Mircea Gh

Munteanu

André Barraco
email: barraco@paris.ensam.fr

A Special Finite Element for Static and Dynamic Study of Mechanical Systems under Large Motion, Part 2

Keywords: finite element method, large motion, mechanical systems. MOTS-CLÉS :méthode des éléments finis, grand mouvement, systèmes mécaniques. 1

In the first part of the paper the theory of the 3D dynamics of mechanical systems composed by elastic beams, structures and mechanisms, was studied. These systems are divided into so-called macro-elements and the movement equations of one macro-element were established. Only the Euler-Rodrigues parameters are used to describe the global motion of the system. In this second part of the paper a special finite element (SFET) having four degrees of freedom per node, the Euler-Rodrigues parameters, is described in details. The stiffness and mass matrices are expressed only in nodal Euler-Rodrigues parameters. The most important aspect of the proposed approach is that the exact equations, written for the deformed configuration, are solved. Therefore an extremely accurate and very fast convergent method results. To validate the SFET finite element finally several 2D and 3D, static and dynamic examples are presented and the accuracy of the results is discussed.

RÉSUMÉ. Dans la première partie de ce papier nous présentions l'approche théorique de l'étude dynamique de systèmes, structures ou mécanismes, composés de poutres élastiques. Ces systèmes étaient décomposés en macro-éléments et les équations du mouvement d'un macro-élément étaient établies. Dans cette seconde partie nous présentons, en détail, un élément fini spécial (SFET) qui a quatre degrés de liberté par noeud, les paramètres d'

Euler-Rodrigues. Les matrices de raideur et de masse ne sont exprimées qu'en fonction de ces degrés de liberté. Le point le plus important à souligner est que les équations exactes, écrites sur la configuration déformée, sont résolues. Ainsi la méthode proposée est-elle très exacte et rapidement convergente. Des exemples en 2D et 3D, statiques et dynamiques sont présentés et démontrent la validité de l'élément SFET, la précision d'un tel élément est discutée et validée.

Introduction

In the first part of this paper, the equations describing the static and dynamic elastic behaviour of a mechanical system were developed. In this work we consider that "mechanical system" denotes a multi body system, in 2D or 3D, which may be structures or mechanisms, both involving large displacements.

The exact differential equations are written for the actual, deformed, configuration of the mechanical system and the equations are exact in the limits of the accepted hypotheses used to find out the constitutive equations. These equations are written in the global axes of the mechanical system. The non-linearity is provided mainly by the large displacements that could be so large that the initial geometry of the system is completely changed (geometrical non-linearity). This firstly means that the rotations are very large and they could not be considered as vectors anymore as in small displacement mechanical systems.

The basic idea of the method is that the actual configuration of any mechanical system might be uniquely described only by the rotations of some points called nodes, of course with the approximation of a rigid body motion. Because in 3D it is very complicated to work in rotations, the authors defined the rotation by using the quaternion or Euler-Rodrigues parameters. The constitutive equations and the variation of the total potential energy were expressed in these Euler-Rodrigues parameters. In the presented approach we try to find a quasi-exact solution of constitutive equations (in static or dynamic analysis) or, that is the same thing, to satisfy the variational principle, that is any geometrically admissible virtual variation of the displacements must lead to a zero total potential energy variation. The problem that we solve using a special finite element (called SFET in the paper) is the dynamic of a Bernoulli-Euler beam, the macro-element shown in the Figure 4 of the first part of this paper. In this second part the special finite element is described in details and finally several examples are presented.

Finite element approach

The proposed solution is based on the finite element method in space and the finite difference method in the time domain. An original special curvilinear finite element type (SFET) was elaborated with 4 degrees of freedom per node in 3D, the four component of Euler-Rodrigues quaternion, and only one degree of freedom per node in plane as the vector n is known (perpendicular to the plane of the problem). The finite element is a curvilinear one and may have several nodes, but very good results were obtained using a two-nodes finite element and, especially, a 3-nodes element (Figure 1). The two-node finite element (SFET2) is very easy to use, but the three-node finite element (SFET3) is extremely accurate even if it is more difficult to program. The stiffness and mass matrices are found out starting from the variation of the total potential energy written for the actual configuration. The shape function of the finite element is:

1 i i h s ) h s 1 ( ) s ( [1]
where e l ˆrepresents the column matrix of nodal unknowns having a number of elements equal to four times the number of element nodes, that is 8 for SFET2 and 12 for SFET3. SFET2, Figure 1, has the shape function for each Euler-Rodrigues parameter:

1 i i h s ) h s 1 ( ) s ( [2]
that is:

1 i i h s ) h s 1 ( ) s ( [3]
with the nodal unknowns matrix 1, we can write:

T 3,2 2,2 ,2 1 0,2 3,1 2,1 1,1 0,1 e ˆl l l l l l l l l and 1 i i h s ) h s 1 ( ) s ( For SFET3, Figure
3 , 2 , 1 , 0 k , ) h 2 S h 2 S ( ) 1 h S ( h 2 S h 2 S ) S ( 1 i , k 1 2 2 1 i , k 2 2 1 1 - k,i 1 2 2 1 1 k l l l l [4]
and in this case N(S 1 ) matrix has the dimension 4x12:

1 i i h s ) h s 1 ( ) s ( [5]
The nodal unknowns of one finite element are grouped in the column matrix:

T 3,3 2,3 1,3 0,3 3,2 2,2 ,2 1 0,2 3,1 2,1 1,1 0,1 e ˆl l l l l l l l l l l l l and ) h 2 S h 2 S ( ) S ( N ) 1 h S ( ) S ( N h 2 S h 2 S ) S ( N 1 2 2 1 1 3 2 2 1 1 2 1 2 2 1 1 1 ; ;
Now we have to establish a relation between the nodal unknowns e l and the column matrix

l l ds d ˆ
starting from the above shape functions:

e 1 ) (S ˆl N l [6]
For SFET2 we get:

1 i i h s ) h s 1 ( ) s (
and then for SFET3:

1 i i h s ) h s 1 ( ) s (
where :

) h 2 1 h S ( ) S ( N h S 2 ) S ( N h 2 1 h S ) S ( N 2 1 1 3 2 1 1 2 2 1 1 1 ; ;
For a finite element, it is possible to write the variation of the curvature along the finite element:

e 1 i 1 1 i 1 T e e 1 i 1 T e i ) S ( ) S ( ) S ( ) S ( 2 1 ) S ( ) S ( ˆl N A N N A N l l N A N l [7]
Or more simply:

e 1 i T e i Ŝ 2 1 l T l [8]
where κ i , i=1…3, represents the i-th element of the matrix κ. The matrices:

) S ( ) S ( ) S ( ) S ( 1 i 1 1 i 1 N A N N A N S T 1 i are symmetrical.
For SFET2 the dimensions of the T i matrices are 8x8 and its terms are polynomials of first degree in S 1 . For SFET3 the dimensions are 12x12 and the terms of T i matrices are polynomials of second degree in S 1 . Obviously we have:

e 1 i T e i Ŝ ˆl T l
The problem of replacing the distributed moment and distributed forces by nodal concentrated ones is discussed below. Firstly, let consider one finite element loaded with a distributed moment c(S 1 ). The following condition must be fulfilled:

n 1 T n , Bb T n T 1 , Bb T 1 T e 1 1 1 T Bb 1 T L T 0 0 0 e ... dS S S S C C G G l c G l
where L e is the length of one finite element (h for SFET2 and 2h for the SFET3) and n is the number of the element nodes. Considering that the above relation must be true for any virtual Euler-Rodrigues parameters we get:

1 1 1 T Bb 1 T L 1 T n 1 T n , Bb T n T 1 , Bb T 1 n , 1 , dS S S S S 0 e 0 0 c G N C C G G C C l l [9]
It is useless to find out the "true" nodal concentrated moments C j , since the moment reduced to the nodal Euler-Rodrigues degrees of freedom:

j T j , Bb T j j , 0 C G C l
can be directly introduced into the expression of the total potential energy. Now let's consider that the macro-element beam is loaded with a distributed force f. We will replace it by nodal concentrated moments C f imposing the condition that the two force systems perform the same virtual work for any virtual displacement field l ˆ :

j T j , Bb T j ns 1 j T j 1 L 1 T 1 S 0 T 0 1 dS S d ˆC G l f G l
The sum in the right hand member of the above equation is performed for all n s nodes of the macro-element, starting with the point 0. Introducing the shape function we get:

1 1 L 0 hs 0 T 1 T T e n 1 m j T j , Bb T j n 1 j T j n 1 j j , T j dS S d m 0 s s f G N l C G l C l l [10]
which obviously must be true for any field of virtual nodal Euler-Rodrigues parameters. In the Equation [10] "n m " is the number of finite elements on the current curvilinear co-ordinate S 1 and therefore

h , h ) 1 m ( S min hs 1
, m being the number of the current finite element. Thus the moments reduced to the Euler-Rodrigues nodal parameters, C l,j , are found out which could be assembled directly to the force vector of the finite element discretisation. The above relation was written for the two-nodes finite element, but it easy to extend it to the three-node finite element. If concentrated moments and forces act on one element, but not in a node, they are replaced with nodal concentrated moments in a similar manner as already shown. Is essential to notice that the presented special finite element type transform any external forces, concentrated or distributed, in nodal moments (see Eq. [9]). Now we substitute the Relations [1], [6] and [8] into variational form of the constitutive equations and take into consideration that distributed moments and forces were replaced by concentrated forces and moments. For the finite element assembly, which replaces the real structure, we get:

0 dS dS ˆj T j , Bb j T j T j j j n L 1 T 1 T e e 3 1 i 1 i L T e e i i n T e 0 j , el e e el C G l F G l l T l l T D l [11]
where n el is the number of the finite elements of the discretization, j means the sum for all concentrated forces or moments acting on the macro-element, n el,j is the number of elements between the origin O and the node in which the concentrated force acts and j l is the vector of the virtual Euler-Rodrigues parameters corresponding to the node j having the curvilinear co-ordinate S 1j , in which the concentrated moment acts, and D i was denoted the i-th diagonal element of the matrix D.

From the first term of the Equation [11] it results the secant stiffness matrix of one finite element:

3 1 i 1 i L T e e i i el dS ê T l l T D k [12]
In the Relation [11], the sum el n of the first term indicates the conventional assembling process of the elemental stiffness matrices into the stiffness matrix of the macro-element. For SFET2 the stiffness matrix has the dimensions 8x8 and the integrand is constant and so, the integration is easy to perform. For STET3 the stiffness matrix is 12x12 and the integrand contains polynomials of forth degree in S 1 . To compute the integral the Newton-Côtes numerical method was chosen because thus the integration points coincide with the nodes. The Newton-Côtes numerical integration method is exact for the polynomials of third degree or smaller, but very good results were obtained as will be shown later.

Static problem

The assembly process of the elemental stiffness matrices, k el into the macroelement stiffness matrix, K, follows the standard procedure. For the static problem the Equation [11] leads to:

0 ˆs s s T s s T s l F l l K l l E l [13]
which has to be true for the any virtual values of nodal unknowns s l geometrically admissible, that is the non-linear system of equations E is got:

0 ˆs s s s l F l l K l E [14]
The force vector F, which in fact contains concentrated moments reduced to the nodal Euler-Rodrigues parameters, is obtained from the last two terms in the Equation [11] and follows the standard procedure of assembling, too. In the above relation the vector of nodal unknowns for the whole macro-element was denoted s l .

It is interesting to notice that the elements of the stiffness matrix K are quadratic polynomials in the nodal unknowns s l .

The four nodal unknown for each node are not independent and therefore the secant stiffness matrix K is singular. We have to impose the following condition for each node, that is:

s j T j n 1 j , 1 ˆ l l [15]
Obviously it results:

s j T j n 1 j , 0 ˆ l l [16]
From the Equation [13] we select only the terms referring to the node j

0 E E E E ˆj , 3 j , 2 j , 1 j , 0 T j 3, j 2, j 1, j 0, j T j l l l l E l we express j 3,
l from the Equation [16] and we replace it in the above equation:

0 E E E E 1 0 0 0 1 0 0 0 1 ˆj j T j 2, j 1, j 0, j , 3 j , 2 j , 1 j , 0 j 3, j 2, j 3, j 1, j 3, j 0, T j 2, j 1, j 0, j T j E P l l l l l l l l l l l l E l [17]
One equation among each four equations was thus eliminated. Because to eliminate one nodal unknown, too, is quite impossible, we preferred to add the Equation [15]. Thus for each node we get the following four non-linear equations:

1 ˆ0 0 0 E E E E 1 0 0 0 1 0 0 0 1 j T j j , 3 j , 2 j , 1 j , 0 j 3, j 2, j 3, j 1, j 3, j 0, l l l l l l l l [18]
In order to avoid the singularities which could occur when l 3,j =0, in the Equations [18], at every iteration cycle and for every node, the maximum absolute value component of the nodal unknowns j l is searched and the corresponding equation is eliminated as shown above and then replaced by the Equation [15].

The boundary conditions for the clamped end of the macro-element, point 0, are:

0 ; 0 ; 0 ; 1 ˆ3 2 1 0 l l l l [19]
and are imposed in the usual conventional way.

A special mention has to be done for the 2D case. In this case 2 cos 0 l and, because the normal n is perpendicular to the plane, it results that

2 sin , 0 , 0 3 2 1 l l l
. So, we could chose two ways to elaborate the finite element: (I) either to consider two degrees of freedom per nodes, 0 l and 3 l , and all we presented above it is easy to particularise for the 2D case, or (II) we can chose an unique function that can describe completely the configuration of the deformed beam, the angle θ(S 1 ) which in this case is the slope of the tangent to the actual configuration in the point of curvilinear co-ordination S 1 . Therefore the finite element will have only one degree of freedom. The first way has the advantage that avoids the calculation of trigonometric functions and use polynomials instead, but leads to two degrees of freedom per node. In addition, it is easy to modify for 2D case the computer code written for the 3D case simply by eliminating the columns and rows corresponding to the 1 l and 2 l nodal unknowns that are known to be zero.

The authors followed the two approaches and the results were identical.

To solve the non-linear system [14] the iterative Newton-Raphson method is applied:

(i) s 1 (i) s 1) (i s l E J l l
where (i) s l is the column matrix of the nodal unknowns after the i-th iteration and J is the Jacobian of the matrix E, E l J s dd , computed for the (i) s l values of the nodal unknowns. The method has proved to be extremely rapid convergent. It is rather easy to compute the Jacobian as the expressions E are polynomials in the nodal unknowns

(i) s l .
In the static domain the finite element type elaborated by the authors is a nonincremental one, usually only one load step is sufficient to reach the final value, but in any case the accuracy of the results does not depend on the number of load steps. This happens because the exact equations (obviously in the limits of the beam theory) were solved and they are written for the actual deformed configuration of the beam system, which is not possible for the classic beam finite element. The accuracy and the very rapid convergence might be explicated by the fact that using only Euler-Rodrigues parameters as nodal unknown the constitutive differential equations become simpler and their order smaller. In addition the obtained finite element equations are polynomials in the nodal unknowns s l .

Dynamic problem

In order to simplify the explanations we will consider that the masses are lumped at the nodes. Let's consider the mass at node j, m j . On this mass, in the actual configuration, the inertia force

j j j , i mR F
acts and therefore we get:

1 S 0 2 0 3 1 3 0 2 1 2 1 2 0 2 2 j j , i dS ) ( 2 ) ( 2 1 - ) ( 2 t m 1 l l l l l l l l l l F
We can write successively:

T 1 T 2 0 0 2 1 3 3 1 3 0 0 3 1 2 2 1 1 1 0 0 2 0 3 1 3 0 2 1 2 1 2 0 [22]
Hence the first variation of the potential of the inertia force of the mass m j is:

1 S 0 3 T 2 T 1 T 3 T 2 T 1 T S 0 1 T 3 T 2 T 1 T j j , i T j dS dS m F j , 1 j , 1 l B l l B l l B l l B l l B l l B l l B l l B l l B l r [23]
where S 1,j is the curvilinear co-ordinate of the mass m j , or in a simpler form:

3 1 p S 0 S 0 1 p T 1 p T S 0 1 p T j j , i T j j , 1j , 1 j , 1 dS dS dS m F l B l l B l l B l r
Now we will apply the Relation [23] to the finite element assembly, that is we will introduce the shape functions N, Relation [1]. We get:

_ dS dS m F l N B N l l N B N l l N B N l l N B N l r
where nj is the number of elements from the point 0 until the mass m j . We could put the above relation written for a concentrated mass m j into a simpler form:

s j , v s j j , i F l M l M
[24]

with: Some concentrated rotational inertia may exist in node, too. For instance in this way we could take into account the transversal dimension of the beam.

For instance let's consider that in the node j we have to add the rotational mass inertia given by: 3 2 1 J 0 0 0 J 0 0 0 J J The above tensor is written in local reference frame R B (G,B i ). The concentrated rotational inertia is converted to the nodal unknowns, in the global axes system R e (O, e 1 , e 2 , e 3 ), according the relation:

j j , Bb T j , Bb T j 0 0 G J G J l
and then assembled to the mass matrix in the conventional way.

Relations [25] and [26] seem to be very complicated, but from programming point of view it is very simply to write a code to compute the two mass matrices.

The variational formulation becomes for the finite element assembly:

0 ) , t ( ˆs s s v s T s l F l K l M C l M l [27]
for any geometrically admissible virtual Euler-Rodrigues nodal parameters. In order to impose the conditions [15] for every node, we will use a similar approach to that in the static case. Thus we will replace the virtual "displacement" column matrix s l by those given by the following relation:

s s l H l
H is a 4n s x 4n s quasi-diagonal matrix, n s being the total number of nodes: where we could consider that 3 l is the absolute maximum of the four Euler-Rodrigues parameters for each node and found out at each iteration cycle. Thus all matrices, M, M v , C, K and F become:

F H F K; H K C; H C ; M H M M; H M T T T v T v
T and will have one zero row at each node corresponding to absolute maximum of the four Euler-Rodrigues parameters of every node. Only in the matrix K instead of this zero row we add the condition [15] for each node. Finally the non-linear system of equations describing the movement of the assembly of SFET becomes:

F l K l M C l M s s v s ˆ [28]
where C is the damping matrix which will not be discussed in this work. The column matrix F is found out in the same manner as in the case of static analysis.

The five matrices M, Mv, C, K and F depend on the configuration of the macroelement. In other words the mentioned matrices depend on the time t. Moreover, the square matrices M, Mv, C and K of the macro-element are full and nonsymmetrical, but this is not necessarily a disadvantage because the number of the finite elements for each macro-element is rather small.

To solve the non-linear system [28] the well-known Newmark method was adopted. At every iteration cycle the following conditions are imposed for each node:

0 ; 0 ˆj T j j T j j T j l l l l l l
If the studied macro-element is not clamped, that is the beam can perform translation movements, up to three extra degrees of freedom and their corresponding equations are to be added to the non-linear system [28]. These supplementary degrees of freedom represent the translation displacement of the point 0. Only the mass matrices M and M v are modified. The matrix F is modified only if external concentrated forces act on the beam, in the node 0.

Examples

Firstly it is interesting to point out that SFET3 is exact for small displacement of straight beams loaded statically with concentrated forces. In 2D case, for small displacements, we have

2 2 sin 3 l
, where θ is the slope of the beam. According to the Equation [4] the slope θ is a polynomial of second degree in the curved coordinate S 1 that is exact following the Euler-Bernoulli beam theory. For instance a cantilever beam loaded with the concentrated force F on the free end has:

1 i i h s ) h s 1 ( ) s (
the free end displacement and slope, respectively. As we expected, only one SFET3 leads to the exact value for the slope. The accuracy for the displacement depends on the numerical method used to compute the displacements starting from nodal unknowns, the slopes, not on SFET finite element, and it could be as small as we want.

To illustrate the performances of the special finite element type (SFET) we present several examples, in static and dynamic regime, in 2D and 3D. The first one is the 2D post-buckling static behaviour of the Euler beam problem. This example was chosen to check the proposed solution to the exact solution given by Timoshenko, [TIM 61]. The number of finite elements was chosen in order to get the displacements for the free end of the beam with a relative error of 10 -4 . The cantilever beam is represented on Figure 3. Figure 4 shows the various positions of the reference lines when the ratio of F over the Euler load F cr is increasing from 1.015 to 9.116. It can be observed that the displacement along the x-axis is almost twice the length of the beam and that the angle of rotation of the final cross section is about π. This example indicates that with space decomposition into 10 finite elements SFET3 (21 nodes) we obtain the exact solution for the free end rotation with a relative precision of 10 -4 . Table 1 gives some numerical results for displacements.

e 1 e 2 O ( )

(l) b 1 b 2 B 1 B 2 F Figure 3. Post-

buckling static behaviour of a cantilever

The computing of the buckling force is an eigenvalue problem that could be solved by SFET, too. For instance the exact Euler buckling force and the corresponding values computed using 10 or 20 SFET3 are respectively:

1 i i h s ) h s 1 ( ) s ( F/F cr 1

Figure 4. Several post-buckling equilibrium configuration

The next example is shown in Figure 5, the cantilever loaded with a force and a moment on the free end. All the data are given in figure. The results were checked using the COSMOS 2.0 finite element code. The tolerance was 1e-5 for both, SFET3 program and COSMOS program. COSMOS could reach only 40% of the final load in 231 load steps, with several iteration cycles being done for each load step, Figure 6a. AUTOSTEP command was activated. For this load the results got by SFET3 and COSMOS are practically the same. On the contrary, SFET3 needed only one load step for 40% of the final load and 3 load steps and 25 iteration cycles for the final load. The final configuration of the beam is given in the Figure 6b. In order to verify the results, the sectional efforts (moments) are computed following two ways. Using the static equation applied on the deformed beam we can write:

1 i i h s ) h s 1 ( ) s (
where R F is the position vector of the force applied on the free end of the beam and

T z 1 T z 1 C 0 0 and ; F 0 0 C F
. Using the physical relation we obtain:

1 i i h s ) h s 1 ( ) s (
where the curvature vector χ(s) is computed by means of the nodal Euler-Rodrigues parameters, l s , found out after solving the FE problem. The average relative error is computed with the formulas:

1 i i h s ) h s 1 ( ) s ( [29] L=100 GJ=2000/2.6= 769.23 EI2=EIy=1500 EI3=EIz=1000 A=5(cross section area) ρ=10 -6 (mass density) 0.1.20 SFET3 F z =0.25 C z =120 x F z L z C z y Figure 5

. Cantilever beam loaded with a force and a moment

The moments were computed in local axis system R B (G,B i ). For the above example and for the final load these errors are ε 1 =0.06% and ε 2 =0.5%. The error ε 2 might seem too big, but we have to consider that the beam is highly distorted, the torsion angle included.

The third statical example is shown in the Figure 7, a helicoidal spring, an initial 3D curved beam subjected to a concentrated force. All the data, the initial configuration and the final one are shown in the Figure 7. Remarkable is that to reach the final value of the load only one load step and 9 iterations were needed, for an absolute tolerance of 10 -5 for the Euler-Rodrigues parameters. In this case the errors computed accordingly the Relations [29] are ε 1 =0.6% and ε 2 =0.8% respectively, that is an extremely good accuracy for stresses (not for displacements). 9 present the accuracy of the method. In Figure 8, the strong nonlinear dependence of force versus displacement at the point of application of the force is represented and shows that very good results are got with only 10 SFET3.

Figure 9 shows how fast the accuracy increases with the number of iterations for 40 SFET3 model: at each iteration step the error diminishes over ten times. Error versus number of iterations plot The first dynamic example is a very simple one, the eigenfrequencies and eigenmodes of the straight cantilever represented in Figure 5. This problem was solved in the small displacements hypothesis, the stiffness and mass matrices were computed as shown in this work. The problem was solved as 3D one, the numerical results using SFET3 being compared to the analytical ones. The results are given in the Table 2 and a extremely good concordance might be noticed. Figure 10

Table 2. Comparison between eigenfrequencies computed numerically (SFET3) and analytically

Figure 11 shows one example extracted from the paper [SIM 86], the "flying spaghetti". All the data are presented in the Figure 11. Firstly the problem was solved exactly like in the work [SIM 86]. The problem is a 2D one as the load consists only in the force F and the moment T z . The comparison between the results from the work [SIM 86] and those obtained using SFET2 is exposed in Figure 12. A very good concordance might be noticed. To get a 3D problem the moment T y was added to the load, Figure 11, and the results are shown in Figures 13 and14.

Figure 10. First eigenmodes ot the cantilever beam computed using SFET3

The last example was studied in [SHA 98] and [TAK 99], the free falling of a flexible pendulum. The pendulum is assumed to fall under the effect of gravity, Figure 15. The beam has a length of 1.2 m, a cross sectional area of 0.0018 m 2 ,a mass density of 5540 Kg/m 3 , a modulus of elasticity of 0.700x10 6 N/m 2 and the moment of inertia of 1.215x10 In order to verify the accuracy of SFET two methods were applied. First the total energy of the pendulum was computed at each time step. It comprises three components: the elastic energy, the potential energy and the kinetic energy. Their sum, the total energy, has to be constant during the motion. Figure 15b shows the variation of all these energies with the time. The error was estimated using the relation:

max k min max E / E E
where E is the total energy and E k is the kinetic energy. Its value depends on the time step and for this example we got 2.65% for 220 time steps, 1.05% for 550 time steps and 0.688% when 1100 time steps were used. The second method for computing the error uses the relations [29], but this time to compute the bending moment the gravity distributed force as well as the inertia forces were considered. The errors ε 1 and ε 2 were evaluated at each time step. The maximum error do not exceed 1…2 % which is a very good accuracy taking into account that the error was computed for the bending moment.

Conclusions

A special finite element (SFET), which has four degrees of freedom per node, the nodal values of the Euler-Rodrigues parameters, was presented in this paper. The main advantage of SFET is that it allows without many complications to solve the exact movement equations of the mechanical systems composed by elastic beams. The proposed finite element may have several nodes, but SFET3 proved to be extremely accurate and simple enough from the programming point of view.

The method is very accurate and very rapidly convergent. This might be explained essentially by the following factors: (I) the exact equations are solved, (II) the unknowns are not the linear displacements, but their derivatives, and also (III) the stiffness and mass matrices are polynomials in Euler-Rodrigues parameters, thus avoiding to use trigonometric functions.

The accuracy for the displacement depends on the numerical method used to compute the displacements starting from nodal unknowns, the Euler-Rodrigues parameters. To compute the displacements the Relation [12] of the first part of the paper has to be used. This integration might be done introducing the shape functions [2] or [4] in the above mentioned relation and then using as many integration points as the accuracy requires. The accuracy for displacement was assessed directly by computing efforts considering both internal and external equilibrium of the beam (See the Relations [29]), and extremely good accuracy was again obtained.

The finite element formulation is written in the global coordinate system, the same for each macro-element in the mechanical system. This approach has several advantages one of the most important is that a simple expression for inertia forces is obtained. In addition due to the good accuracy and convergence of the method, the computer codes elaborated by the authors are very fast, a computer time of several minutes are enough for all 3D examples presented in the work. All the computer codes used in this paper and applying SFET were written by the authors in MATLAB 5.3.

Several examples are presented in this paper and they confirm the accuracy of the SFET. It is important to notice that the computed errors concern the efforts (bending and/or torsion moments) and not the nodal unknowns. The bending and torsion moments are expressed in derivatives of Euler-Rodrigues parameters. The errors for derivatives are much bigger than the errors for the functions, as it is known! In the static domain the finite element type elaborated by the authors is a nonincremental one, usually only one load step is sufficient to reach the final value, but in any case the accuracy of the results does not depend on the number of load steps. This happens because the exact equations (obviously in the limits of the beam theory) were solved and they are written the actual deformed configuration for of the beam system. This is not possible for the classic beam finite element.

Figure 1 .

 1 Figure 1. Two-node and three-node finite elementsThe shape function of the finite element is:

Figure 2 .

 2 Figure 2. Topology of the matrices M j and M v,j

Figure 6 .Figure 7 .

 67 Figure 6. Deformed configuration of the cantilever R=10 (spring radius) Thread=10 n=2 (number of threads) F=10 (vertical direction) GJ=2000/2.6=769.23 EI2=1500 EI3=1000 40 SFET3

Figure 8 .

 8 Figure 8. Force-displacement dependenceFigure 9. Error versus number of iterations plotThe first dynamic example is a very simple one, the eigenfrequencies and eigenmodes of the straight cantilever represented in Figure5. This problem was solved in the small displacements hypothesis, the stiffness and mass matrices were computed as shown in this work. The problem was solved as 3D one, the numerical results using SFET3 being compared to the analytical ones. The results are given in the Table2and a extremely good concordance might be noticed. Figure10contains the first 6 eigenmodes in the xy plane.

Figure 11 .Figure 12 .

 1112 Figure 11. The "flying spaghetti"

Figure 13 .

 13 Figure 13. 3D "flying spaghetti": the first 15 s of the movement

  Figure 15. Flexible pendulumThe errors ε 1 and ε 2 were evaluated at each time step. The maximum error do not exceed 1…2 % which is a very good accuracy taking into account that the error was computed for the bending moment.

Table 1 .

 1 Comparison between analytical and numerical SFET results

			.015 1.152 1.518 1.884 2.541 4.029 9.116
		α	20	60	100	120	140	160	176
	Timoshenk	u/L 0.0300 0.259 0.651 0.877 1.107 1.1340 1.577
	o	v/L 0.220 0.593 0.792 0.803 0.750 0.625 0.421
	10 SFET3	u/L 0.0292 0.2585 0.6504 0.8763 1.1066 1.3403 1.5774
	21 nodes	v/L 0.2157 0.5928 0.7914 0.8032 0.7504 0.6244 0.4209
					14

  contains the first 6 eigenmodes in the xy plane.

	Mode	#1	#2	#3	#4	#5	#6	#7	#8	#9	#10
	Anal.	4.972 6.090 31.16 38.17 87.24 106.8 171.0 209.4 282.6 346.1
	SFET3 4.971 6.089 31.15 38.15 87.24 106.9 171.0 209.5 283.1 346.6

  -6 m 4 .

			EI 2 =EI 3 =100
			GJ=40
			A =1
	y		I	=10
	8		T z (t)
		T y
			80
	x	F(t)	T z (t)	2.5 s	t
	6			F(t)=T z (t)/10 T y (t)=T z (t)/8

[20] with:

It is obvious that in a similar way as Relation [20] we can write:


Elen Le Foll
email: elefoll@uni-koeln.de

The Potential Impact of EFL Textbook Language on Learner English: A Triangulated Corpus Study *

Keywords: causative constructions, English as a Foreign Language, materials design, input, textbook corpus, learner corpus, construction learning

 input-dependent L2 acquisition hypothesis which suggests that differences between EFL and ESL learner corpus data may be (partially) attributable to EFL learners' minimal exposure to natural language input. For instance, by far the most frequent construction in the examined textbooks, [X MAKE Y Vinf], was previously shown to be vastly overused by EFL learners (e.g., Wong 1983; Gilquin 2016), whilst constructions that are underused by learners as compared to ENL data (e.g., Gilquin 2012; 2016) are very rare in the textbook data. Low frequencies of specific constructions in textbooks also correlate with EFL learners producing high rates of syntactic errors when attempting these constructions.

Introduction

In instructed foreign language learning contexts and, in particular, at lower secondary school level, a considerable proportion of learners' language input comes from their textbooks (Volkmann 2010: 235). Given their relevance, it will come as no surprise that there is now a growing body of corpus-linguistic studies analysing what will be referred to here as "Textbook English". Most studies to date have compared Textbook English to that of naturally-occurring or "authentic" English as a Native Language (ENL) and many have reported major discrepancies in the use of individual lexico-grammatical features such as future constructions (e.g., [START_REF] Mindt | Sprache, Grammatik, Unterrichtsgrammatik: futurischer Zeitbezug im Englischen (Schule und Forschung)[END_REF], the progressive (e.g., [START_REF] Römer | Progressives, Patterns, Pedagogy: A Corpus-Driven Approach to English Progressive Forms, Functions, Contexts, and Didactics[END_REF], and the phraseological patterns of highfrequency verbs (e.g., Le Foll 2022a). Far fewer studies, however, have explored potential correlations between Textbook English and the language produced by learners of English as a Foreign Language (EFL) as the recipients of these textbooks. Exceptions include triangulated investigations of if-conditionals (e.g., [START_REF] Gabrielatos | If-conditionals in ICLE and the BNC: A success story for teaching or learning[END_REF][START_REF] Möller | From pedagogical input to learner output: Conditionals in EFL and CLIL teaching materials and learner language[END_REF][START_REF] Winter | Testing the Pedagogical Norm: Comparing Ifconditionals in EFL Textbooks, Learner Writing and English Outside the Classroom[END_REF], which have, on the one hand, shown that the representations of if-conditionals in EFL textbooks are far removed from naturalistic ENL use and, on the other, that EFL learners are influenced by the types of if-sentences featured in such pedagogical materials, even at higher levels of English proficiency.

Since they entail the comparison of three corpora, such triangulated, Textbook English vs. ENL vs. Learner English, studies are rarely conducted. The present study shows how the results of existing EFL learner vs. ENL corpus research may be drawn upon to explore whether and, if so, how textbook language input may influence learners' syntactic and lexical choices. As an example of this influence, the current study investigates the potential impact of Textbook English on EFL learners' use of causative constructions. Causative constructions were chosen as an example of a construction that straddles textbooks' traditional binary distinctions between vocabulary and grammar and because they have been the focus of a number of learner corpus studies that have pointed to persistent issues in learners' use of this construction (e.g., Wong 1983; [START_REF] Altenberg | The grammatical and lexical patterning of MAKE in native and non-native student writing[END_REF][START_REF] Liu | Investigating learner vocabulary: A possible approach to looking at EFL/ESL learners' qualitative knowledge of the word[END_REF][START_REF] Gilquin | Lexical infelicity in causative constructions. Comparing native and learner collostructions[END_REF][START_REF] Gilquin | Input-dependent L2 acquisition: Causative constructions in English as a foreign and second language[END_REF].

The following section introduces the theoretical background and summarises insights from learner English corpus studies on causatives. Section 3 presents the textbook corpus data queried as part of this investigation, the procedure adopted to process the data, and the method employed for the phraseological analysis. Section 4 begins with the results of the qualitative analysis of the descriptions of causative constructions featured in the textbook grammar sections and continues with the corpus-based analysis of the causatives implicitly featured within the texts of the same textbooks. The representations of causative constructions in textbooks are then compared to the results of previous corpus-based learner English studies with the aim of teasing out the potential impact of Textbook English on EFL learners' language production, before showing how corpora can be used to supplement textbook materials.

Theoretical background

Causatives in the Construction Grammar framework

The present study focuses on EFL textbooks' representations of periphrastic causative constructions with non-finite verb complements such as: (2) What situations cause people to make faces like that? <TEC: Green Line New 5>

(3) Josh got me to try bungee-jumping. <TEC: Piece of Cake 3 e > (4) She had her old phone repaired, even though she expected to get a new one.

<TEC: Solutions intermediate>

Causative constructions in English have been extensively described within a number of theoretical frameworks (2016: 115). The present study adopts a Construction Grammar (e.g., † In this and the following examples, the code between angle brackets denotes the source of the example, i.e. the abbreviation of the corpus followed by the relevant subcorpus (e.g., textbook volume, learner L1 language) or document ID. Unless otherwise stated, emphasis was added. [START_REF] Goldberg | Constructions: A construction grammar approach to argument structure[END_REF][START_REF] Goldberg | Constructions at work: The nature of generalization in language[END_REF] approach. Within this framework, constructions are, at the most basic level, understood as "conventionalized pairings of form and function" (Goldberg 2006: 3) and are seen as the basic units of language. Traditional English grammars have usually presented the different types of English causative constructions as more or less synonymous (see Gilquin 2010: 98-99 for overview). However, within a Cognitive Linguistics (e.g., [START_REF] Langacker | Cognitive Grammar: A Basic Introduction[END_REF] and, more specifically, Construction Grammar framework, the principle of "one form, one meaning" prevails. In other words, "[i]f two constructions are syntactically distinct, they must be semantically or pragmatically distinct", too (Goldberg 1995: 67; Construction Grammar assumes that speakers' language knowledge consists of "a network of learned pairings of form and function, or constructions" (Goldberg 2005: 17). These range from morphemes to individual words, to collocations and verb-argument structures. Whilst the idea that language can be broken down to constructions that span right across the traditional grammar-lexicon divide is no longer new, it remains controversial in many a linguistic circle (Langacker 2008: 22). In foreign language pedagogy, however, approaches advocating remarkably compatible ideas go back a long way (e.g., [START_REF] Hausmann | Wortschatzlernen ist Kollokationslernen. Zum Lehren und Lernen französischer Wortverbindungen[END_REF]). Yet such approaches have yet to be reflected in most of the EFL textbooks currently in use in secondary schools, which still tend to apply a strict (but often largely arbitrary) distinction between grammar units on the one hand and vocabulary on the other.

Causatives in English Language Teaching and Learner English

Periphrastic causative constructions are both very frequent and of high communicative value; however, they form complex syntactic patterns which often prove problematic for learners of English. [START_REF] Gilquin | Lexical infelicity in causative constructions. Comparing native and learner collostructions[END_REF] shows with data from the International Corpus of Learner English (ICLE; [START_REF] Granger | International Corpus of Learner English[END_REF]) that even proficient learners of English frequently confuse the different verb complements of causative verbs (e.g., using a to-infinitive instead of the bare infinitive with MAKE as in ( 6)) or struggle with the placement of the causee (e.g., ( 7)).

( most EFL users will have, at least at first, learnt English as a school subject with few additional language inputs other than their textbooks, ESL users have usually completed large parts of their education in English and/or live in countries where English is an official language (e.g., India, Nigeria, Singapore) and are therefore exposed to much more non-textbook materials than most EFL learners.

Gilquin (2012) identifies three possible causes for learners' unidiomatic use of certain periphrastic causative constructions: a) lack of register awareness, b) negative L1 transfer, and c) the inadequacy of teaching materials. The influence of this third potential factor (also identified by Altenberg & Granger [2001: 184] as a potential source of learners' misuse of MAKE causatives) was further explored in [START_REF] Gilquin | Input-dependent L2 acquisition: Causative constructions in English as a foreign and second language[END_REF], whose results suggest that different types of language input may indeed be partially responsible for the differences observed between EFL and ESL learners' use of causatives. [START_REF] Gilquin | Input-dependent L2 acquisition: Causative constructions in English as a foreign and second language[END_REF] results support her inputdependent L2 acquisition hypothesis which states that:

considering that (i) EFL learners learn English mainly in an instructional setting and receive relatively little exposure to the target language, and (ii) ESL learners, in addition, also acquire English in a natural setting and thus receive more exposure to the target language, it can be expected that ESL learners will have been exposed to more (authentic) instances of a given construction than EFL learners and will thus have better integrated its schema, resulting in a better, more native-like command of the construction. (Gilquin 2016: 119) This hypothesis, which raises the question of the (in)adequacy of teaching materials in EFL instructional settings, is at the heart of the present study. It draws on a large corpus of EFL textbooks to explore the distributional patterns of the syntactic, semantic, and phraseological characteristics of the causative constructions to which learners are frequently exposed to via their textbooks. Moreover, it explores whether textbooks provide learners with adequate input that will help them produce these constructions idiomatically in context.

Data and methodology

Corpus data

The Textbook English Corpus (TEC; cf. Le Foll 2021a; 2022a; 2022b) aims to capture the language input that lower secondary school pupils in France, Germany and Spain obtain from their English textbooks. The corpus consists of nine series of best-selling EFL textbooks from eight publishers. The textbooks have been manually annotated to enable fine-grained comparisons between learner levels, learner L1 and text registers (see Appendix I). Seven register subcorpora of the TEC were included in the present analysis: Informative, Instructional, Narrative, Other, Personal, Poetry and Spoken (see Appendix II). Together, these subcorpora comprise 1,8 million words (hereafter referred to collectively as the TEC-T).

Data extraction and processing

This section describes how the causative constructions of the TEC-T were extracted for the second part of the analysis. In the context of a pedagogically-oriented study, it made sense to focus on the most frequent causative verbs: MAKE, CAUSE and GET. ‡ The first step was thus to locate and analyse all the causative constructions of the TEC-T with these three verbs. To this ‡ The extraction of causative HAVE constructions was not attempted because these are relatively rare and their extraction would have required a lot of manual work. Indeed, Gilquin (2010: 37) reported that only ca. 0.35% of all occurrences of HAVE are instances of causative constructions.

end, the corpus was uploaded onto the web-based corpus analysis tool Sketch Engine [START_REF] Kilgarriff | The Sketch Engine: Ten Years On[END_REF]. Sketch Engine provides a layer of automatic linguistic analysis which makes it possible to query the data on the basis of part-of-speech information (e.g., whether the word make is a noun, a verb in infinitive form, or present tense) and lemmas (e.g., make, makes, making and made are all forms of the lemma MAKE). Due to the high frequency of the verbs MAKE and GET, the extraction of these constructions was divided into two phases: search queries were used to automatically extract all possible instances of the constructions and these were then manually sorted to narrow the results down to actual occurrences of causative constructions. Thus, for the first phase, it was necessary to find queries that would be precise enough to considerably reduce the second time-consuming phrase but, at the same time, would have the highest possible recall rate so as not to miss any relevant constructions. After some experimentation, the following queries were chosen for the automatic extraction phase:

(8) [lemma="make" & tag="V.*"] [tag!="SENT|,"]{0,5} [tag="V.*"] (9) [lemma="get"] [tag!="IN|V.N|V.G" & word!="into|to"] [tag="N.*|PP.?"]{1,5} [tag="V.*"]{1,7} (10)
[lemma="cause" & tag="V.*"]
The queries are formulated in Corpus Query Language (CQL). The first, (8), returns all occurrences of the verb MAKE followed by another verb form (operationalised by [tag="V.*"])

with up to five intervening tokens in between to allow for longer nominal phrases as direct objects. A token is a word-like unit which, however, can also include punctuation marks which is why query (8) specifies that the five intervening tokens may be of any kind except end-ofsentence punctuation or commas. The second query for GET causatives, (9), is more complex because it attempts to exclude a number of other frequent GET constructions such as passives (e.g., He got fined) and constructions of intransitive motion (e.g., How did she get into this?).

As CAUSE is a relatively infrequent verb, its CQL query, (10), is much simpler: it simply retrieves occurrences of CAUSE as a verb and, since they are only 105 in the TEC-T, these were then manually disambiguated.

All causative constructions were subsequently manually annotated for their intermediate-level construction following Gilquin's (2010: 222) There is nothing secret that will not be made known. <BNC1994: A18>

Collostructional analysis

Collostructional analysis explores the interactions between grammatical constructions and the content words (lexemes) associated with them (cf. [START_REF] Stefanowitsch | Collostructions: Investigating the interaction of words and constructions[END_REF]. As such, the method also follows a construction-based approach to language since it assumes that grammar, just like the lexicon, consists of pairings of linguistic forms with linguistic meaning.

Collostructional analysis takes a construction as a starting point and measures which lexemes are attracted to or repelled by the construction. To do so, it evaluates whether lexemes occur more or less frequently than their expected frequency on the basis of the total lexeme and construction counts in the corpus under study. The lexemes associated with a particular construction are referred to as its collexemes (Stefanowitsch & Gries 2003: 215).

The analysis was carried out using the Coll.analysis 3.2a package for R [START_REF] Gries | Coll.analysis 3.2a package for R[END_REF]. As recommended by Stefanowitsch and Gries (2003: 218), the association measure adopted here is Fisher's exact test because it does not rely on or make any distributional assumptions (which are frequently not met when working with corpus data), nor does it impose specific sample size demands. Nonetheless, like all phraseological measures, collostructional analysis requires relatively large sample sizes to yield significant results. The causative constructions data extracted from the TEC-T only provided enough data to perform a collexeme analysis on the most frequent causative construction: [X MAKE Y Vinf].

Causatives in EFL textbooks

Causatives in textbook grammars

The term causative is mentioned in five out of the nine textbook series of the TEC (see → MAKE exprime plus la contrainte que HAVE.

Appendix I).

• HAVE + participle passé

We've had the lock changed.

[…]

→ On ne nous dit pas qui a changé la serrure mais seulement que la serrure a été changée. Le participe passé a un sens passive. MAKE est impossible dans ce cas.

• Traduction de FAIRE FAIRE

Il faut se demander si l'énoncé a un sens passif ou actif.

■ Sens passif : HAVE + participe passé […]

» She's had some food delivered.

■ Sens actif (quelqu'un accomplit une action à la demande de quelqu'un d'autre):

MAKE ou HAVE + base verbale […]

» She made / had them go out of the room.

In its grammar point on causative constructions, New Missions 2 e also takes the translation of faire faire as its starting point. It, too, focuses on the differences in the syntactic patterns of the passive and active forms. In addition, it draws a semantic continuum across MAKE, HAVE and LET from the most constraint to the least (where "authorisation" is considered to be the least constrainedthough it may be argued that authorisation does not necessarily imply causation) (see Table 2). All of its example sentences are drawn from the semantic field of law. This section has provided an overview of how causative constructions are explicitly presented in the grammar sections of five EFL textbook series. The following sections investigate the nature of the MAKE, CAUSE and GET causatives that learners are exposed to implicitly when learning English with the textbooks of the TEC. The results are then compared to the results of existing EFL learner corpus research in Section 5.

Syntactic analysis of causative constructions in the TEC-T

Collostructional analysis of the [X MAKE Y Vinf] construction in the TEC-T

The potential impact of EFL textbooks on EFL writing

In this section, the relative frequencies of causative constructions in the TEC-T are compared to the relative frequencies found in corpora of EFL and ESL learner essay writing.

EFL learners' choice of causative constructions

Gilquin (2016: 127) (Gilquin 2010: 37).

Although the learners of the learner corpus studies cited above presumably did not learn with exactly the textbooks of the TEC-T, they can be assumed to have been exposed to very similar pedagogical materials; thus, these results suggest that how frequently a construction is presented in typical textbook materials influences learners' choices in their own language production (see also [START_REF] Winter | Testing the Pedagogical Norm: Comparing Ifconditionals in EFL Textbooks, Learner Writing and English Outside the Classroom[END_REF]. Indeed, by far the most frequent causative construction in the TEC-T, [X MAKE Y Vinf], is seemingly overused by EFL learners, whilst constructions that are underused either have a relative frequency that is considerably lower in the TEC-T than in ENL writing, or are entirely absent from the 43 textbooks of the TEC.

Unidiomatic syntax in EFL writing

Not only does textbook-based language input appear to influence EFL learners' choice of construction, the results also suggest that low exposure from textbooks corresponds to an increase in EFL users' production of unidiomatic syntax. Whereas periphrastic constructions with CAUSE occur more than 30 times pmw in ENL and EFL writing (Gilquin 2012: 48), in 4.2

we saw that they are very rare in the TEC-T. This finding sheds new light on Gilquin's (2012: 50) conclusion that the causative constructions which trigger the most syntactic errors in EFL writing are those involving the verb CAUSE: nearly a quarter of CAUSE constructions in ICLE are reported to feature verb-and/or causee-related syntactic errors such as those illustrated in ( 17) and ( 18) (taken from Gilquin 2012: 49-50).

(15)

The ease payment of credit card cause the students buying too much and too quickly. Missions 2 e where it is mentioned once, but with no example sentence or additional explanation, see 4.1).

Inversely, the results of the present corpus-based analysis of Textbook English suggest that high exposure through textbook input may have a positive impact on learners' production.

Active [X MAKE Y Vinf] constructions have been shown to be considerably overused by EFL learners [START_REF] Gilquin | Lexical infelicity in causative constructions. Comparing native and learner collostructions[END_REF][START_REF] Gilquin | Input-dependent L2 acquisition: Causative constructions in English as a foreign and second language[END_REF] and, indeed, this study shows (see 4.3) that they are, by far, the most frequent causative constructions in the nine textbook series under study. In addition, Gilquin's (2016: 128-137) syntactic error analysis in ESL and EFL essays reveals that, with [X MAKE Y Vinf] constructions, EFL learners generally fare better than ESL students. Together, these findings suggest that EFL learners' generally good assimilation of the syntactic peculiarity of the [X MAKE Y Vinf] constructionwhich is unusual in that it takes a bare infinitivemay be, at least partially, the result of high exposure to this construction in EFL teaching materials.

The passive [X BE made Vto-inf] construction, on the other hand, was shown to be largely absent from the TEC-T. If the syntactically unusual active [X MAKE Y Vinf] construction is featured so prominently in textbooks, but the passive construction is more or less ignored, this is likely to have an impact on EFL learners' idiomaticity when they do attempt the passive construction.

Indeed, Gilquin (2016: 131) reports that whilst ESL students, who are likely exposed to more naturalistic language input, use native-like syntax in 100% of attested cases, some EFL learners, on the other hand, seemingly overgeneralise the rule from the active construction and thus produce sentences such (17) (from Gilquin 2016: 133).

(17) Czech students majoring in languages, for example, are often made believe that grammar is the most important part of language […]. <ICLE: Czech L1> Gilquin (2016: 136) hypothesises that MAKE, often perceived as the most prototypical causative verb (see also Altenberg 2002: 99), may "serve as a pathbreaking verb in instruction, that is, a verb that is used to introduce the general characteristics of the construction, before the construction is extended to other verbs." Whilst this pedagogical strategy appears to bear fruit for the active [X make Y Vinf] construction, it may be misleading some learners into assuming that all causative constructions follow this syntactic pattern with a bare infinitive. The present results thus underline the inseparability of grammar and vocabulary in language teaching and learning: each causative construction needs to be taught as a lexico-grammatical unit with its own syntactic patterns, semantics, and, as will be discussed in the following section, frequent lexical associations.

EFL learners' use of the [X MAKE Y Vinf] construction

The collexeme analysis in 4.3 shows that [X MAKE Y FEEL] is by far the most salient instantiation of causative constructions in the TEC-T. Interestingly, both [START_REF] Liu | Investigating learner vocabulary: A possible approach to looking at EFL/ESL learners' qualitative knowledge of the word[END_REF] and [START_REF] Gilquin | Lexical infelicity in causative constructions. Comparing native and learner collostructions[END_REF][START_REF] Gilquin | Input-dependent L2 acquisition: Causative constructions in English as a foreign and second language[END_REF] report that FEEL is used significantly more frequently by EFL than ENL users. In addition, BE and BECOME are also overused in this construction by learners [START_REF] Liu | Investigating learner vocabulary: A possible approach to looking at EFL/ESL learners' qualitative knowledge of the word[END_REF]: 179, Gilquin 2016: 139). In both cases, however, such an overuse of BE and BECOME is likely to be the result of learners using a verb complement where an adjective complement would have sufficed, see example (20) from Gilquin (2012: 14), and, as such, we would not expect textbooks to feature such infelicitous constructions.

(18) That will make it be more popular. <ICLE: Tswana L1> Indeed, the results in Table 4 show that BE is the verb lemma most strongly repelled by the [X MAKE Y Vinf] construction. Thus, unlike the use of FEEL, learners' overuse of BE and BECOME in this construction does not appear to be textboook-induced. This finding also supports the input-dependent L2 acquisition hypothesis as Gilquin (2016: 139-140) observes an overuse of BE and BECOME for both ESL and EFL learners, whereas the overuse of FEEL is seemingly exclusive to EFL learners who can be assumed to rely much more on textbook input than ESL users.

Pedagogical applications

In response to the results presented in this chapter, teachers may want to supplement textbook materials with their own corpus-informed materials. This could be achieved by using the webbased corpus platform Sketch Engine to create data-driven learning activities. Figure 1 shows how to search for MAKE causative constructions in the British National Corpus (BNC). Under the "Advanced" tab (a), Sketch Engine's concordance function has a CQL "Query type" option (b). Having selected this option, the CQL queries formulated in 3.2 can be copied into the search field (c). If desired, the "Text types" drop-down menu (d) can be used to select a register subcorpus of interest, e.g., "Spoken demographic" (e) for everyday conversation.

Conclusions

This study has examined the representations of causative constructions in school EFL textbooks, focusing on periphrastic constructions involving three of the most frequent causative verbs: CAUSE, GET and MAKE. The results have been compared to the findings of previous learner corpus studies with the aim of further exploring the input-dependent L2 acquisition hypothesis [START_REF] Gilquin | Input-dependent L2 acquisition: Causative constructions in English as a foreign and second language[END_REF]) which suggests that EFL learners' use of specific constructions is likely to be influenced by the distributions of these same constructions in their main source of L2 input, namely their textbooks. Consequently, EFL teachers may want to draw on ENL corpus data to expose their students to constructions that are underrepresented in teaching materials (e.g., GET causatives). Additional, learner corpora can be used to raise awareness of frequent difficulties that learners of specific L1s face in attempting to produce particular constructions.

As in all corpus-based investigations, the observations made in this study are limited to the data included in the TEC-T and may only very tentatively be generalised to the entire population, in this case, of all lower secondary school EFL textbooks. Even more tentative are the hypotheses drawn from comparisons between this study's results and those obtained from learner corpora because they are based on different text registers. The learner corpora queried in [START_REF] Gilquin | Lexical infelicity in causative constructions. Comparing native and learner collostructions[END_REF][START_REF] Gilquin | Input-dependent L2 acquisition: Causative constructions in English as a foreign and second language[END_REF], [START_REF] Altenberg | The grammatical and lexical patterning of MAKE in native and non-native student writing[END_REF] and [START_REF] Liu | Investigating learner vocabulary: A possible approach to looking at EFL/ESL learners' qualitative knowledge of the word[END_REF] all consist of academic essays, a register which is not widely represented in secondary school EFL textbooks.

Medium and register, however, have been shown to have a major influence on the distribution and phraseological patterns of causative constructions in English. For instance, the [X MAKE Y Vinf] construction has been shown to be about four times more frequent in spoken than in written language (Gilquin 2010: 226). Moreover, Gilquin (2012: 58)

  made you consider changing the way you use your mobile phone in public? <TEC: Achievers B2> † These constructions consist of a causative verb and a non-finite verb complement clause. The causative verb (e.g., in example (1), make) is governed by the causer (this lesson) and controls the complement clause which defines both the causee (you) and the effect exerted by the causer (consider changing the way you use your mobile phone in public). The most frequent causative verbs are MAKE (1), CAUSE (2), GET (3) and HAVE (4):

<ICLE

  It causes to lose their good emotions such as love, mercy and the other good ones. <ICLE: Turkish L1> Furthermore,Gilquin (2016: 127-128) demonstrates that ESL speakers produce CAUSE constructions which follow the prototypical syntactic patterns used by ENL speakers significantly more frequently than EFL learners. Gilquin interprets this finding as a confirmation of the input-dependent L2 acquisition hypothesis because ESL speakers are likely to have much richer, extra-curricular language input than EFL students. The results of the present study support this hypothesis by pointing to the extremely low frequency of CAUSE constructions in school EFL textbooks. Moreover, CAUSE is not featured in the explicit treatment of causative constructions in the nine textbook series investigated (except for New

Figure 1 :

 1 Figure 1: Screenshot of the advanced concordance search function on sketchengine.eu

Figure 2 :

 2 Figure 2: Screenshot of a random sample of query results with three selected concordance lines

Figure 3

 3 Figure 3 shows how, under the "Text types" (a) options of the OCLC, it is possible to narrow down the search to learners of a specific L1, e.g., Spanish (b), education level (c), CEFR level (d), etc.

Figure 3 :

 3 Figure 3: Screenshot of the text type options when querying the OCLC

Table 1 :

 1 Intermediate-level causative constructions with MAKE

	Intermediate-level construction	Example
	[MAKEact Vact]	This is often done deliberately to make a product appear more popular.
		<TEC: Achievers B1+>
	[MAKEact Vpas]	How can you make yourself heard? <TEC: Piece of Cake 4 e >
	[MAKEpas Vact]	It was made to keep the devil away. <TEC: Join the Team 5 e >
	[MAKEpas Vpas]

  New Bridges 2 e (see extract (11) from p. 196; emphasis original) focuses on potential syntactic difficulties when translating the French construction faire faire and, at the end of the explanation, points to a semantic difference between MAKE and HAVE, arguing that MAKE implies more of a constraint than HAVE.

	(11)	« faire faire »
		• MAKE ou HAVE + base verbale
		The police made the terrorist talk. […]
		Juliet had us talk all night. […]

The term is found in the following volumes: Piece of Cake 4 e , New Bridges 2 e , New Missions 2 e , English in Mind 3 and Solutions B2 Intermediate Plus. The first three represent one volume from each of the three French textbook series of the TEC. This is because, in France, causative constructions are expected to be explicitly taught as part of the lower secondary school EFL curriculum.

Table 2 :

 2 Extract from New Missions 2 e , p. 208 (emphasis original) (p. 208) also lists allow, ask, cause, encourage/incite/urge, force/oblige, get, order and persuade as causative verbs. However, these are not further explained or illustrated with any example sentences.

	verbe	construction	degré de pression
		(base verbale)
	make	someone do	Her lawyer made her admit the crime.
		something	Forte contrainte
	have		He had the prisoner pay for his crime.
			Pression moins forte
	let		He let his lawyer talk.
			Autorisation
	New Missions 2 e Piece of Cake 4

e (p. 166) features a relatively detailed description of causative constructions with the verbs HAVE, MAKE and LET. Aside from analysing their syntactic structures, it also contrasts a number of semantic aspects. In particular, the authors claim that HAVE causatives are used to delegate actions, whilst MAKE causatives imply that a (forced) action or an emotion is provoked.

The two non-French textbooks that explicitly mention causative constructions only do so fleetingly. English in Mind 3 (pp. 49 and 53) focuses on HAVE constructions in the sense of HAVE (something) done and its exercises on the subject suggest that the semantics of the construction are limited to services provided to individuals (the highlighted prototypical example being: having your hair done), whilst Solutions B2 Intermediate Plus (p. 130) simply lists GET and HAVE as verbs entering causative constructions in the following pattern verb + object + past participle without providing any further information.

  Table3displays the raw and relative frequencies (per million words, hereafter pmw) of the causative constructions extracted from the TEC-T with the combination of the automatic corpus queries and manual sorting (see 3.2). A cursory glance suffices to conclude that one construction is featured much more prominently than any other: [X MAKE Y Vinf], e.g., (1). It accounts for 97% of MAKE causative constructions in the TEC-T. Whilst some of the textbooks focus their grammar sections on causatives on the syntactic difficulty involved in the use the causative verb MAKE in the passive (see 4.1), the [X BE made Vto-inf] construction is largely absent from the actual language input delivered by the textbooks: it is featured just once in two textbook series (Access and Join the team) and twice in Green Line New. No single occurrence of [MAKEpas Vpas] was observed across the entire TEC-T.

Table 3 :

 3 Causative constructions in the TEC-T

	Construction	Raw frequency	Relative frequency
		in the TEC-T	in the TEC-T (pmw)
	[X CAUSE Y V to-inf ]	8	5.98
	[X GET Y Vpp]	13	9.71
	[X GET Y Vprp]	2	1.49
	[X MAKE Y Vinf]	289	215.88
	[X MAKE Y Vpp]	5	3.73
	[X BE made Vto-inf]	4	2.99

Table 4

 4

		lists the collexemes of the [X MAKE Y Vinf] construction with the strongest
	collostructional strength (see Appendix III for full results). The collostructional strength
	column indicates whether the co-occurrence frequency of a verb collexeme in the construction
	is significantly different to what we would expect under the assumption of no association given
	the construction's and collexeme's overall frequencies within the TEC-T (these correspond to
	the expected frequencies). The higher the value, the more strongly this verb is associated with
	the construction. The verb most strongly associated to the [X MAKE Y Vinf] construction in the
	TEC-T is, by far, FEEL, e.g.:
	(12)	Hobbies are excellent ways to make you feel happy and they can also be a way of making
		friends. <TEC: Hi There 5 e >

Table 4 :

 4 Most significant results from the collexeme analysis of [X MAKE Y Vinf] in the TEC-T (for full results see Appendix III)

	Collexeme lemma Lemma	Observed	Expected	Relation	Collostructional
		frequency in the	frequency in	frequency		strength
		TEC-T	[X MAKE Y Vinf]
	FEEL	1,412	65	1.27	attraction	88.74
	LAUGH	252	14	0.23	attraction	20.36
	SOUND	344	10	0.31	attraction	11.90
	THINK	5,432	25	4.88	attraction	10.34
	PROMISE	87	4	0.08	attraction	5.87
	WONDER	133	4	0.12	attraction	5.14
	SEEM	301	5	0.27	attraction	5.04
	SNEEZE	14	2	0.01	attraction	4.14
	CONSIDER	122	3	0.11	attraction	3.71
	CRY	139	3	0.12	attraction	3.54
	UNDERSTAND	654	5	0.58	attraction	3.46
	WANT	3,280	10	2.94	attraction	3.05
	DO	19,247	6	17.30	repulsion	-2.88
	BE	73,499	1	66.06	repulsion	-30.80

  reports that the relative frequency of the [X MAKE Y Vinf] construction is above 400 pmw in advanced EFL speakers (cf. Gilquin 2012: 8) and around 125 pmw in advanced ESL speakers. At 216 pmw, the relative frequency of this particular causative construction in the TEC-T is more than four times higher than inGilquin's (2012: 8) reference ENL corpus (50.77 pmw). It is also considerably higher than in her ESL corpus data. EFL learners use it to approximately the same extent as ENL writers (cf. Gilquin 2012: 48). The verb CAUSE is not infrequent in the TEC-T, but only 8% of occurrences are in periphrastic constructions, compared to 44% in British ENL data

Only 13 instances of the [X GET Y Vpp] construction were found in the TEC-T. Interestingly, the construction's relative frequency in the TEC-T (9.71 pmw) is noticeably closer to that found in

Gilquin's (2012: 8)

EFL data (7.73 pmw in ICLE) than in the reference ENL data (3.80 pmw). The [X GET Y Vprp] construction has a similarly low frequency across all corpora.

By contrast, Gilquin (2012) reported a significant underuse of the [X GET Y Vto-inf] construction by EFL learners and, indeed, not a single occurrence of this construction is found in the textbooks under study.

The second most frequently used causative construction in

[START_REF] Gilquin | Input-dependent L2 acquisition: Causative constructions in English as a foreign and second language[END_REF]

ENL data is [X CAUSE

Y Vto-inf]

. This construction is considerably less frequently in the TEC-T than in the ENL data. This difference may well be due to text register differences and would warrant further analysis with a comparable reference corpus. At the same time, however, it is worth noting that, according to

[START_REF] Gilquin | Input-dependent L2 acquisition: Causative constructions in English as a foreign and second language[END_REF]

, the [X CAUSE Y Vto-inf] construction is the only causative construction which ESL learners significantly overuse compared to ENL writers. By contrast,

  points out that [X MAKE Y FEEL] is highly frequent in spoken English, and demonstrates using distinctive collexeme ://doi.org/10.1075/ijlcr.20021.win. Wong, S.C. 1983. Overproduction & Unidiomatic Usage in the Make Causatives of Chinese Speakers: A Cause for Flexibility in Interlanguage analysis. .freq: frequency of the word in the corpus obs.freq: observed frequency of the word with/in causative_MAKE exp.freq: expected frequency of the word with/in causative_MAKE faith: percentage of how many instances of the word occur with/in causative_MAKE relation: relation of the word to causative_MAKE delta.p.constr.to.word: delta p: how much does the word/construction help guess the word? delta.p.constr.to.word: delta p: how much does the construction help guess the word/construction? coll.strength: index of collocational/collostructional strength: -log10 (Fisher-Yates exact, one-tailed), the higher, the stronger

	International Appendix III itch 3 1	Journal 0.0027	of attraction Learner 0.3333 0.0036 Corpus	4	Research D 0.3324	8(1).	31-66. 2017 2.5696
	sweat	3		1	0.0027	attraction	0.3333 0.0036		5	0.3324	E	2018 2.5696
										1	A	2013
	come throw up Communication 2(2). 151-165. 2638 8 2.371 attraction Cornelsen 5 1 0.0045 attraction	0.003 Access G 0.2	0.0201 0.0036		2 3	Language Learning & B 0.0022 2.5318 2014 C 2015 0.1991 2.3481
	work	768		4	0.6903	attraction	0.0052 0.0118		4	0.0043	D	2016 2.2696
										5	E	2017
	ache	6		1	0.0054	attraction	0.1667 0.0036	A1+	0.1658	A	2.2692
	suspect	6		1	0.0054	attraction	0.1667 0.0036	A2	0.1658	B	2.2692
	poke	7		1	Richmond 0.0063	attraction	Achievers 0.1429 0.0036	B1	0.142	C	2015 2.2024
									B1+	D
	stand out	7		1	0.0063	attraction	0.1429 0.0036	B2	0.142	E	2.2024
	disappear	136		2	0.1222	attraction	0.0147 0.0067	Starter	0.0138	A	2.1659
	Spain change words realize feel appear	843 word. 168 1412 freq 172	Cambridge University Press 4 0.7577 attraction obs. exp. relation 2 0.151 attraction 65 1.2691 attraction freq freq 2 0.1546 attraction	1 2 3 4 constr.to.word 0.0047 0.0116 English in Mind faith delta.p. 0.0119 0.0066 0.046 0.2278 0.0116 0.0066	B C delta.p. 0.0039 0.011 D 0.0453 E word.to.constr 0.0107	2010 2.13 coll.strength 1.9899 2011 88.7419 1.9704
	stretch	18		1	0.0162	attraction	Elementary 0.0556 0.0035	0.0547	A	2014 1.7944
	laugh	252		14	0.2265	attraction	Pre-Intermediate 0.0556 0.0492 0.0547	B	2016 20.355
	sound appreciate	344 19	Oxford University Press 10 0.3092 attraction 1 0.0171 attraction	Solutions 0.0291 0.0346 Intermediate 0.0526 0.0035	0.0282 0.0517	C	11.8997 2017 1.7711
	flow think sign	20 5432 22		1 25 1	0.018 4.8822 0.0198	attraction attraction attraction	0.05 0.0046 0.0719 0.0035 Intermediate Plus 0.0455 0.0035	0.0491 0.0038 0.0446	D	1.749 2017 10.3363 1.708
	promise	87		4	0.0782	attraction	0.046	0.014			0.0451	5.8727
	react	242		2	0.2175	attraction	0.0083 0.0064			0.0074	1.6909
	wonder	133		4	0.1195	attraction	0.0301 0.0139			0.0292	5.1389
	regret	24		1	0.0216	attraction	0.0417 0.0035			0.0408	1.6706
	seem	301		5	0.2705	attraction	0.0166 0.0169			0.0157	5.0425
	correspond	31		1	0.0279	attraction	0.0323 0.0035			0.0314	1.5608
	sneeze	14		2	0.0126	attraction	0.1429 0.0071			0.142	4.1383
	grow	285		2	0.2562	attraction	0.007	0.0062			0.0061	1.5595
	consider	122		3	0.1097	attraction	0.0246 0.0103			0.0237	3.7079
	dream	47		1	0.0422	attraction	0.0213 0.0034			0.0204	1.3832
	cry	139		3	0.1249	attraction	0.0216 0.0103			0.0207	3.5415
	bite	54		1	0.0485	attraction	0.0185 0.0034			0.0176	1.3242
	understand	654		5	0.5878	attraction	0.0076 0.0158			0.0068	3.4614
	tidy	55		1	0.0494	attraction	0.0182 0.0034			0.0173	1.3164
	want	3280		10	2.948	attraction	0.003	0.0252			0.0022	3.0519
	stand up	63		1	0.0566	attraction	0.0159 0.0034			0.015	1.259
	cringe	1		1	0.0009	attraction	1	0.0036			0.9991	3.0463
	pay	420		2	0.3775	attraction	0.0048 0.0058			0.0039	1.2562
	empathise	1		1	0.0009	attraction	1	0.0036			0.9991	3.0463
	go	7419		11	6.6681	attraction	0.0015 0.0155			0.0006	1.1301
	wince	1		1	0.0009	attraction	1	0.0036			0.9991	3.0463
	look	5761		9	5.1779	attraction	0.0016 0.0137			0.0007	1.106
	stop	826		5	0.7424	attraction	0.0061 0.0152			0.0052	3.0072
	rise	95		1	0.0854	attraction	0.0105 0.0033			0.0096	1.0868
	happen	1387		6	1.2466	attraction	0.0043 0.0170			0.0034	2.759

analysis that the frequency of use of [X MAKE Y FEEL] in EFL academic essays is closer to its frequency in ENL speech than in ENL essay writing. It may therefore be tempting to conclude that school EFL textbooks are more representative of spoken than written language; however, httpsword

this hypothesis does not entirely hold, since it would also imply that causative constructions with GET ought to be far more frequent than they were shown to be in 4.2. Register-based variation ought to be considered when teaching causatives (see also Le Foll 2021a;2022a;2022b). Since current EFL textbooks do not do so adequately, Section 7 showed how textbook authors and teachers can use Sketch Engine to explore how causative constructions vary in different registers, e.g., academic writing vs. conversation.

Appendix I

To best compare pedagogical materials used in different educational systems, each textbook was labelled for proficiency level on a universal scale of A to E: level A textbooks correspond to the first year of EFL instruction at secondary level, in other words, beginner level to roughly A1 on the CEFR scale, whilst level E corresponds to the fifth year (CEFR B1-B2). Note that French textbook series only cover the first four years of secondary school (which take place at Collèges), which is why, whenever possible, a textbook from the same publisher corresponding to the fifth year of instruction (the first year of Lycée) was added. At the time of corpus compilation, Le Livre Scolaire did not produce any textbooks for Lycées.

The full bibliographic metadata is available on doi.org/10.5281/zenodo.4922819.


Elen Le Foll

Keywords:

Even at the higher proficiency levels, many learners continue to rely on the same limited range of basic expressions to express opinions, persuade and present arguments (e.g., I think, X is very important, because of this…) or are not aware of register differences and thus frequently use words and phrases typical of written discourse in their spoken production (e.g., it has been argued that, moreover…). This chapter outlines how advanced students can explore a corpus of transcripts of discussions and debates from the European Parliament to tap into a wealth of useful expressions and find out how they are used in an authentic context. The proposed data-driven learning (DDL) lesson thus involves a hands-on corpus-based activity aimed at enabling students to improve their speaking skills. It relies on the EUROPARL subcorpus of the multilingual InterCorp corpus via the free corpus interface KonText. In the second half of the lesson, students are then encouraged to try out their newly acquired expressions in a series of mini debates on familiar topics to be chosen by the teacher.

Introduction

Many advanced learners continue to rely on the same limited range of basic expressions to express opinions, persuade and present arguments (e.g. I think, X is very important, because of this). In this lesson, students will explore a corpus of transcripts of discussions from the European Parliament to tap into a wealth of useful expressions that they can immediately try out in a series of mini debates. This is a post-peer-review, pre-copyedit version of a paper published as: Le Foll, Elen (in press/2022). Whilst I do not object, I strongly believe... -Exploring spoken argumentative and persuasive discourse. In Viana, V. (Ed.) New Ways in Teaching with Corpora. Abingdon, Oxon; New York, NY: Routledge. https://www.routledge.com/Teaching-English-with-Corpora-A-Resource-Book/Viana/p/book/9781032252971 Steps 1. Divide the class into two teams. 2. Ask half of the class to brainstorm arguments in favour of using smartphones in the English classroom and the other half to compile a list of arguments against it. Students should briefly discuss their ideas in their teams and, if necessary, take a few notes. 3. Tell students that you would like to hear their arguments to see which team is most convincing. 4. Award a point per valid argument. At this stage, do not correct language use, but discreetly make a note of the words, phrases and constructions that students use to express their opinions, present arguments, and persuade (e.g. we believe, if... then..., it's obvious that). 5. Declare the team with the most points the winner. 6. Write the expressions that students used to express opinions, present arguments and persuade on the board. 7. Introduce the lesson's aim: to learn a broader range of idiomatic phrases to give opinions and present persuasive arguments. 8. Divide students into pairs or groups of three (if feasible, roughly according to their proficiency levels). 9. Explain that students will, in their small groups, examine authentic debates and discussions from the European Parliament in order to learn new phrases in context. 10. Ensure that each group has one computer or tablet with Internet access. 11. Instruct students to connect to the KonText web interface (https://KonText.korpus.cz) [START_REF] Machálek | KonText: Advanced and flexible corpus query interface[END_REF]. If the interface is not automatically displayed in English, students should click on "English" in the top-right corner. 12. Instruct students to click on "syn2020" (A -see Figure 1), then on "All corpora" (B), and finally on "InterCorp" (C). In the list of corpora, they should select the latest version of the English InterCorp [START_REF] Čermák | The case of InterCorp, a multilingual parallel corpus[END_REF]) -e.g. "InterCorp v13 -English" (D). 15. In the section "text.txtype", instruct students to tick "discussions -transcripts". This restricts the search to transcripts of authentic discussions from the European Parliament. 16. Instruct students to scroll down to the bottom of the page and click on the "Search" button. 17. Instruct students to look for phrases and constructions that they did not use in the first discussion activity. The results page shows a list of excerpts from spoken discussions in the European Parliament containing the word I (see Figure 2). Students can click on the back and forward buttons (A) to display more results. Clicking on any pink instance of "I" (B) will open an additional window displaying more context. This may well be necessary to understand the meaning of the expression fully. If students are curious, they can click on "_EUROPARL" (C) to find out, among other things, the name of the speaker who uttered the phrase displayed in the same line. 18. Ask students to make a list of new expressions that they would like to be able to use to express opinions, persuade and present arguments in future discussions and debates (e.g. "I would like to emphasise that", "I would like to point out"). 19. Encourage students to infer the meaning of the new phrases from the given context (remind them that more context can be visualised by clicking on I in each line, see B in Figure 2) and make a note of the context in which their new expressions occur. 20. Clarify that not all the phrases displayed will be relevant (e.g. "I congratulate" in Figure 2). 21. Check that students consult several pages of results and that they remember to read the expanded co-text for their selected lines. 22. Once students have made note of a certain number of expressions or after ca. 15 minutes, instruct students to log off the computers. 23. Divide students into new groups of four. Hand each group a card with a pro-contra discussion topic and, if necessary, a few key words (see example card in Figure 4). The core vocabulary for these topics should already have been covered in previous lessons. For example, if over the past weeks, you have explored the topic of climate change, procontra discussion topics could include:

• banning diesel vehicles;

• heavily taxing short-haul flights;

Points for consideration and alternative steps

• It is not necessary to register at KonText for this lesson. Registering is free but is time consuming and therefore unnecessary for this lesson. Simply instruct students to ignore the messages to register. • Remind students that many Members of the European Parliament are not speakers of English as a first language and that many of the English texts are translations (though largely from highly qualified and experienced translators). If students are in any doubt, they should check with you whether the expressions they find are genuinely idiomatic. • Note also that the InterCorp -English corpus largely reflects British English usage.

• It is possible to repeat the lesson using we as the search term to find further expressions to give opinions, present arguments and persuade. • Some students may struggle to make immediate use of the expressions that they have selected from the corpus work in the debate activity. This is why it is important that students are familiar with the debate topic(s). Weaker students may be provided with debate topic cards that list more relevant keywords and collocations. • The debate activity can also be gamified by first compiling a class list of all the corpusinformed expressions students have collected and then students awarding each other one point per expression used in the debate activity.

Figure 1 :

 1 Figure 1: Corpus selection

Figure 2 :

 2 Figure 2: Results page for I in InterCorp -English

•

  boosting wind and solar energy production.

Figure 4 :

 4 Figure 4: Sample discussion topic card


Elen Le Foll

Raising awareness of first-language interference using parallel corpora of subtitles

Negative first-language (L1) transfer is a common source of unidiomatic language use as learners often assume that words, phrases, collocations and other idiomatic expressions can be translated word for word. This chapter outlines a fun group activity that introduces learners to parallel corpora to tackle such unidiomatic language usage. The proposed paperbased data-driven learning (DDL) activity encourages learners to find more idiomatic ways to translate difficult words or phrases from their L1 into English by analysing concordance lines from TV and film subtitles. The chapter features a few examples, and the appendix includes detailed instructions for teachers to create their own worksheets on any language issue of relevance to their students. The activity draws on the subtitle subcorpus of the multilingual InterCorp corpus via the free corpus interface KonText. It can be adapted to the full range of proficiency levels and for more than 40 different learner L1s. Optionally, more advanced students can be introduced to the web interface to conduct their own parallel corpus searches online.

Levels: Elementary and above

Aims:

• Raise learners' awareness of issues of L1-L2 equivalence • Practise conversation skills Class time: 45 minutes Preparation time: 30 minutes Resources: Worksheet(s) designed by teacher (see online support materials)

Introduction

Negative first-language (L1) transfer is a common source of unidiomatic language use since learners often assume that words, phrases, collocations and other idiomatic expressions can be translated word for word. The fun group activity proposed in this plan is a stimulating way to introduce learners to parallel corpora offline and to show learners how to find more idiomatic ways to translate difficult words or phrases from their L1 into English. Drawing on the multilingual nature of the class, the lesson can be delivered in English, in the students' first language or using a combination of both. This is a post-peer-review, pre-copyedit version of a paper published as: Le Foll, Elen (in press/2022). Raising awareness of first-language interference using parallel corpora of subtitles. In Viana, V. (Ed.) New Ways in Teaching with Corpora. Abingdon, Oxon; New York, NY: Routledge. https://www.routledge.com/Teaching-Englishwith-Corpora-A-Resource-Book/Viana/p/book/9781032252971

Steps 1. Write a sentence featuring an L1-induced error that your students often make on the board. For instance, if you teach English to Spanish L1 speakers, a typical L1-induced error might be the use of the general noun people with a definite article and a verb in the singular form. Thus, Spanish L1 speakers may produce the following: *I don't think that a local public transport website is a good idea because the people does not have Internet access in the region. 2. Ask students to discuss what is unidiomatic in this sentence in pairs. For example, in this context, people usually occurs without an article and is used with the plural verb form. 3. As a whole class, encourage students to think of reasons why this unidiomatic usage is common in the speech of Spanish L1 speakers. Here, this is likely to be because, in Spanish, gente (people) is typically used with the definite article la (the) and a verb in the singular form (e.g., La gente no tiene acceso a Internet). Therefore, this unidiomatic usage is likely due to L1 interference. 4. Group students in pairs or groups of three. 5. Hand out the subtitle worksheet(s) you have designed for the student groups (see online support materials for preparation procedure). 6. Explain that the sentences displayed come from film and TV subtitles in the students' L1 and in English. 7. Ensure that all students are aware of what subtitles are. 8. In their groups, ask students to identify different ways in which the focus word/phrase has been translated in the subtitles. Ensure that students pay particular attention to articles, verb forms, prepositions, and word order. 9. Tell students that there will likely be more than one idiomatic way to translate the focus word/phrase, so they should consider the context(s) in which each solution is most frequent. For instance, in the case of la gente (as suggested in the online support materials), students will see that it is usually translated by people without an article. However, some of the examples feature an adjective specifying a particular group of people, and, in these cases, an article is used (e.g., And when the wrong people get shot?). 10. Pool answers together as a class and summarise the information on the board. 11. Highlight the different ways to idiomatically translate the focus word/phrase. For instance, there are cases in which the phrase la gente is translated without using the word people at all (e.g., Probaré que la gente del Sur ve la verdad , no el color de la piel is translated as I'll show the men and women of the South will look past color and see the truth and Es mejor hablar a la gente as It's better to talk to the crowd). 12. Let students in each group choose two or more examples from the worksheet that they particularly like and ask them to imagine a film scene in which the subtitles will feature in the dialogue. 13. Instruct students to write the dialogues of their film scenes. 14. Allow time for students to create and practise their dialogues. Every student in the group should have an active role. 15. Invite students to perform their film scenes in front of the rest of the class.

Points for consideration and alternative steps

• You can easily create bespoke worksheets for each group (see online support materials), thus adapting the activity to various levels of proficiency or L1 backgrounds within one class.

• This activity does require the teacher to spend some time preparing the worksheet(s); however, from the second worksheet onwards, the procedure will be relatively quick and is well worth the time investment since the KonText interface (Machálek 2020) allows for the creation of bespoke worksheet(s) created on the basis of authentic spoken-like language. • If you teach older and more advanced students and have access to at least one computer/tablet with Internet access per pair of students, students can also work with KonText directly. They should follow the procedure described in the online support materials. The advantage is that they will learn to search parallel corpora and have access to many more examples than you can provide on a single worksheet. In future, they may also be able to use this method to perform other queries and solve other L1-L2 equivalence issues. Note, however, that this option is probably not suitable for young learners due to the strong language and inappropriate content of some of the films featured in the subtitle datasets of the InterCorp corpora. • With a monolingual class, it is also possible to extend the by requiring students to write subtitles to their film scene in their L1 to accompany the live performances of their scenario. This provides further opportunities for students to gain greater awareness of L1-L2 equivalence issues and can help foster interlingual mediation skills. o Use Microsoft PowerPoint or LibreOffice Impress to create a plain slide presentation with just one long text box at the bottom of each slide. o Ask students to write one subtitle per slide. o At the same time as they perform their film scene, display the subtitles using a projector and allow a student to click through the slides at the appropriate moment.

Online support materials

Procedure for teacher preparation 1. Open a web browser and connect to KonText (https://kontext.korpus.cz). It is not necessary to register at KonText to prepare this lesson but registering is free and will give you access to some useful additional options such as being able to save your searches. 2. If the interface is not automatically displayed in English, click on "English" in the top-right corner. 3. Click on "syn2020", which is the default corpus at the time of writing, then click on "All corpora" and on "parallel" (see Figure 1). At the time of writing, InterCorp parallel corpora were currently available on the KonText web interface in Albanian, Arabic, Belarusian, Bulgarian, Catalan, Chinese, Croatian, Czech, Danish, Dutch, English, Estonian, Finnish, French, German, Greek, Hebrew, Hindi, Hungarian, Icelandic, Italian, Japanese, Latvian, Lithuanian, Macedonian, Malay, Maltese, Norwegian, Polish, Portuguese, Romani, Romanian, Russian, Slovak, Slovenian, Serbian, Spanish, Swedish, Turkish, Ukrainian, and Vietnamese. 5. Choose a language issue caused by L1 transfer that your students regularly face and that you would like them to improve on. This could be negative L1 transfer derived from a single word, collocation, phrase, or construction.

• For example, German native speakers often struggle to translate the ubiquitous word selbst, which can mean automatic(ally), even, in person, self-, (by) myself/yourself/herself etc. Negative L1 transfer can lead to the production of utterances such as: *This is a self-made cake. • In French, many constructions use the verb faire and, as a result, L1 French speakers tend to overuse the English verb make and produce unidiomatic collocations such as: *I made my homework, or *I always used to make nightmares. • Spanish uses the phrase la gente to refer to people in general. Consequently, Spanish L1 speakers will often use people with an article and with a verb in singular form, e.g., *The people here is very friendly. 6. Type the language item(s) you have selected in the search box.

• If the issue is a single invariable word (e.g., selbst in German), type selbst in the search box.

• If the issue stems from a construction containing more than one word in the learners' L1, such as la gente in Spanish, type the full phrase (e.g., la gente) in the search box. 7. In the search box, click on the option "Aligned corpora" and, in the drop-down menu, select the latest version of the English InterCorp (e.g., "InterCorp v13 -English"). 8. Click on the option "Restrict search". In the section "text.txtype", tick "subtitles". This restricts the search to texts drawn from aligned film and TV series subtitle data. 9. Scroll down to the bottom of the page and click on the "Search" button. 10. Take a moment to explore the results page (see Figure 2) which shows a list of subtitles containing the L1 word or phrase queried on the left, and the English versions of the subtitles on the right. The number of hits (A) corresponds to the number of times the queried word, lemma or phrase was found in subtitles in InterCorp. The results are displayed in a random order. Clicking on a subtitle on the right-hand side (E) will open an additional window displaying the co-text in which the subtitle is found in English. 12. By ticking the corresponding boxes (F), select subtitles which:

• have been accurately and idiomatically translated (bearing in mind that, even though many of the subtitles come from DVDs, some of the translations fail to meet professional standards); • contain lexical and grammatical features suitable for your students' level;

• are appropriate for your students in terms of content (especially since strong language is not uncommon in films and TV series). 13. Once you have made your selection, click on the blue link "[number] selected" (see G in Figure 2) and choose "Keep only selected lines" from the "Action" drop-down menu. This opens a new window containing only the subtitles that you have selected. 14. Print your selection or save as a PDF (landscape is recommended). This is your subtitle worksheet.

Figure 1 :

 1 Figure 1: Corpus selection

Figure 2 :

 2 Figure 2: Results page for la gente in InterCorp Spanish and English