λ y ) = [φ q1 (λ y ) • • • φ qn (λ y )] , Φ q (λ y ) = [φ q1 (λ y ) • • • φ qn (λ y )] Φ F (λ y ) = [φ F1 (λ y ) • • • φ Fn (λ y )] , Φ F (λ y ) = [φ F1 (λ y ) • • • φ Fn (λ y )]. ( 15
)
in which φ qj is the solution of the eigenproblem (13) and φ Fj = t t f j LB t f j L the associated force vector.

3. Building eigenvectors with free boundary 3.1. Reduction to one-dimensional problems Consider now a rectangular domain which undergoes forces over its left/right and bottom/top edges, as depicted in Figure 2 where the loads are shown as red lines on the boundary while free sides are depicted as green lines. The global loads can be decomposed as loads on the vertical sides F 1 B and loads on the horizontal sides 2 B such that the displacements of the global problem can be recovered as the sum of the displacements of problems 1 and 2. So we have

F B = (F B ) 1 + (F B ) 2 ⇒ q B = (q B ) 1 + (q B ) 2 . ( 16
)
We focus now on one of the two problems defined previously. Consider for instance the first problem as the second problem can be solved in the same way. From the knowledge of the propagation constants and wave modes for one-dimensional WFE, the vectors of displacements and forces of an assembly of N x substructures like the one displayed in Figure 2 are expressed on global left and right boundaries in terms of wave shapes, as follows:

q L = Φ q Q + Φ q µ Nx Q , -F L = Φ F Q + Φ F µ Nx Q q R = Φ q µ Nx Q + Φ q Q , F R = Φ F µ Nx Q + Φ F Q (17)
where Q and Q are vectors of wave amplitudes which are respectively defined at the left and right edges of the whole domain. The elements of the diagonal matrix µ have modulus less or equal to one. The main problem is to find the wave modes Φ F and Φ F with free boundaries (no force on the bottom and top boundaries).

Symmetric solution

We suppose now that the geometry and the mechanical parameters of a period are invariant by substituting l y -y to y. Consider a solution defined by the vectors

  q l q B q T   and   f l f B f T   (18)
Now, denoting q(x, y) the displacement associated to this solution, one defines the symmetric solution such that q(x, y) = 1 λ y q(x, l y -y)

So the symmetric solution is such that qB = q B , and qT = 1

λ y qB = 1 λ y q B ( 20
)
Concerning the forces, one has

fB = D Bl ql + D BB qB + D BT qT fT = D T l ql + D T B qB + D T T qT (21)
We make the hypothesis that the symmetry of the structure leads to the following relations on the submatrices.

D BB = D T T , D BT = D T B , D T B = D BT (22)
From this, one can prove that the symmetric solution satisfies

fB = -f B , fT = - 1 λ y fB ( 23
)
This shows that the symmetric solution is associated to the propagation constant 1 λy . One can also check that

λ x W T 0 (λ y ) + W T 1 (λ y ) D l ( 1 λ y ) W 0 ( 1 λ y ) + λ x W 1 ( 1 λ y ) qr = 0 ( 24
)
and qr is associated to the propagation constant λ x in the x direction and to 1 λy in the y direction.

Fast solution for the two-dimensional periodic structures 4.1. Decomposition of force

The solution is searched under the form of waves associated to propagation constants λ y,n = e iπn/Ny along y, for -N y ≤ n ≤ N y -1, leading to the propagation constants λ x+ and λ x-along x. Along x one has propagation constants λ x,m = e iπm/Nx , for -N x ≤ m ≤ N x -1, leading to the propagation constants λ y+ and λ y-along y. For instance, the force on the global bottom boundary can then be decomposed as

f B (p B ) = n=Ny-1 n=-Ny F x+ Bn (Λ x+ n ) p B a x+ n + F x- Bn (Λ x- n ) p B -(Nx-1) a x- n + m=Nx-1 m=-Nx e iπp B m/Nx F y+ Bm a y+ m + F y- Bm (Λ y- m ) -(Ny-1) a y- m (25) for 0 ≤ p B ≤ N x -
Λ x+ n = diag λ x+ nj j=1...J + n , Λ x- n = diag λ x- nj j=1...J - n F x+ Xn = f x+ Xn1 , ..., f x+ Xnj , ..., f x+ XnJ + n , F x- Xn = f x- Xn1 , ..., f x- Xnj ..., f x- XnJ - n a x+ n = a x+ n1 , ..., a x+ nj , ..., a x+ nJ + n T , a x- n = a x- n1 , ..., a x- nj , ..., a x- nJ - n T (26)
From relations (13), one has in a general way,

Λ x+ n = (Λ x- -n ) -1 , Λ y+ m = (Λ y- -m ) -1 (27)
Supposing that the substructure is symmetric, we also have

Λ x+ -n = Λ x+ n , Λ x- n = Λ x- -n , Λ y+ -n = Λ y+ n , Λ y- n = Λ y- -n F x+ B(-n) = -F x+ Bn , F x- B(-n) = -F x- Bn ( 28
)
One can choose the decomposition such that

a x+ -n = a x+ n , a x- -n = a x- n , a y+ -m = a y+ m , a y- -m = a y- m ( 29
)
Inserting relations ( 27), ( 28) and ( 29) into (25) leads to

f B (p B ) = m=Nx-1 m=-Nx e iπp B m/Nx F y+ Bm a y+ m + F y- Bm (Λ y- m ) -(Ny-1) a y- m Using the relation F y- T m + F y- Bm Λ y- m = 0 (30)
we finally get

f B (p B ) = m=Nx-1 m=-Nx e iπp B m/Nx F y+ Bm a y+ m -F y- T m (Λ y- m ) -Ny a y- m (31)
And similar relations for the three other parts of the boundary.

System to solve

The precedent relations can be written a more compact form as

F B = ω x F y+ B a y+ -ω x F y- T (Λ y-) -Ny a y- F R = -ω y F x+ L (Λ x+ ) Nx a x+ + ω y F x- R a x- F T = -ω x F y+ B (Λ y+ ) Ny a y+ + ω x F y- T a y- F L = ω y F x+ L a x+ -ω y F x- R (Λ x-) -Nx a x- (32)
with for instance

F L =      f L (0) f L (1) . . . f L (N y -1)      F x+ L = diag F x+ L(-Ny) , ..., F x+ Ln , ..., F x+ LNy-1 , F x- L = diag F x- R(-Ny) , ..., F x- Rn , ..., F x- RNy-1 Λ x-= diag Λ x- -Ny , ..., Λ x- n , ..., Λ x- Ny-1 , ω y = λ nm y I nm 0≤n≤Ny-1,-Ny≤m≤Ny-1 (33)
with λ y = e iπ/Ny , I nm = I n l ×n l and n l = n LB + n L the number of reduced dofs.

Symmetric relations

Consider relation (31), one has

f B (p B ) = m=Nx-1 m=-Nx e iπp B m/Nx F y+ Bm a y+ m -F y- T m (Λ y- m ) -Ny a y- m = m=Nx-1 m=-Nx e -iπp B m/Nx F y+ B(-m) a y+ m -F y- T (-m) (Λ y- m ) -Ny a y- m (34)
because λ x,-Nx = λ x,Nx = -1, and for instance F y+ B(-Nx) = F y+ BNx . Note also that, for instance,

F y+ B(-m) = 1 λx,m
Fy+ B(m) , and a similar relation for the top component, so that we finally gets

fB (p B ) = m=Nx-1 m=-Nx e -iπ(p B +1)m/Nx F y+ Bm a y+ m -F y- T m (Λ y- m ) -Ny a y- m with fB (p B ) obtained from f B (p B
) by inverting the dofs on the bottom boundary on each substructure between 0 and l x . The function f B (p B ) can now be defined for

-N x ≤ p B ≤ N x -1 with f B (-p B -1) = fB (p B
). The global system to solve is then

            F B FB F R FR F T FT F L FL             =              O O ω + x F y+ B -ω + x F y- T (Λ y-) -Ny O O ω - x F y+ B -ω - x F y- T (Λ y-) -Ny -ω + y F x+ L (Λ x+ ) Nx ω + y F x- R O O -ω - y F x+ L (Λ x+ ) Nx ω - y F x- R O O O O -ω + x F y+ B (Λ y+ ) Ny ω + x F y- T O O -ω - x F y+ B (Λ y+ ) Ny ω - x F y- T ω + y F x+ L -ω + y F x- R (Λ x-) -Nx O O ω - y F x+ L -ω - y F x- R (Λ x-) -Nx O O                  a x+ a x- a y+ a y-     (35) with FB
=    FB (N x -1) . . FB (0)    , F B =    F B (0) . . . F B (N x -1)    ω - x =       
e -iπNxm/Nx e -iπ(Nx-1)m/Nx . . .

e -i2πm/Nx e -iπm/Nx        -Nx≤m≤Nx-1 , ω + x =        1 e iπm/Nx
. . .

e iπ(Nx-2)m/Nx e iπ(Nx-1)m/Nx        -Nx≤m≤Nx-1 (36)
4.4. Use of FFT for an efficient solution Taking the Fast Fourier Transform of both sides of (35) for each of the four parts of the boundary,

noting that F F T ω + x ω - x = 2N
x I and that after the FFT the right hand side is made of block matrices, the system can be decomposed as the following subsystems for each component of the FFT

F Rn F Ln = 2N x -F x+ Ln (Λ x+ n ) Nx F x- Rn F x+ Ln -F x- Rn (Λ x- n ) -Nx a x+ n a x- n (37)
with for instance F Rn the n th component of f f t F R FR . The solution of this small size system gives to n th component of the amplitude and the whole solution can then be rebuilt.

Numerical results

Validation with a homogeneous structure

We first consider the case of the Helmholtz equation (modelling acoustic or a membrane vibration) on a homogeneous medium divided into rectangular substructures. The equation is ∆p + k 2 p = 0 in Ω, ∂p ∂n = q 0 over ∂Ω

with k = ω/c, ω the circular frequency and c = 343m/s the sound velocity. The structure Ω is divided into N x ×N y substructures of size L x ×L y . q 0 is a given function on the boundary. In this case it is simple to find an analytical solution to estimate the accuracy of the proposed method. For example a solution is p 0 (x) = i 4 H 0 (k|x -x s |) for a point source x s outside the domain Ω and q 0 = ∂ ∂n p 0 . We compare the results of the analytical solution, a full Finite Element (FEM) solution and the present two-dimensional WFE method in table 1 for structures with 5 × 5 and 25 × 25 substructures. The substructures are meshed with 10 × 10 quadratic 8 nodes elements. The sizes are L x = L y = 0.1m. The position of the source is at x s = (-0.5m, -0.5m). It can be observed in table 1 that there is a perfect agreement for low frequencies and that for high frequencies the accuracy of the 2D WFE is similar to the full FEM.

Structure with holes

We consider now a substructure with a central hole of radius 0.02m as in figure 3. All the other properties are the same as before. The mesh of a substructure is shown in figure 3 and is made of 654 nodes with linear triangular elements obtained from gmsh. The boundary condition Table 1. Comparison of the solution for different methods, number of frequencies and computation points. is obtained by the normal derivative of a plane wave of direction (1/ √ 2, 1/ √ 2). The case of a structure made of 25 × 25 substructures computed at 1000Hz is shown in figure 5 for the full FEM computation and in figure 6 for the 2D WFE. A perfect agreement between the two computations is observed. The same computation is also made for structures of increasing sizes between N × N = 10 × 10 substructures and N × N = 1000 × 1000. The computing time is shown in figure 4 for the 2D WFE. For the FEM the computation is only done up to 100 × 100 substructures. The computing time is strongly increasing for the FEM while it increases only mildly for the 2D WFE. Note that for the 1000 × 1000 structure the number of nodes for a classical FEM computation would be about 650 millions. This problem is solved by the present 2D WFE in 9 seconds with Matlab on a personal computer with a memory of 16Go and a processor Intel Core(TM) i7-9700.

Conclusion

We have developed a new method to compute two-dimensional periodic media for the case of a scalar wave equation. This is based on the Wave Finite Element Method by considering propagation along the two directions parallel to the boundary of a substructure. For a boundary excitation the method proves to be very efficient and allows to get the solution for structures made of millions of substructures. Future works should try to extend the method for the case of two-dimensional elasticity meaning structures described by vectorial equations and substructures having no special symmetry. The case of loading on the internal dofs should also be considered.

Figure 1 .Figure 2 .

 12 Figure 1. A substructure in the periodic medium.

Figure 3 .

 3 Figure 3. Substructure with a central hole.Figure 4. Computing times.

Figure 4 .

 4 Figure 3. Substructure with a central hole.Figure 4. Computing times.

Figure 5 .

 5 Figure 5. FEM solution at 1000z.

Figure 6 .

 6 Figure 6. 2D WFE at


Sylvie Démurger
email: demurger@gate.cnrs.fr

Martin Fournier
email: fournier@gate.cnrs.fr

Weiyong Yang
email: weiyongyang@gmail.com

Rural households decisions towards income diversification: Evidence from a township in northern China

Keywords: income-source diversification, agricultural households, off-farm employment, China. JEL codes: J2, R2, Q1, O53

Economic reforms in rural China have brought opportunities to diversify both within-farm activities and off-farm activities. Participation in these activities plays an important role in increasing rural households' income. This paper analyzes the factors that drive rural households and individuals in their income-source diversification choices for ten villages in Northern China. At the household level, we distinguish three types of diversification as opposed to grain production only: within farm (nongrain production) activities, local off-farm activities, and migration. At the individual level, we analyze the determinants of participation in three different types of jobs as compared to agricultural work: local off-farm employment, local self-employment and migration. At the household level, we find that land and labor availability stimulates on-farm diversification. Local off-farm activities are mostly driven by household wealth and credit constraints, while migration decisions strongly depend on the household age and composition. At the individual level, we find a clear gender and age bias in access to off-farm activities that are mostly undertaken by male and by young people. More surprisingly, education is found to play a role for accessing local wage employment but not in migration decision. As at the household level, the household assets position is found to strongly affect participation in any off-farm activity.

a township in northern China

Introduction

Over the past decades, there has been an outstanding trend of activity diversification in rural areas in developing countries. A rich related literature suggests that rural households adjust their activities either to exploit new opportunities created by market liberalization [START_REF] Delgado | Rural Economy and Farm Income Diversification in Developing Countries[END_REF] or to cope with livelihood risks (Barrett et al. 2001a;[START_REF] Carter | Environment, Technology, and the Social Articulation of Risk in West African Agriculture[END_REF]. These adjustments are found to have an important impact on income, income distribution and welfare across rural households [START_REF] Block | The dynamics of livelihood diversification in post-famine Ethiopia[END_REF][START_REF] Canagarajah | Non-farm income, gender, and inequality: Evidence from rural Ghana and Uganda[END_REF][START_REF] De Janvry | Income Strategies Among Rural Households in Mexico: The Role of Off-farm Activities[END_REF][START_REF] Ellis | Survey article: Household strategies and rural livelihood diversification in developing countries[END_REF][START_REF] Ellis | Rural Livelihoods and Diversity in Developing Countries[END_REF][START_REF] Hoogeveen | Income Risk, Consumption Security and the Poor[END_REF][START_REF] Reardon | Effects of nonfarm employment on rural income inequality in developing countries: An investment perspective[END_REF].

In China, the launching of economic reforms from the end of the 1970s has led to important changes in productive activities in rural areas. Pre-reform central planning and regional economic self-sufficiency policies had resulted in specialization patterns quite frequently disconnected from local comparative advantages. As a consequence, the rural economy was overwhelmingly dominated by agricultural activities, with grain crops accounting for more than 80 per cent of total sown area. Economic reforms have brought incentives and opportunities for rural households to diversify both within-farm activities and off-farm activities. First, the Household Responsibility System led to the dismantling of the People's Communes and made it progressively possible for rural households to take private decisions regarding their economic activities. At the same time, the emergence of market mechanisms through price reforms and the development of free markets encouraged profitoriented activities. And more recently, China's joining the WTO has accelerated structural adjustments, from land-intensive grain production to more labor-intensive activities, including fruits and vegetables crops, animal husbandry and rural industrialization.

China's rural economy has been diversifying at various levels. First, the agricultural production itself has been diversified with a constant decline of farming and a steady rise of husbandry, forestry and fishery in terms of output value. Second, the importance of grain in the farming sector has dropped rapidly in favor of cash crops, whose share increased from 20 per cent of total sown area in 1978 to 35 per cent in 2003. Third, non-farm activities have thriven as illustrated by both the prosperity of rural enterprises and the huge flows of rural migrants to urban areas. Using macroeconomic data from Chinese provinces between 1985 and 2001, [START_REF] Yang | Economic Structural Changes and Rural Income: Evidence from Chinese Provinces during the Reform Period[END_REF] has shown that this production restructuring from grain crops to cash crops and the labor shift from cropping to non-agricultural activities have both significantly contributed to rural income increase and income stability in China.

The diversification of activities in rural areas is of great concern for rural development not only because of its expected impact on income and poverty reduction, but also because of its importance for the understanding of out-migration movements and rural exodus. Given the size of the rural population in China, gaining a good understanding of how the transition is taking place is essential for the design of adequate rural and urban development policies. A key question is thus whether or not opportunities to develop nonagricultural activities are big enough to foster the expansion of middle-size cities and towns in rural areas, or if one should continue to encourage huge flows of rural population into big cities.

The purpose of this paper is to highlight the main factors that drive rural households as well as individuals in their decision to diversify their economic activities. We use an Although the focus is on a small area, we hope that the analysis presented below may give useful insights as to how income-source diversification is working in rural China.

The rapid increase of off-farm opportunities in rural China, illustrated by the development of rural enterprises and a soaring rural-urban migration, has motivated an array of empirical research on the determinants of participation in off-farm activities in China.

Most papers tend to focus on a particular choice among the different alternatives, especially on the determinants of migration (e.g. Zhao, 1999;Zhu, 2002), or to consider off-farm activities as a group without separating the different types of jobs [START_REF] Zhang | Employment, emerging labor markets, and the role of education in rural China[END_REF]. [START_REF] De Brauw | The evolution of China's rural labor market during the reforms[END_REF] and [START_REF] Shi | Choices between different off-farm employment subcategories: An empirical analysis for Jiangxi Province, China[END_REF] that offer detailed analyses of the determinants of individual participation in various sub-categories of off-farm jobs. However, less emphasis has been given to household level choices and especially to the explanation of differences of strategies among households in terms of income-source diversification. Our paper intends to contribute to the literature by analyzing economic diversification strategies as a whole. As a consequence, we not only focus on individual labor allocation between farm and off-farm activities, but we also consider household level decisions that include within-farm diversification strategies.

Noticeable exceptions are De

The questions of interest in studying income-source diversification are the following: The paper is organized as follows. Section 2 highlights diversification patterns observed in Labagoumen township over the recent years. Section 3 discusses the determinants of rural households' decision to diversify their economic activities. Section 4 provides an econometric analysis of the determinants of rural income diversification behaviors at both the household level and the individual level. Section 5 concludes. The rural economy in this township has been traditionally dominated by farm activities. Surrounded by high and steep mountains, it does not enjoy favorable endowments in arable land. At the township level, arable land only accounts for three per cent of total land while forestland represents 83 per cent. The population pressure upon land is also severe, with an average farm size of less than 0.5 hectare per household. Until recently, the agricultural sector alone was employing most of the active population, and the area was relying on subsistence agriculture and the production of corn for seed. Agricultural households surveyed in 2003 allocated on average 51 per cent of their arable land to food crops2 , the proportion rising up to 82 per cent when corn for seed is included. Land scarcity and land fragmentation strongly constrain crop diversification: with corn being the main crop, it may appear more rational to keep the current production pattern on a network of plots belonging to different households rather than to diversify the production structure on such small plots.

Study area: diversification patterns in Labagoumen township

Our

In recent years however, the township has started to move to a wider range of activities, both within and out of the agricultural sector. At the household level, three types of diversification can be identified: on-farm diversification within agriculture, local off-farm activities, and rural-urban migration. A household falls into the category of "on-farm diversification" if it cultivates products of higher value-added than grain (like ginseng, liquorice, or fruit trees), "local off-farm diversification" if it has at least one member involved in local off-farm work and "migration" if it has at least one member with a migration experience over the last 5 years. Our data show that although 34.1% of the surveyed households had not yet engaged in any economic diversification in 2003, 30% undertook onfarm diversification, 29.3% were involved in local off-farm activities and 25.6% in migration.

Since some households may engage into more than one diversification activity, the total of the percentages given here does not sum up to 100%. In the rest of the paper, we make an additional assumption regarding the "degree" of diversification so that every household can fall into one and only one category. We assume that engaging in migration somehow reflects a higher involvement in diversification as compared to all other alternatives and that engaging in local off-farm activities reflects a higher diversification than on-farm diversification only. As a consequence, the categories are nested as follows: "migration" comprises all households for which at least one member has had a migration experience over the last five years, whether or not the household is engaged in on-farm diversification and/or local off-farm activities; "local off-farm activities" includes households for which at least one member is working off-farm in the local area whether or not the household is engaged in on-farm diversification, but excludes those households also involved in migration; "on-farm diversification" includes households only engaged in on-farm diversification. With this restricted definition, the respective percentages of households engaged in income-source diversification are 16.7% for on-farm diversification, 23.6% for local off-farm and 25.6% for migration (Table 1).

Concerning on-farm diversification, the market development that characterized China over the past three decades has led to the commercialization of agricultural production, which is thus becoming more profit-oriented and increasingly guided by market conditions. Therefore, farmers in the township have started to convert some land from corn crop to higher value-added agricultural products, including American ginseng, liquorice and fruit trees. These productions are better related to the local comparative advantages in terms of climatic and topographic conditions, and they benefit from a direct access to the market through the regular inflow of tourists in the area. Moreover, farmers who turned to these new activities have benefited from various sets of preferential policies and subsidies 3 .

As for non-farm activities, two main occupational choices can be distinguished: Since there are very few manufacturing enterprises in Labagoumen township, local non-farm activities are mostly related to services. One village (Sunzazhi) benefits from a particular position at the entrance of a Nature Reserve, which recently favored the 3 Preferential policies and subsidies vary across villages. In Xiahebei village, government subsidies of 500 yuan, plus 50kg of cereal per mu have been given to households who started American ginseng culture. In Zhongyudian village, besides a subsidy of 400-500 yuan per mu, interest-free loans were also offered to households starting American ginseng cultivation. The implementation of the Sloping Land Conversion Program in the township since 2001 also brought subsidies for tree planting. On average, 26% of agricultural land had been converted by the end of 2003. Most of the land conversion (68% of converted land) concerns fruit trees, mainly chestnut trees, apricot trees and hawthorn trees (Crataegus laevigata). development of tourism. The establishment of the Nature Reserve in 1999 has led to the opening of family hotels, restaurants and the development of related tourist activities. Other non-farm activities include working for public services, local government, etc.

Even though farmers are free to choose the crops they grow since the early 1980s, when the Household Responsibility System was introduced, their choice in terms of diversification still largely depends on villages' strategies. As can be seen from Table 1, there are large differences across villages in the speed of activity restructuring. These differences come from differences in natural endowments as well as from differences in the villages' dynamism. Important disparities in resource endowments condition the villages' ability to create income opportunities out of traditional cropping. Some fortunate villages are endowed with specific tourist sights or specific land characteristics, which attract outside investors and provide them with favorable initial conditions. Moreover, activity diversification also strongly depends on policies implemented at the local level to promote economic restructuring. Some dynamic villages did actively promote alternative activities by providing villagers with information as well as incentives, while some conservative villages even forbade the process by imposing grain production to all households. In dynamic villages, the range of measures that has been adopted to encourage activity restructuring includes cash subsidies, longer land-use right, favorable terms, and training.

By providing additional income sources independent of the agricultural cycle, offfarm activities can increase both the level and the stability of household income [START_REF] Ellis | Rural Livelihoods and Diversity in Developing Countries[END_REF][START_REF] Hoogeveen | Income Risk, Consumption Security and the Poor[END_REF][START_REF] Alderman | Do the Poor Insure? A Synthesis of the Literature on Risk and Consumption in Developing Countries[END_REF][START_REF] Yang | Economic Structural Changes and Rural Income: Evidence from Chinese Provinces during the Reform Period[END_REF]. As shown in Table 2, although farm income still represents more than one third of households' annual income, for those households with income from a specific source, off-farm income is by far the most remunerative. In particular, while remittances account for a rather small share in households' income for the total sample (7.6%), they represent half of the income for households with migrant members. Tourism can also be a worthwhile alternative to favor rural economic development in the township since the average annual income is as high as 8,000 yuan among households involved in tourism activities. Moreover, Table 3 shows a clear relationship between diversification patterns and per capita households' income. Although the causality can be of a bidirectional nature, it clearly reveals an over-representation of non-diversifying households in the poorest quartiles and an over-representation of households engaged in (local and non-local) off-farm activities in the richest quartiles. In sum, the remunerative nature of off-farm activities calls for a better understanding of the conditions for accessing these activities.

The determinants of households' diversification behaviors: theoretical linkages

Various explanations for diversification behaviors can be found in the economic literature to explain both incentives and disincentives for rural households to combine traditional crops with new crops [START_REF] Norman | Rationalising Mixed Cropping under Indigenous Conditions: The Example of Northern Nigeria[END_REF], agricultural crops with animal husbandry or forestry activities [START_REF] Kurosaki | Risk and Insurance in a Household Economy: Role of Livestock in Mixed Farming[END_REF][START_REF] Kurosaki | Production Risk and Advantages of Mixed Farming in the Pakistan Punjab[END_REF], and/or agricultural activities with off-farm activities such as migration and tourism [START_REF] Barrett | Income diversification, poverty traps and policy shocks in Cote d'Ivoire and Kenya[END_REF][START_REF] Murphy | Return Migrant Entrepreneurs and Economic Diversification in Two Countries in South Jiangxi, China[END_REF]. On one hand, in a changing economic and institutional environment, agricultural households have incentives to find alternative income sources in order to secure their livelihood. But on the other hand, several factors such as risk aversion and barriers to entry can also hold them back from engaging into new activities.

A rich literature on income diversification in rural areas has identified a wide range of explanatory factors for activity restructuring out of subsistence farming at the household level [START_REF] Abdulai | Determinants of Income Diversification amongst Rural Households in Southern Mali[END_REF][START_REF] Smith | Livelihood diversification in Uganda: Patterns and determinants of change across two rural districts[END_REF][START_REF] Ellis | Survey article: Household strategies and rural livelihood diversification in developing countries[END_REF]. The motives are usually divided into two categories: "pull factors" and "push factors" [START_REF] Barrett | Nonfarm income diversification and household livelihood strategies in rural Africa: Concepts, issues, and policy implications[END_REF]. Pull factors include benefits from complementarities between activities [START_REF] Norman | Rationalising Mixed Cropping under Indigenous Conditions: The Example of Northern Nigeria[END_REF], new income opportunities created by market development [START_REF] Davis | The rural nonfarm economy in Central and Eastern Europe[END_REF], improvement of infrastructure [START_REF] Jalan | Geographic poverty traps?[END_REF], and diversification for asset accumulation (Hart, 1994). Push factors include ex ante risk management [START_REF] Hoogeveen | Income Risk, Consumption Security and the Poor[END_REF][START_REF] Alderman | Do the Poor Insure? A Synthesis of the Literature on Risk and Consumption in Developing Countries[END_REF], ex post risk coping [START_REF] Carter | Environment, Technology, and the Social Articulation of Risk in West African Agriculture[END_REF], high transaction costs [START_REF] Omamo | Transport Costs and Smallholder Cropping Choices: An Application to Siaya District, Kenya[END_REF], liquidity constraint and credit market failure [START_REF] Reardon | Links between nonfarm income and farm investment in African households: Adding the capital market perspective[END_REF], and the seasonality of agricultural activity [START_REF] Sahn | Seasonal Variability in Third World Agriculture: The Consequences for Food Security[END_REF]. Household livelihood strategies are jointly determined by these two sets of factors. Market development encourages households to reallocate their productive resources to higher-return activities [START_REF] Xia | Diversify and Prosper: Peasant Households Participating in Emerging Markets in Northeast Rural China[END_REF], while poor resource endowments, agricultural seasonality, frequent climatic hazards, and poor access to credit may all push rural households to undertake a wider range of activities in order to secure their livelihood.

Risks play a key role in the activity diversification process. Since they strongly influence rural production, income and welfare, risks are major "push" factors that encourage households to turn to a more diversified portfolio of activities [START_REF] Carter | Environment, Technology, and the Social Articulation of Risk in West African Agriculture[END_REF][START_REF] Reardon | Determinants and Effects of Income Diversification Amongst Farm Households in Burkina Faso[END_REF]. Both on-farm and off-farm diversification can thus be seen as efficient mechanisms for households to reduce income risks [START_REF] Ellis | Survey article: Household strategies and rural livelihood diversification in developing countries[END_REF][START_REF] Ellis | Rural Livelihoods and Diversity in Developing Countries[END_REF][START_REF] Hoogeveen | Income Risk, Consumption Security and the Poor[END_REF]. However, in a rapidly changing and volatile environment, uncertainty may also make agricultural households more reluctant to engage in new activities. This is particularly the case for poor households who typically have a higher absolute risk aversion [START_REF] Rosenzweig | Wealth, Weather Risk and the Composition and Profitability of Agricultural Investments[END_REF]. In a poor area, agricultural households may prefer to stick to traditional crops for which risks are known, even though expected returns associated with alternative activities are higher and a more diversified portfolio of activities would certainly reduce the expected hazard of total income. In sum, risk aversion combined with poverty traps, an ageing population and a massive migration of young people may strongly reduce incentives for poor agricultural households to allocate a higher portion of their land to non-food crops.

Risks are abundant in rural China, and given the lack of credit and insurance markets, the risk incidence is heavy for agricultural households. In the heart of a semi-dry area of northern China, Labagoumen township is exposed to climatic risks, especially droughts.

Additional sources of risk in the area come from numerous market imperfections brought by institutional reforms. Given the uneven development process as well as the incompleteness of the reforms, price risks have become a prominent risk for agricultural households. Indeed, in contrast to stable State procurement prices prevailing before the reforms started, farmers now face volatile and unpredictable market prices for agricultural products, especially for "new" products whose market is often very thin and highly fragmented.

Many studies have also shown that the rural poor have less access to lucrative alternative activities than their better-off counterparts because of high barriers to entry associated with these activities (Barrett et al., 2001a, b;[START_REF] Abdulai | Determinants of Income Diversification amongst Rural Households in Southern Mali[END_REF][START_REF] Woldenhanna | Income Diversification and Entry Barriers: Evidence from the Tigray Region of Northern Ethiopia[END_REF]. One of the most important barriers to entry is credit constraint. Indeed, a restricted access to credit and financial savings can impede high initial investments as well as the acquisition of assets that are essential to most non-farm activities [START_REF] Barrett | Food Marketing Liberalization and Trader Entry: Evidence from Madagascar[END_REF] 4 . In Labagoumen township, credit constraint may be an important factor to explain households decisions to engage in diversification. As reported in Table 4, 44% of the surveyed households declared having faced a credit constraint over the last 5 years. Among those who had borrowed money over the last 5 years, the vast majority borrowed from parents or friends, and only a few borrowed from institutional lenders such as credit cooperatives (4%), banks (5%) and local communities (3%). Among those households who reported the reasons why they did not borrow from financial institutions, 62% reported excessive collaterals required for official loans as well as restrictive and inflexible lending conditions as the main reason, followed by refusals from financial institutions (24%), high interest rates (13%) and the lack of financial institutions (1%).

In addition to financial constraints, another important barrier to entry to betterremunerated activities lies in skills and education constraints [START_REF] Smith | Livelihood diversification in Uganda: Patterns and determinants of change across two rural districts[END_REF]. In Labagoumen township, the educational attainment of farmers is low, with only 4.7 years of schooling for an average worker. Unskilled poor have no choice but to stick to activities with low education requirements and this low education level may reinforce inertia in terms of diversification behaviors. The situation may get even worse because of the sharp increase in educational costs that occurred over the past two decades in rural China. Data from our survey indicate that the average annual cost of education per child is about 2,000 yuan for children aged below 16 and jumps to more than 8,000 yuan for university-aged children.

Most households cannot afford such educational costs, which leads to early dropout, even before the nine years of compulsory education. In the long run, the high cost of education may strongly limit the rural population's ability to enter into more skilled-labor intensive activities.

Econometric evidence

Methodology

The determinants of rural income diversification can be modeled through a simple model of participation choice. Considering alternative activities j (j = 1,... J) , the choice of the household/individual (i) between J alternatives is determined by the utility level U ij derived from choosing activity j:

ij j i ij v Z U   
where Z measures household/individual characteristics.

The chosen (observed) activity j * is the one that maximizes the utility level:

  j j ij U U max * 
We use a multinomial logit model to estimate the probability of choosing a particular activity.

Within this framework, it can be shown that the probability to choose activity j can be expressed as follows:

    J k k i j i i i Z Z Z j y P 1 ) exp( ) exp( ) / (  
To identify the model,  k is normalized to 0 for activity j=1. This yields the following choice probabilities: As de Janvry and Sadoulet ( 2001), we assume that individual decisions are not independent across members of a given household. As a consequence, multinomial logit estimations for the determinants of individual participation in off-farm activities are run allowing for intra-household correlations through a cluster effect.

     J k k i i i Z Z y P 2 ) exp( 1 1 ) / 1 (  and      J k k i j i i i Z Z Z j y P 2 ) exp (
Tables 5 and6 report the estimation results, respectively for the household decision and the individual decision. The reported coefficients are the exponential values that can be easily interpreted in terms of "relative risk ratios": for each variable z, the relative risk ratio tells us how the probability of choosing j relative to the baseline alternative changes if z increases by one unit.

Results for the household economic diversification choice

Table 5 reports the estimation results for the household income-source diversification choice. At the household level, human and social capital variables do not appear to have a strong impact on decisions to diversify. We find that older households have a slightly higher probability (5% more) to participate in on-farm diversification as compared to no diversification and on the opposite, a slightly lower probability (5% less) to participate in migration. This suggest that older households are more experienced in farming, which favors on-farm diversification, while they are more reluctant to migrate into cities certainly because of a higher risk aversion with old age. Our empirical study at individual level confirms below that most migrants are young people.

The average education level of household members does not have any significant role on diversification decisions. The only, indirect, impact of education is to be found in the education level of the household head's father, which increases the likelihood (by 10%) of the household to undertake on-farm diversification activities. The potential "social capital" measured by communist party membership of the household head does not appear to have any significant impact at the household level.

Household composition seems more influential in terms of diversification strategies.

First, the household size is found to have a positive and significant impact on the decision to diversify agricultural activities: every additional member in a household increases the probability of undertaking on-farm diversification by a factor of 1.66. Second, a larger male labor force is associated with a much higher probability of both local off-farm diversification and migration (the relative risk ratios are respectively 2.84 and 7.40). These results can be explained by increasing returns to scale in household chores for households with a larger size and more labor availability that makes it easier for them to let some members engage in offfarm activities. Dercon and Krishnan (1996) on Ethiopa and Tanzania and [START_REF] Micevska | Rural Nonfarm Employment and Incomes in the Himalayas[END_REF] on India find similar results. Third, the presence of old members strongly reduces the likelihood of households to participate in migration (and to a lesser extent to engage in on-farm diversification): a higher dependency ratio of the household reduces the labor availability for migration. In sum, we find a stronger impact of the household composition on migration choice as compared to local off-farm, which is consistent with [START_REF] Shi | Choices between different off-farm employment subcategories: An empirical analysis for Jiangxi Province, China[END_REF] finding for southern China.

Among household assets variables, credit constraint is by far the one with the strongest impact on households' decisions to diversify. A constrained access to credit strongly reduces the probability of engaging in any diversification strategy, both on-farm and off-farm, locally or elsewhere. This result is consistent with the empirical literature showing that activity diversification is associated with entry-barriers and that financial resources or access to credit are needed to cover the starting-up investment of new activities (e.g. Barrett et al., 2001a, b;[START_REF] Abdulai | Determinants of Income Diversification amongst Rural Households in Southern Mali[END_REF][START_REF] Woldenhanna | Income Diversification and Entry Barriers: Evidence from the Tigray Region of Northern Ethiopia[END_REF]. As for the decision to diversify agricultural production, more arable land per adult also increases the likelihood of on-farm diversification.

A third measure of household assets includes its wealth, computed as a linear combination of household assets indicators through principal components analysis [START_REF] Filmer | Estimating Wealth Effects without Expenditure Data --or Tears: An Application to Educational Enrollments in States of India. World Bank Policy Research Working Paper[END_REF]. The wealth composite index is meant to measure an ex ante level of wealth that is supposedly less subject to endogeneity problems than a simple measure of household income. A set of six indicators has been selected to reflect the household possession of durable goods: ownership of a bicycle, a motorcycle, a color TV, a VCD-DVD player, a refrigerator and a washing machine. The wealth index is defined using the first principal component weights (with the first principal component accounting for 35% of the variance of our indicators). As for credit access, Table 5 shows that wealth plays an important role for the decision to undertake local off-farm activities, while it does not appear to be significant for migration decision or on-farm diversification. As a general rule, local off-farm decision is driven by households' asset position rather than by human/social capital or household composition.

At the individual level, the decision to participate in non-farm activities versus family farming work (reference choice) is analyzed in Table 6. Individual characteristics influence participation decisions. As De [START_REF] De Brauw | The evolution of China's rural labor market during the reforms[END_REF], we find a clear gender bias in participation in off-farm activities. Men have a much higher probability to participate in local wage employment, local self-employment and migration than do women. Likewise, young adults are more engaged in all types of non-farm activities than older individuals. Being the child of the household head also increases the likelihood of out-migration while it reduces participation in local self-employment.

Education has a contrasting role on decisions to participate in off-farm activities. A higher education level increases the probability to engage in a local wage work, while it has no impact on migration decision. Compared to empirical evidence in other developing countries, this result is somehow surprising. As noted by Miceska and Rahut (2008), "empirical evidence overwhelmingly finds positive effects of education on participation in non-farm activities". In our case, the fact that education does not significantly affect participation in migration can be related to the nature of jobs offered to rural migrants in Chinese cities. A well-documented feature of the urban labor market in China is that it is highly segmented between urban residents and rural migrants [START_REF] Démurger | Migrants as second-class workers in urban China? A decomposition analysis[END_REF][START_REF] Knight | Towards a Labour Market in China[END_REF]. Jobs taken by rural migrants in urban areas are mostly low-skilled jobs, with no specific requirement in terms of education. Although education has been found to be an increasing determinant of individual participation in migration over the 1990s (e.g. Zhang et al., 2002;[START_REF] De Brauw | The evolution of China's rural labor market during the reforms[END_REF][START_REF] Shi | Choices between different off-farm employment subcategories: An empirical analysis for Jiangxi Province, China[END_REF], the restricted access to better-paid jobs in urban areas may still hinder the importance of education for migration decision. On the other hand, the level of education in rural China is not zero, which implies that primary school education is largely sufficient to take urban low-skilled jobs with no incentives for individuals to get a higher education level. De Brauw and Giles ( 2008) have highlighted the trade-off between education and migration opportunity. With already high educational costs for rural households, higher expected wages in urban areas increase the opportunity cost of education in the short run as compared to the long-run expected returns to investment in education.

Household asset positions are found to strongly affect individual participation in offfarm activities while household composition does not have much impact on individual decision. The only exception is the number of elderly that negatively influences the decision to migrate. As at the household level, credit constraint strongly reduces participation in off-farm activities. Likewise, household wealth increases the likelihood to engage in local offfarm activities, with a stronger effect for self-employment that requires initial investment.

Finally, more arable land per adult significantly reduces the probability to migrate: more land will retain more people in agriculture and thus reduce labor availability for migration.

Last, village dummies confirm disparities across villages in activity diversification.

As an example, Sunzhazi village located at the entrance of the Nature Reserve is found to be quite dynamic in off-farm diversification, especially in local self-employment that reflects the vigorous development of family hotels and restaurants. In contrast, the remote Maoshan village is found to be significantly conservative in terms of local wage employment and migration.

Conclusion

The purpose of this paper was to highlight the main factors driving rural households and individuals in their decision to diversify economic activities. In developing countries, income-source diversification is a key livelihood strategy for rural households [START_REF] Ellis | Survey article: Household strategies and rural livelihood diversification in developing countries[END_REF] and as such, a good understanding of the determinants of access to off-farm sources of income across households is essential for the design of rural development policies.

In spite of the fact that one third of the households in the studied township have not engaged yet in any form of economic diversification, both non-grain cropping and off-farm activities contribute to an increased average total household income in the area. As in many parts of western China, villages in this mountainous region are characterized by land scarcity and by the absence of any strong comparative advantage in agricultural activities. Among offfarm activities, tourism and migration are by far the most remunerative activities for households in the township. Hence, increasing rural income and reducing rural poverty strongly relies upon the development of off-farm activities, including the development of a local rural industry, tourism as well as migration.

Conditions for success are based on the ability to increase access to off-farm activities for all rural households, particularly for households with little human, land and monetary assets. Our econometric analysis of both households' strategies and individual choices shows that key determinants of success are to be found in a better access to credit as well as in wellfunctioning labor markets in both rural and urban areas. On the credit side, we observed that a constrained access to credit (that affects 40% of the surveyed households) deeply reduces the likelihood to participate in any off-farm activity at both the household and the individual level. Hence, to overcome this important barrier to enter into more remunerative off-farm activities, massive efforts are required to develop rural infrastructure and financial markets.

Adequate local credit institutions serving small-scale rural investments are essential to release the constraint that most rural households face.

Regarding education, State intervention through the lowering of education costs is also required. As highlighted by De Brauw & Rozelle (2008), China is lagging far behind of its Asian neighbors in terms of both investment in rural education and educational attainment.

Hence, although the average level of education attainment has increased over time in rural China, it remains quite low (only 6.13 years according to De Brauw & Rozelle, 2008) in view of the nine-year compulsory education goal. Our results confirm this very low level of education, while better educated people are able to take more remunerative local wageearning jobs. Together with the need for higher investment in rural education, our results also suggest that on the supply side, efforts must be done in urban areas to give better access to skilled jobs to rural migrants. If migrants were to be given an equal access to urban skilled jobs as compared to urban residents, higher expected returns to education would probably pull more educated people out of rural jobs. Notes: The three types of economic diversification are defined as follows: i) "migration" comprises all households for which at least one member has had a migration experience over the last five years, whether or not the household is engaged in on-farm diversification and/or local off-farm activities; ii) "local off-farm activities" includes households for which at least one member is working off-farm in the local area whether or not the household is engaged in on-farm diversification, but excludes those households also involved in migration; iii) "on-farm diversification" includes households only engaged in on-farm diversification. Notes: The reference choice is "no diversification" (91 observations). Robust standard errors. The relative risk ratio for a one-unit change in a variable is the exponential value of the corresponding coefficient (exp(b) rather than b). Standard errors and confidence intervals are similarly transformed.

Map 1 -Beijing Municipality and Labagoumen Township

Labagoumen township

  original household survey conducted in December 2003 in 10 villages of Labagoumen township, a rural township located at the northern border of Beijing municipality. Traditional activity in Labagoumen township used to be grain production. Economic reforms have increased opportunities for households in the township to start new activities both on farm and outside farm. The survey provides detailed information on 293 households and 627 working individuals, including information on diversification behaviors and activity choices.

i)

  What types of on-farm and off-farm activities do rural households engage in? ii) What determines individual participation in the various off-farm activities? iii) What policy implications can be drawn to reduce rural poverty? To answer these questions, we proceed in two steps. First, we analyze household level decisions to diversify among several alternatives: within farm activities (towards non-grain production), local off-farm activities, and migration, where no diversification is the reference choice. Second, we analyze the determinants of individual participation in three sub-categories of off-farm activities: local off-farm employment, local self-employment and migration, where agricultural work is the reference choice.

  research is based upon a household survey conducted in December 2003 and covering 322 households, with 293 households engaged in productive activities and 627 working individuals. The surveyed area comprises ten villages in Labagoumen township located in the north of Huairou county, Beijing municipality. The township government is situated 160 km away from Beijing city and 93 km from Huairou county seat (Map 1). With 302 square kilometers, Labagoumen is the biggest township in Huairou county. It is composed of 15 administrative villages, with a population of almost 7,000 inhabitants among which one third of Manchu minority. Although it belongs to the rich municipality of Beijing, the township is a rather poor area as compared to both neighboring townships and provinces 1 .

  individuals can either leave the farm and take a local non-agricultural work, or migrate to towns and cities. As shown by De Brauw et al. (2002), choices have shifted over time towards an increased migration, which had become the most prevalent form of off-farm activity in rural China by 2000. Although migration was not so much widespread in our research area in 2003, villagers, mostly young people, are engaged in migration, with migrants accounting for 15.6 % of the total active population. Owing to proximity to big cities such as Huairou and Beijing and to a relatively well-developed infrastructure network in the region (most villages are served by relatively well-maintained roads), a quarter of rural households had at least a member with a migration experience over the last 5 years, mostly within Beijing municipality (about 23% of migrants work within their county, 30% in neighboring counties, and 35% in Beijing city).

  model is based on the strong assumption of independence of irrelevant alternatives (IIA). We have tested the IIA hypothesis using the Hausman-McFadden test based on comparing the parameters obtained with the multinomial logit with the parameters obtained by excluding the alternatives one by one. The IIA hypothesis cannot be rejected at conventional levels, which allows us to use a multinomial logit model. As indicated above, we consider two levels of analysis: i) the household choice of activity portfolio, and ii) the individual choice of participation in off-farm activities. The dependant variables are defined as follows. At the household level, we distinguish three types of activity diversification (as opposed to grain production only, the reference category): onfarm diversification (shift from grain to non-grain crops), local off-farm activities, and migration. At the individual level, we analyze the determinants of participation in three types of off-farm jobs (as opposed to agricultural work, the reference category) 5 : local off-farm employment, local self-employment and migration. The set of determinants for these two types of choice include human and social capital variables, household composition variables, household assets variables, and local institutions and village characteristics. For the household level choice, human and social capital variables include the household's average age and education level, communist party membership of the household head, as well as the education level of the household head's father. Household composition refers to household size, the number of elderly, and the number of male adults. Household assets include arable land per adult, a wealth composite indicator, credit constraint, and the constrained access to agricultural equipment. Local institutions and village characteristics are taken into account through village fixed effects. For the individual choice, most variables are similar to the household choice regressions, except individual characteristics that include age, education, communist party membership, gender, and the relationship to the household head. A brief description of explanatory variables' definition and construction is given in Appendix.

Table 1 -Households' diversification strategies across villages

 1

		No diversification	On-farm diversification	Local off-farm	Migration
	Total	34.1%	16.7%	23.6%	25.6%
	Dadianzi	39.3%	10.7%	17.9%	32.1%
	Dongcha	39.3%	32.1%	10.7%	17.9%
	Huying	39.3%	17.9%	17.9%	25%
	Labagoumen	34.6%	15.4%	26.9%	23.1%
	Maoshan	55.6%	11.1%	11.1%	22.2%
	Miaoying	10.7%	35.7%	28.6%	25%
	Sidaoxue	46.4%	17.9%	14.3%	21.4%
	Sunzhazi	16.7%	2.1%	54.2%	27.1%
	Xiahebei	26.9%	34.6%	11.5%	26.9%
	Zhongyudian	46.1%	0%	19.2%	34.6%

Source: Household survey conducted by the authors in 2003.

Table 2 -Diversification and household income by source, 2003

 2 Household survey conducted by the authors in 2003.

		Mean	Share in total income	% of households with income from the source	Mean among households with income from that source
	Total income in yuan	6,015
	Per capita income	1,935
	Farm income	1,645	36.7%	58%	2,818
	From grain crop	917	25.5%	46%	2,006
	Off-farm income	3,434	44.3%	52%	6,663
	From tourism	464	3.8%	6%	8,000
	Remittances	718	7.6%	12%	5,847
	Other income	936	19%	27%	3,429
	Source:

Table 3 -Household diversification behavior across income per capita quartile, 2003

 3 Household survey conducted by the authors in 2003. Notes: see Table1. Household survey conducted by the authors in 2003. Notes: a. Share of households who have borrowed money during the last 5 years in the total number of surveyed households.b. The sum of all lenders may exceed 100%, since households can borrow from more than one source. c. Share of households facing credit constraint in the total number of surveyed households.

		Total	First (poorest)	Second	Third	Fourth (richest)
	No diversification	34.1%	47.1%	45%	25%	18.8%
	On-farm diversification	16.7%	33.8%	15%	6.6%	13%
	Local off-farm	23.6%	2.9%	23.7%	31.6%	34.8%
	Migration	25.6%	16.2%	16.3%	36.8%	33.3%
	Source:

Table 5 -Determinants of economic diversification at the household level

 5 Household survey conducted by the authors in 2003.

		Mean value	On-farm	Local off-farm	Migration
			Relative risk ratio	P-value	Relative risk ratio	P-value	Relative risk ratio	P-value
	Household characteristics
	Average age	46	1.054	0.033	.9635	0.236	.9482	0.066
	Average education	4.5	1.07	0.427	1.036	0.667	.997	0.981
	Household head party membership	26%	1.208	0.673	1.293	0.595	.9235	0.874
	Household head father's education	2.2	1.106	0.115	1.053	0.445	1.035	0.594
	Household composition
	Household size	3.1	1.658	0.089	.8048	0.583	1.42	0.371
	# Male adults	1.4	.970	0.949	2.845	0.045	7.400	0.000
	# Elderly	0.5	.5543	0.123	.6308	0.358	.1874	0.017
	Household assets position
	Arable land/adult	2.4	1.254	0.064	1.040	0.802	.858	0.327
	Constrained access to agricultural equipment	16%	.6363	0.456	.9060	0.871	.788	0.660
	Constrained access to credit	40%	.502	0.107	.2578	0.003	.3780	0.025
	Wealth		.7929	0.183	1.670	0.001	.9740	0.869
	Village characteristics
	Sunzhazi village		1.006	0.996	7.404	0.001	1.702	0.433
	Dongcha village		2.816	0.086	.2193	0.119	.5159	0.439
	Maoshan village		.9338	0.924	.3874	0.121	.4809	0.273
	Miaoying village		10.93	0.001	3.389	0.138	2.042	0.403
	Xiahebei village		5.962	0.003	1.282	0.780	2.732	0.145
	Number of observations in the category	269	48		60		70
	Pseudo R²	0.28
	Source:

In terms of per capita GDP, Labagoumen township is the second poorest township in Huairou county, with 5,668 Yuan (approx.

2,715 PPP$) per capita per year in 1999.

Food crops include corn, soybean, sorghum, sweet potato, rice, millet, peas and wheat.

Even for farming activities, a restricted access to credit may reduce the incentive to invest in new activities. In Labagoumen township, most of the higher profit agricultural alternatives such as American ginseng, liquorice and tree planting require initial investment and long delays. This implies that households have to find the required initial amount plus additional income to live on before they can receive any return from their investment (a few years for ginseng, more for fruit trees, and decades for timber plantation).

There is no agricultural wage employment in our sample. This implies that off-farm activities refer to non-farm sectors only.

Appendix -Explanatory variables definition


Pham-Truong Son

Tien Hoang
email: tien.hoang@enpc.fr

Denis Duhamel
email: denis.duhamel@enpc.fr

Gilles Foret
email: gilles.foret@enpc.fr

Franziska Schmidt
email: franziska.schmidt@univ-eiffel.fr

François-Baptiste Cartiaux
email: cartiaux@osmos-group.com

Véronique Le Corvec
email: lecorvec@osmos-group.com

Load identification and damage evaluation of a beam subjected to moving loads by using co-located strain and acceleration measurements Bien centrer les noms

Keywords: Damaged Identification, Moving Load, Bridge, Strain, Acceleration

2 be determined. Numerical results with several scenarios will be presented to demonstrate the efficiency and the robustness of the method.

ABSTRACT

Bridge structural health monitoring and assessment are important subjects nowadays. In this paper, we propose a novel method which allows to evaluate both the bridge structural health and its loads based on strains and accelerations measured at the same location. The bridge is considered as governed by the Euler-Bernoulli beam equation. The strain can be obtained directly from sensors such as fiber optics. This type of sensors has many advantages such as high precision, high sampling frequency, durability… so that the dynamic strain component can be acquired easily. By decomposing the obtained signals into static and dynamic components, the correlation between strain and acceleration for the same type of component is clarified. The proportional coefficients, which reflect the intrinsic state of the beam and its load, can be obtained by optimization. From the variation of these coefficients, information on the damage of the bridge can be obtained. Moreover, in case of multiple loads (multi-axle vehicle), the distance between the axles of the vehicle can also

INTRODUCTION

Since most of bridges have been constructed in the 20 th century, bridge Structural Health Monitoring (SHM) and assessment is an important subject nowadays. Many studies about vibration-based structural damage identification have been carried out the last decades. The presence of damage decreases the global rigidity of the structure and leads to a reduction of its natural frequencies. Modal properties such as natural frequencies and mode shapes have been used as damage detection criteria [START_REF] Ostachowicz | Analysis of the effect of cracks on the natural frequencies of a cantilever beam[END_REF][START_REF] Rizos | Identification of crack location and magnitude in a cantilever beam from the vibration modes[END_REF]. However, the change in natural frequencies is relatively small, and the measure is influenced by environmental conditions such as temperature, humidity… [START_REF] Fan | Vibration-based damage identification methods: A review and comparative study[END_REF]. Allemang et al. [START_REF] Allemang | The modal assurance criterion -Twenty years of use and abuse[END_REF] presented the Modal Assurance Criterion (MAC) that studies the correlation between vibration modes of a structure by two different approaches. Based on MAC, Lieven et al. [START_REF] Lieven | Spatial correlation of mode shapes, the coordinate modal assurance criterion (COMAC)[END_REF] proposed the COMAC (Coordinate Modal Assurance Criterion) as an extension of MAC. In reality, these methods require a large number of sensors.

The advantage of a moving load is that the excitation does not depend only on time, but also on space. From the dynamic response of the beam, we can find out its modal properties. Zhu and Law [START_REF] Zhu | Wavelet-based crack identification of bridge beam from operational deflection time history[END_REF] presented a method that allows to identify the location of the damage from a single deflection time history at the midpoint of a simply supported beam, by using the Continuous Wavelet Transform (CWT). Nie et al. [START_REF] Nie | Bridge condition monitoring under moving loads using two sensor measurements[END_REF] used the time series of two distinct measured displacements in order to locate the damage. These methods work well when the velocity of the moving load is very small compared to the first critical velocity (i.e. resonance velocity [START_REF] Son | Dynamic response of a cracked multi-span continuous beam subjected to a moving multi-axle vehicle load[END_REF]) of the beam.

Nowadays, thanks to advanced technology, bridges can be monitored continuously. Unlike the traditionally electrical strain gauge, a long-gauge optical strand measures the average strain over its length. Hence, it has a higher probability to cover the damage. Moreover, Optical Strand sensors have many advantages such as high-sample strain measurements over long periods, high precision and durability [START_REF] Cartiaux | Traffic and Temperature Effects Monitoring on Bridges by Optical Strands Strain Sensors[END_REF].

In this paper, we present firstly a novel method that allows to evaluate the beam-like bridge structural health and eventually the loads of the vehicle in the absence of damage, by using the proportional coefficients between the measured signals (strain, acceleration). Moreover, this method can be used to determine the distances between the axles of the vehicle. In order to validate the method, we apply the optimization procedure to the passage of a 3-axle vehicle over a simply supported beam and re-evaluate the intrinsic properties of the beam and the vehicle. Then, a parametric study is carried out to study the dependence of the proposed coefficients to the properties of the damage (location, damage level, damage length…).

FORMULATION AND IDENTIFICATION METHODS

Analytical formulation

The study examines a simply supported beam of length L ? as shown in Fig. 1. We assume that the beam follows the Euler-Bernoulli beam theory, the governing equation of the beam which describes the vertical displacement is [START_REF] Svedholm | Vibration of damped uniform beams with general end conditions under moving loads[END_REF]:

(

) 1
where EI is the bending stiffness, ρS is the mass per unit length of the beam. In Eq. 1, the term c(∂u/∂t) represents the external damping mechanism whereas the term represents the internal damping mechanism, denotes the Dirac distribution. The beam is simply supported at its two ends, which leads to the following boundary conditions:

(2) where is the bending moment of the beam. The prime operator denotes the derivative of the function with respect to x. Moreover, before the time when the moving load enters the beam, it is considered at rest at all space points. Therefore, the following initial conditions are taken:

(3)

The dot operator denotes the derivative of the function with respect to t.

Forced responses of the beam under moving load

By modal decomposition, the solution of Eq. ( 1) can be expressed as a linear combination of the mode shapes and the generalized coordinated of the n th mode:

From the initial conditions (Eq. 2), and mode shapes which are normalized such that , where is the Kronecker delta, the form of for a simply supported beam can be expressed explicitly as: [START_REF] Lieven | Spatial correlation of mode shapes, the coordinate modal assurance criterion (COMAC)[END_REF] in which and . Since the mode shapes are orthonormal, the generalized coordinates are solutions of the following differential equation: [START_REF] Zhu | Wavelet-based crack identification of bridge beam from operational deflection time history[END_REF] where are the angular frequencies of the undamped system and the damping ratio. The initial conditions that are described in Eq. ( 3) can be written as: [START_REF] Nie | Bridge condition monitoring under moving loads using two sensor measurements[END_REF] The time-history of the strain at a given point located at a distance below the neutral axis of the beam is given by: [START_REF] Son | Dynamic response of a cracked multi-span continuous beam subjected to a moving multi-axle vehicle load[END_REF] As defined, the acceleration at this point is the second derivative of the displacement with respect to time: [START_REF] Cartiaux | Traffic and Temperature Effects Monitoring on Bridges by Optical Strands Strain Sensors[END_REF] Particular and homogeneous solutions. The general solution of Eq. ( 6) is the sum of a particular solution and the homogeneous solution: [START_REF] Svedholm | Vibration of damped uniform beams with general end conditions under moving loads[END_REF] Substitute Eq. ( 10) into the Eq. ( 4), we have:

By the same concept as the displacement, the strain and the acceleration are also decomposed into two components: the particular component and the homogeneous one, in which only the timedependent party is decomposed:

(12) (13)
Particular component. Hereafter, for simplification, we neglect the damping effect. By using the method of undetermined coefficients, a particular solution of Eq. ( 6) is given by the following expression [START_REF] Son | Dynamic response of a cracked multi-span continuous beam subjected to a moving multi-axle vehicle load[END_REF]:

(14)
where and . We denote the critical velocity which corresponds to the n th vibration mode of the beam.

for some small value of and . The particular component of strain and acceleration are then calculated:

(15)

From Eqs. (15), ( 16), we deduce the linear relation between the particular acceleration and the particular strain:

(17) Homogeneous components. The right hand side of the Eq. ( 6) is set to be zero. Its homogeneous solution can be written as:

(18) with the damped angular frequency of the system for the n th natural mode. The coefficients are determined from the initial conditions in Eq. ( 7). We deduce also:

(19)
When the load passes the beam at small velocity ( ) and the damping ratio is also small ( , we can assume that:

. We have:

Note that , we deduce the following relation between the homogeneous acceleration and the homogeneous strain for the n th mode:

(22)

Load and damage identification by optimization

Load and damage identification. Now, by suitable sensors placement, we study only the first mode of the homogeneous components. The dynamic strain and acceleration can be written as:

(23)

From the Eqs. ( 23), (24) and the proportional relations that are described in Eqs. ( 17), ( 22), we have:

(25)
We introduce the quasi-static strain, which can be defined as the limit of the particular strain when the velocity of the load approaches zero. As only the first mode is retained, for a small speed of the load ( ), from Eq. ( 14), we have:

Essayer de clarifier comment on obtient cette relation à partir de (14)

On the other hand, the quasi-static strain can be expressed as: where is the bending moment influence line ordinate at strain sensor's location in time . Hence, the Eq. (25) becomes:

(28) where and . For the measured signal , we can define the objective function as the sum of squared difference between the measured and the approximated strain signal:

. (29)

We search the coefficients that minimize . In case of an intact beam, the coefficient is proportional to the moving load. Otherwise, when the beam is damaged, both and is inversely proportional to the bending stiffness EI. When the load of the vehicle is unknown, we can use to evaluate the damage of the beam.

Axle spacing identification.

In case of the passage of a multiple-axle vehicle, the quasi-static strain is determined by superposition of the quasi-static strain which is caused by each individual force. Assume that a vehicle of M axles passes over the beam with a constant velocity . Let's denote the distance between the i th and (i+1) th axle and is the distance between the first axle and the i th axle (i≥2). By convention, . The quasi-static strain is given by: (30)

The objective function in the Eq. ( 23) now becomes:

(31) where . For a set of given distances , we search firstly a set of values which minimize Eq. ( 22). Then, we optimize the following objective function :

(32)

Once the distances that minimize the function are determined, we have the associated coefficients that have the same meaning as mentioned before.

NUMERICAL RESULTS AND DISCUSSION

Intact beam: Loads and axle distances identification

In this section, a simply supported Euler-Bernoulli beam is studied. The length of the beam is . The cross section is square of dimensions . The beam is made of steel, which has the modulus of elasticity and the density . The damping ratio of steel is equal to 2%. The fundamental frequency of the beam is . The beam is subjected to a 3-axle vehicle whose loads are and . The distances between the loads P 1 -P 2 and P 2 -P 3 are respectively and (Fig. 2). At , the vehicle enters the beam with constant speed , which is about 11% v c1 . Two sensors are placed at the midpoint of the beam, namely an accelerometer and an optical strand (long fiber optic strain sensor) of length 2 m. The responses are calculated analytically and the 12 first frequencies are taken into account, so that the cumulative effective mass factor is 90,4%. Then, these responses are filtered by a lowpass filter with a cut-off frequency that is equal to 1.7f 0 . Then, they are served as input of optimization process.

Table 1 presents the results of the optimization process in comparison with the reference for some parameters. The errors of the estimated values are relatively small.

Damage beam: Parametric study

For the parametric study, we use the same steel beam as the previous example. A two-axle vehicle and passes over the beam with constant speed , which is equal to 11 % the first critical velocity of the beam. The distance between these loads is (Fig. 3). The damage is modelled as a uniform decrease of bending stiffness over the length of the damaged zone [START_REF] Friswell | Crack modeling for structural health monitoring[END_REF]. In this study, we take a damaged zone of length , with various locations and various reductions of bending stiffness (the orange segment). The location of the damage is characterized by the position of its centre ( ). Moreover, we denote the overlaying length of the damaged and the strain sensor (the green segment). An accelerometer and an optical strand of length 2 m are placed at the midpoint of the beam, like the previous example. The beam is modelled in code_Aster, an Open Source FEM software. The FEM model has 40 linear beam elements, which follows the Euler-Bernoulli beam theory. The transient response is calculated on a generalized basis, using Newmark's scheme for temporal integration with and . The maximum frequency taken into account is up to 100 Hz. The sampling frequency of the responses is set to be 100 Hz, which is common to most optical strands in service. In this parametric study, we investigate 2 cases:

1. The damaged zone is located at the center of the beam:

. Its length and the reduction of stiffness are of various values. 2. The length of the damage is fixed at , its position and the reduction of stiffness change (Fig. 3).

Aligner à droite le paragraphe ci-dessous Figure 4 shows the dependence of relative change on as a function of the length of the damage and the reduction of the stiffness in case 1. When the length of the damage excesses the length of the optical strand, the coefficient remains constant (Fig. 4a). We can also see this on In both cases, the estimated distance obtained from the optimization procedure is m, The accuracy is equal to 1.67 %.

CONCLUSION

In this paper, we have presented a novel method that allows to detect the presence of a damage on a simply supported beam and the distance between axles of a vehicle. Numerical results demonstrate the sensitivity of the proposed method to damage. In the case of an intact structure, the method can be used to determine the vehicle's load.

Figure 1 :

 1 Figure 1: A simply supported beam subjected to a single moving force.

Figure 2 :

 2 Figure 2: Three-axle vehicle passing a simply supported beam at constant speed.

Figure 3 :

 3 Figure 3: Damaged beam subjected to a 2-alxle moving load.

Figure 4 :

 4 Figure 4: Case 1: Relative change of the coefficient as a function of (a): the length of the damage and (b): reduction of stiffness .

Figure 5

 5 Figure 5 shows the results of case 2. When the damage is located at midspan of the beam, the change on is maximal (Fig 5b.). On Fig 5b., we see that the curves corresponding to or are very close, because they have the same overlaying length (). When the damage is not covered by the optical strand, the coefficient does not change.

Figure 5 :

 5 Figure 5: Case 2: Relative change of the coefficient as a function of (a): the position of the damage and (b): reduction of stiffness .

Table 1 :

 1 Results of optimization

	Parameter	Notation	Unit	Reference values Estimated values	Error (%)
	1 st critical velocity	v c1	m/s	47.06	47.29	0.50
	Distance P 1 -P 2	d 1	m	3.50	3.50	0.02
	Distance P 1 -P 2	d 2	m	1.40	1.42	1.39
	Fist axle load	P 1	N	1000	993.15	0.68
	Second axle load	P 2	N	2000	1975.55	1.22
	Third axle load	P 3	N	2000	2032.04	1.60

ACKNOWLEDGMENTS

This work has been developed in the partnership between OSMOS Group and Ecole des Ponts ParisTech. The supports from the personnel of OSMOS Group are acknowledged as well.


Simone Ebert

Thomas Buffet

B Semihchan Sermet

Olivier Marre

Bruno Cessac

Temporal Pattern Recognition in Retinal Ganglion Cells is mediated by Dynamical Inhibitory Synapses

Electrophysiological recordings to stimulation with dark flashes of different frequencies

• pharmacological blockade of glycinergic transmission.

The retina encodes complex and rapidly moving visual scenes via its hierarchical, multi-layered structure where most neurons respond by a graded variation of their membrane potential without spiking (except ganglion cells). Here, we show that the retina (A) forms error signals via excitation and slower inhibition (B) while short-term plasticity refines temporal predictions of future stimuli.

Besides temporal predictions on a spatially uniform stimulus, the retina has been shown to anticipate motion and to respond to changes in trajectories. Can we extrapolate the mechanism proposed here to a more general setting, where amacrine cells enable the anticipation of a moving object [4] while short-term plasticity enables a temporal refinement of speed estimation?

1. Convolution of of the stimulus 𝑠(𝑡) and a temporal kernel

𝛼 ! 𝐹 ! 𝑡 = 𝛼 ! * 𝑠 𝑡 , "# ! "$ = - # ! % ! + 𝐹 ! 𝑡 . 𝑑𝑉 & 𝑑𝑡 = - 𝑉 & 𝜏 & + . 𝑤 ! 𝑝 (𝑉 ! , 𝜃 ! ).
• error signal via disinhibition from ON components

𝑑𝑛 𝑑𝑡 = 1 -𝑛 𝑘 '() -𝛽 𝑘 '(* 𝑝 𝑉 + "#$ %&&, 𝜃 + "#$ %&& 𝑛. 𝑤 + "#$ %&& t = n t : 𝑤 + "#$ %&& 𝑋 = 𝐸 ,-, 𝐼 ,-, 𝐼 &*. ,// .
4. Short-term depression in glycinergic amacrine cells by modulation of 𝑤 " !"# $%% via a dynamic variable n which obeys the kinetic equation [3] where

Computational model replicates experimental finding

Acknowledgements

• Latency shift flattens for shorter stimuli • In the model, this is due to lacking steady state of synaptic strength n(t)

STA of cells wtith OSR

The retina emits an Omitted Stimulus Response (OSR) [2] when an expected flash in a sequence is missing. The latency of this response scales with the period of the stimulus 1-to-1, thus signalling the timepoint when the missing stimulus was expected. This is an example for predictive coding in the retina, with a temporal expectation of future stimuli .

This work has been funded by the Neuromod Institute, Université Cote d'Azur.

Experimental Setup

• ON cells exhibit an OSR to dark flashes

• Glycinergic Amacrine cells are necessary to form temporal expectations.

Results

Removing synaptic plasticity removes latency scling

Internal variables of the model

•

  1. Hosoya, T.,Baccus, S. A., and Meister, M. (2005). Nature, 436:71-77. 2. Schwartz, G., Harris, R., Shrom, D., and Berry, M. J. (2007a). Nat Neurosci, 10:552-554. 3. Hennig, M. (2013) Frontiers in Computational Neuroscience, 10.3389/fncom.2013.00045. 4. Souihel, S., Cessac B. (2021). Journal of Mathematical Neuroscience ,10.1186/s13408-020-00101-z. • Depression reduces inhibition and thus latency shift • Higher depression for fast frequency yields shorter a latency • Little depression for slow frequency yields longer latency bioRxiv simone.ebert@inria.fr Glycinergic amacrine cells are necessary for latency shift 2. Signal integration via a dynamical system where 𝑉 ! is the voltage of cell 𝑋 with characteristic time 𝜏 ! 3. A RGC pools over the network where each input has a synaptic weight 𝑤 ! Computational model The latency shift remains unchanged when the intervals between flashes are brighter • Latency shift is lost when the flashes change duration with frequency.


Nina Colitti
email: nina.colitti@inserm.fr

Franck Desmoulin

Wafae Labriji

Alice Le Friec

Lorenne Robert

Amandine Michaux

Fabrice Conchou

Carla Cirillo

Isabelle Loubinoux

Adpd

FIVE-WEEK NERVE GROWTH FACTOR TREATMENT IS SAFE FOR BRAIN REPAIR

whether they are published or not. The documents may come

Introduction Materials & Methods

Malonate

Donnan GA et al., Stroke, 2008 ;Arvidsson A et al., Nat Med, 2002 ;Khelif Y et al., Theranostics, 2018 ;Zhu W et al., Drug Deliv, 2011 ;R. G. THORNE et al., Neuroscience, 2004 ;Davoust C et al., Stem Cells Res Ther, 2017

Results

Discussion

➢ Stroke is the leading cause of acquired severe motor disability in adults. ➢ The brain has limited ability to reconstruct itself by generating new neurons from stem cells. ➢ New neurons survival is too low (0.2%) to induce recover. ➢ Low survival is also due to the lack of growth factors.

This study focuses on Nerve Growth Factor (NGF) : neuroprotective/anti-inflammatory/stimulating neurogenesis effects.

The intranasal (IN) pathway was chosen as delivery route.

Malonate was used to induce M1 cortex and adjacent structure lesion in rats (n=19)

Sensorimotor function was evaluated by: 1/ grip strength test -2/ neurological severity score assessment -3/ limb-use asymmetry test

Conclusion

In this preclinical study, we show that fiveweek intranasal NGF treatment is safe and promotes tissue regeneration with a significantly higher proportion of neurons observed twelve weeks after brain injury.

  Lesion volume and RT identification were evaluated by: -MRI T2 weighted images -Nissl staining coloration (RT calculated on Case viewer software ; % of RT is normalized to the size of the lesion) -Immunostaining: GFAP (glial scar, astrocytes) Iba1 (microglia) PDGFRb ( vessels, pericytes) Doublecortin (neurons progenitors) beta 3 tubulin (immature neurons) NeuN (mature neurons) ❖ MRI T2 weighted might be a reliable biomarker for reconstructed tissue evaluation ❖ Migration pathways coming from the anterior part of the ventricle until the edge of the lesion ❖ Long-term NGF treatment promotes tissue reconstruction and remodeling ❖ NGF induces a significant increase in the percentage of mature neurons in the reconstructed tissue compared to PS (p=0.0043) ❖ Five-week NGF treatment is safe; however ten-week treatment retarded motor recovery (p=0.01) • First study with long-term NGF treatment • Reconstructed tissue characterized for the first time at 3 month post injury • The nose-to-brain pathway is a valid strategy for repeated and non-invasive administrations : Hope for chronic stroke treatment Treatment: IN administration of NGF (n=9) Control: IN administration of Physiological Serum (PS ; n=9)


Quentin Sourisseau

Maxime Deydier

Stéphane Paboeuf

Emilie Lepretre

Sylvain Chataigner

Xavier Chapeleau

Sylvain

Development of Composite patch for Offshore Steel Repair

Development of Composite patch for Offshore Steel Repair

Q Sourisseau 1, 3 , M Deydier 2 , S Paboeuf 2 , E Lepretre 1 , S Chataigner 1 , X Chapeleau 3 1 University Gustave Eiffel, MAST-SMC Bouguenais, France, University Gustave Eiffel.

2 Bureau Veritas Marine & Offshore, Research Department, Nantes, France.

3 University Gustave Eiffel, Inria, COSYS-SII, I4S Team, Bouguenais, France.

The number of floating production, storage and offloading units (FPSO) around the globe is in continuous increase and a relatively high number of them are now almost 20 years aged. The general geographical layout, being in tropical area makes the corrosion a fundamental ageing problem of the steel structures in structural area, like decks or side shell but also inner structure.

Therefore, there is a strong need for proposing repair solutions having low impact on their exploitation. Such repair solutions ("cold repair" in contrast with "hot works"), like adhesively bonded FRP (Fiber Reinforced Polymer) requires additional development, in particular in the preliminary characterization and design step, and regarding the durability issues.

Use of composite to build onsite repair seems adequate to solve this issue as they require limited heat (80°C) and can easily be installed on various shape, position and surfaces. However, the lack of application cases and design method lead to limited references on the best way to install composite patch repair. By design, the stress concentration at the border of the patch tends to decrease the overall strength and failure load that adhesives can sustain. Therefore, patch geometry must be addressed with care to obtain high performance patches. The overall capacity of the patch is highly related to the energy that can wisthand the bonded interface with the steel adherent, and so, a particular attention to the surface preparation must be taken. Finally, the patch layup controls the overall stiffness and strains in the patch, which must lead to a particular design to obtain the required properties. This paper presents the design methodology, the surface preparation protocol study and the manufacturing protocol of a composite patch developed during the Joint Industrial Project Strength Bond Offshore. The results of the static and fatigue test campaign in tension and bending are also presented. The assessment of the overall capacities of the composite patch repair are compared to a simpler bonded steel repair. The use of distributed strain measurement optical fiber as the new patch monitoring technique following [1] applied to composite patch are developed and highlighted.

[1] Q. Sourisseau, E. Lepretre, S. Chataigner, X. Chapeleau, L. Mouton and S. Paboeuf, Int J of Adh and Adh, Elsevier, 2022, 115, pp.1-20.