Amir M Nia

Uwe Fuhr

Natig Gassanov

Erland Erdmann

Fikret Er
email: fikret.er@uk-koeln.de

Torsades de pointes tachycardia induced by common cold compound medication containing chlorpheniramine

Keywords: chlorpheniramine, long QT syndrome, torsade des pointes, potassium current

Torsades de pointes tachycardia induced by common cold compound medication containing chlorpheniramine

Chlorpheniramine (CPM) is a first generation histamine-receptor-1 antagonist registered in the World Health Organization Model List of Essential Medicines [START_REF] Rumore | Clinical pharmacokinetics of chlorpheniramine[END_REF]. Due to its moderate degree of sedative effects it is frequently used as a component of compound preparations for symptomatic treatment of common cold and influenza symptoms.

CPM shows pronounced interindividual variability in pharmacokinetics, with an elimination half-life ranging from 2 to 43 hours because of its individual hepatic metabolism and the different activity of the involved cytochrome CYP2D6 which has a pronounced genetic polymorphism [START_REF] Yasuda | The roles of CYP2D6 and stereoselectivity in the clinical pharmacokinetics of chlorpheniramine[END_REF][START_REF] Paton | Clinical pharmacokinetics of H1-receptor antagonists (the antihistamines)[END_REF]. Beside this mechanism, there are still unknown metabolic pathways which are involved in the disposition of CPM in humans [START_REF] Yasuda | The roles of CYP2D6 and stereoselectivity in the clinical pharmacokinetics of chlorpheniramine[END_REF]. CPM and its metabolites are excreted pH dependent in the urine with individual excretion dosages of unchanged CPM ranging from 0% to 34% [START_REF] Rumore | Clinical pharmacokinetics of chlorpheniramine[END_REF][START_REF] Huang | Pharmacokinetics of chlorpheniramine after intravenous and oral administration in normal adults[END_REF] (see also drug information).

A 40-year-old woman (63 kg body weight; BMI 21 kg/m²; Caucasian), otherwise healthy, was initially admitted to a primary care hospital after torsades de pointes tachycardia (TdP) that necessitated cardiopulmonary resuscitation. There repeated TdP was monitored (Figure 1A) in the first 24 hours after administration, which were treated successfully by magnesium infusions. At the second day after onset of TdP the patient was referred to the cardiological department of the University of Cologne for further diagnostics. Laboratory values, including creatinine were in normal ranges except mildly elevated alanine aminotransferase and aspartate aminotransferase. The main pathological finding was detected by the retrospective evaluation of the documented electrocardiogram (ECG) during the in-hospital observed TdP tachycardia: the corrected QT interval (QTc) using Bazett`s formula was prolonged up to 524 ms (Figure 1A). Therefore the diagnosis of long-QT-syndrome (LQTS) was made. The ECGs documented in our hospital revealed a borderline QTc of 440 ms (Figure 1B). All other diagnostic procedures were normal including coronary angiogram, ventriculography and complete invasive electrophysiological testing.

The patient was on no permanent medication, denied abusing any other substances, completely unremarkable on physical examination, and had no prior history of cardiac arrhythmias. She smoked regularly up to 5 cigarettes a day, drank little amounts of wine and suffered for a common cold for some days which she selfmedicated with a compound medication taking two capsules three times a day. She had stopped self-medication the evening before the event. Each capsule contained 200 mg of acetaminophen, 150 mg of ascorbic acid, 25 mg of caffeine and 2.5 mg of CPM. In animal experiments CPM has been shown a dose dependent block of the delayed rectifier potassium channel I Kr, and to lengthen the action potential, slow cardiac repolarization and [START_REF] Hong | Block of HERG k channel by classic histamine h(1) receptor antagonist chlorpheniramine[END_REF][START_REF] Salata | Cardiac electrophysiological actions of the histamine H1-receptor antagonists astemizole and terfenadine compared with chlorpheniramine and pyrilamine[END_REF][START_REF] Taglialatela | Cardiac ion channels and antihistamines: possible mechanisms of cardiotoxicity[END_REF][START_REF] Taglialatela | Cardiotoxic potential and CNS effects of firstgeneration antihistamines[END_REF][START_REF] Wang | Conventional" antihistamines slow cardiac repolarization in isolated perfused (Langendorff) feline hearts[END_REF]. The CPM-like drug diphenhydramine also significantly prolonged QTc intervall in healthy volunteers and in patients with coronary heart diseases [START_REF] Khalifa | Block of potassium currents in guinea pig ventricular myocytes and lengthening of cardiac repolarization in man by the histamine H1 receptor antagonist diphenhydramine[END_REF].

The coincidence between drug intake and prolongation of the QT interval in our patient is highly suggestive of CPM-induced TdP. One may speculate that the patient is a poor metabolizer with respect to CYP2D6 activity, which might have contributed to an extensive exposure, however, the patient did not agree to genotyping. The almost normal QTc interval two days after cessation of drug intake supports the causal link to CPM since serum concentration may have been decreased considerably at this time even in case of a long elimination half-life as expected for poor metabolizers.

In 1978 the putative first case of was reported. A young women on long-term thioridazine medication took a compound preparation containing CPM (Contac C ® ) and died due to ventricular arrhythmia [START_REF] Chouinard | Death attributed to ventricular arrhythmia induced by thioridazine in combination with a single Contac C capsule[END_REF]. The authors assumed the combination of the neuroleptic with ephedrine as fatal, but we speculate that CPM might have been played a more prominent role inducing fatal tachycardia than suspected. Indeed, thioridazine is both a substrate and inhibitor to CYP2D6, and also prolongs the QTc interval [START_REF] Drolet | Thioridazine lengthens repolarization of cardiac ventricular myocytes by blocking the delayed rectifier potassium current[END_REF][START_REF] Shin | Effect of antipsychotic drugs on human liver cytochrome P-450 (CYP) isoforms in vitro: preferential inhibition of CYP2D6[END_REF][START_REF] Von Bahr | Plasma levels of thioridazine and metabolites are influenced by the debrisoquin hydroxylation phenotype[END_REF]. Concomitant medication with the two drugs thus may have resulted in a drug-drug interaction at both the pharmacokinetic and the pharmacodynamic level.

To be on the safe side, in our patient a cardioverter defibrillator was implanted and we recommended her to avoid any known QT-interval interacting medications (see list at www.azcert.org) and drugs containing CPM.

Certainly this report is limited by the missing evaluation of the initial drug concentration. Thus we cannot clarify the exact causal link between CPM intake and the occurrence of TdP. Especially, the underlying cold of the patient might have been jointly responsible for the QT prolongation [START_REF] Pripp | Epiglottitis and torsade de pointes tachycardia[END_REF][START_REF] Marques | Electrical storm induced by H1N1 A influenza infection[END_REF][START_REF] Reinsch | Prevalence and risk factors of prolonged QTc interval in HIV-infected patients: results of the HIV-HEART study[END_REF].

Nonetheless it is important to report this case, because several levels of evidence suggest that second generation antihistamines interfere with the QT interval while CPM might be overlooked [START_REF] Craft | Torsade de pointes after astemizole overdose[END_REF][START_REF] Monahan | Torsades de pointes occurring in association with terfenadine use[END_REF]. On the other hand, CPM is used worldwide in compound preparations, usually over-the-counter drugs and available for selfmedication without the necessity of a prescription. In particular patients with known LQTS should avoid such compounds containing antihistamines until harmlessness in these high-risk patients has been proven.

Figure 1 .

 1 Figure legend Figure 1. Electrocardiograms documented in the acute phase (A, 25 mm/s) with calculated QTc of 524 ms and TdP tachycardia and two days later (B, 50 mm/s) with almost normal QTc of 440 ms interval.

European Journal of Clinical Pharmacology


E J Sapountzakis

V J Tsipiras

Shear Deformable Bars of Doubly Symmetrical Cross Section under Nonlinear Nonuniform Torsional Vibrations -Application to Torsional Postbuckling Configurations and Primary Resonance Excitations

Keywords: shear deformation, secondary twisting moment deformation effect, independent warping parameter, bar, beam, boundary element method, nonuniform torsion, nonlinear vibrations, torsional vibrations, torsional postbuckling, primary resonance

Shear deformable bars of doubly symmetrical cross section under nonlinear nonuniform torsional vibrations-application to torsional postbuckling configurations and primary resonance excitations

INTRODUCTION

When arbitrary torsional boundary conditions are applied either at the edges or at any other interior point of a bar due to construction requirements, this bar under the action of general twisting loading is leaded to nonuniform torsion. In this case, apart from the well known primary (St. Venant) shear stress distribution, normal and secondary (warping) shear stresses arise formulating the warping moment (bimoment) and secondary twisting moment (bishear), respectively [START_REF] Sapountzakis | Warping shear stresses in nonuniform torsion by BEM[END_REF][START_REF] Simo | A Geometrically-exact rod model incorporating shear and torsion-warping deformation[END_REF]. Warping shear stresses can be estimated by formulating a boundary value problem with respect to a secondary warping function [START_REF] Sapountzakis | Warping shear stresses in nonuniform torsion by BEM[END_REF][START_REF] Sapountzakis | Warping shear stresses in nonlinear nonuniform torsional vibrations of bars by BEM[END_REF][START_REF] Schulz | Generalized Warping Torsion Formulation[END_REF] or by studying the equilibrium equations of a small segment of an elementary slice of the bar [START_REF] Vlasov | Thin-walled elastic beams, Israel Program for Scientific Translations[END_REF]. However, the aforementioned techniques do not achieve to include the warping shear stresses in the global equilibrium of the bar and to perform an accurate analysis of bars of closed shaped cross sections [START_REF] Murín | An effective finite element for torsion of constant crosssections including warping with secondary torsion moment deformation effect[END_REF], that is to account for the secondary twisting moment deformation effect (STMDE). This effect generally necessitates the use of an independent warping parameter in the kinematical components of the bar (along with the angle of twist), increasing the difficulty of the problem at hand. Besides, since weight saving is of paramount importance in many engineering fields, frequently used thin-walled open sections have low torsional stiffness and their torsional deformations can be of such magnitudes that it is not adequate to treat the angles of cross section rotation as small. In these cases, the study of nonlinear effects on these members becomes essential, where this non-linearity results from retaining the nonlinear terms in the strain-displacement relations (finite displacement -small strain theory). When finite twist rotation angles are considered, the nonuniform torsional dynamic analysis of bars becomes much more complicated, leading to the formulation of coupled and nonlinear torsional and axial equations of motion. When the twist rotation angles of a member are small, a wide range of linear analysis tools, such as modal analysis, can be used and some analytical results are possible. As these rotation angles become larger, the induced geometrical nonlinearities result in effects that are not observed in linear systems. In such situations the possibility of an analytical solution method is significantly reduced and is restricted to special cases of boundary conditions or loading.

During the past few years, the linear static and linear free vibration analysis of shear deformable bars undergoing twisting or general deformations have been thoroughly studied [START_REF] Murín | An effective finite element for torsion of constant crosssections including warping with secondary torsion moment deformation effect[END_REF][START_REF] Sapountzakis | Secondary torsional moment deformation effect by BEM[END_REF][START_REF] El Fatmi | Non-uniform warping including the effects of torsion and shear forces. Part I: A general beam theory[END_REF][START_REF] El Fatmi | Non-uniform warping including the effects of torsion and shear forces. Part II: Analytical and numerical applications[END_REF][START_REF] Kim | Exact dynamic/static stiffness matrices of non-symmetric thinwalled beams considering coupled shear deformation effects[END_REF][START_REF] Back | A shear-flexible element with warping for thin-walled open beams[END_REF][START_REF] Chen | A C0 finite element formulation for thin-walled beams[END_REF][START_REF] Hu | A finite element model for static and dynamic analysis of thin-walled beams with asymmetric cross-sections[END_REF][START_REF] Saadé | Non-uniform torsional behavior and stability of thin-walled elastic beams with arbitrary cross sections[END_REF][START_REF] Gendy | Generalized thin-walled beam models for flexural-torsional analysis[END_REF][START_REF] Laudiero | Shear strain effects in flexure and torsion of thin-walled beams with open or closed cross-section[END_REF][START_REF] Laudiero | The shear strain influence on the dynamics of thinwalled beams[END_REF][START_REF] Prokic | Dynamic analysis of thin-walled closed-section beams[END_REF][START_REF] Kollár | Flexural-torsional vibration of open section composite beams with shear deformation[END_REF][START_REF] Park | A finite element analysis of discontinuous thinwalled beams considering nonuniform shear warping deformation[END_REF] and theoretical statements and/or numerical comparisons between beam theories ignoring or taking into account the STMDE have been presented [6-7, 9-10, 19]. However, this is not the case for vibration analysis of bars taking into account both shear effects and geometrical nonlinearities. Cortinez and Piovan [START_REF] Cortinez | Vibration and buckling of composite thin-walled beams with shear deformability[END_REF] and Machado and Cortinez [START_REF] Machado | Free vibration of thin-walled composite beams with static initial stresses and deformations[END_REF] performed buckling and vibration analysis of composite beams of open and closed cross sections with orthotropic laminates, Minghini et al. [START_REF] Minghini | Vibration analysis with second-order effects of pultruded FRP frames using locking-free elements[END_REF] analyzed pultruded FRP beams and frames by developing lockingfree elements while Vo and Lee [START_REF] Vo | Flexural-torsional coupled vibration and buckling of thin-walled open section composite beams using shear-deformable beam theory[END_REF] performed buckling and vibration analysis of composite beams of open cross sections with arbitrary lay ups. However, in the aforementioned contributions the analyzed cross sections are thin-walled ones, forced vibrations are not investigated and geometrical nonlinearities are considered only for static initial stresses and deformations.

During the past few years, the nonlinear nonuniform torsional dynamic analysis of bars has received a good amount of attention in the literature. Apart from research efforts that neglect torsional warping (Da Silva in [START_REF] Da Silva | Non-linear flexural-flexural-torsional-extensional dynamics of beams--I. Formulation[END_REF][START_REF] Da Silva | Non-linear flexural-flexural-torsional-extensional dynamics of beams--II. Response analysis[END_REF], Pai and Nayfeh in [START_REF] Pai | Three-dimensional nonlinear vibrations of composite beams -I. Equations of motion[END_REF][START_REF] Pai | Three-dimensional nonlinear vibrations of composite beams -II. flapwise excitations[END_REF][START_REF] Pai | Three-dimensional nonlinear vibrations of composite beams -III. Chordwise excitations[END_REF]), Di Egidio et al. in [START_REF] Di Egidio | A non-linear model for the dynamics of open cross-section thin-walled beams--Part I: formulation[END_REF][START_REF] Di Egidio | A non-linear model for the dynamics of open cross-section thin-walled beams--Part II: forced motion[END_REF] presented a FEM solution to the nonlinear flexuraltorsional vibrations of thin-walled open beams taking into account in-plane and out-ofplane warpings and neglecting warping inertia. In these papers, the torsionalextensional coupling is taken into account but the inextensionality assumption leads to the fact that the axial boundary conditions are not general. Moreover, Simo and Vu-Quoc in [START_REF] Simo | A Geometrically-exact rod model incorporating shear and torsion-warping deformation[END_REF] presented a FEM solution to a fully nonlinear (small or large strains, hyperelastic material) three dimensional rod model based on a geometrically exact description of the kinematics of deformation. Pai and Nayfeh in [START_REF] Pai | A fully nonlinear theory of curved and twisted composite rotor blades accounting for warpings and three-dimensional stress effects[END_REF] studied a geometrically exact nonlinear curved beam model for solid composite rotor blades using the concept of local engineering stress and strain measures and taking into account the in-plane and out-of-plane warpings. In the last two research efforts, the outof-plane buckling of a framed structure and a helical spring have been analyzed respectively, thus the extensional-torsional coupling is not discussed. Recently, Avramov et. al. [START_REF] Avramov | Analysis of flexural-flexural-torsional nonlinear vibrations of twisted rotating beams with cross-sectional deplanation[END_REF] analyzed flexural-flexural-torsional free vibrations of twisted rotating beams employing nonlinear normal modes. Rozmarynowski and Szymczak [START_REF] Rozmarynowski | Non-linear free torsional vibrations of thinwalled beams with bisymmetric cross-section[END_REF] and Sapountzakis and Tsipiras [START_REF] Sapountzakis | Nonlinear nonuniform torsional vibrations of bars by the boundary element method[END_REF] focus to the problem of nonlinear torsional vibrations. Nevertheless, dynamic analysis of bars at a torsional post-buckled state, as this is presented for the static case in [START_REF] Szymczak | Buckling and initial post-buckling behavior of thin-walled I columns[END_REF], is not performed. Although free or forced vibrations of bars at flexural postbuckling configurations are well studied both numerically and experimentally [START_REF] Emam | Nonlinear Responses of Buckled Beams to Subharmonic-Resonance Excitations[END_REF][START_REF] Emam | On the Nonlinear Dynamics of a Buckled Beam Subjected to a Primary-Resonance Excitation[END_REF][START_REF] Nayfeh | Exact solution and stability of postbuckling configurations of beams[END_REF][START_REF] Emam | Postbuckling and free vibrations of composite beams[END_REF], however this is not the case for buckled bars at a torsional post-buckled state. To the authors' knowledge, only Mohri et. al. [START_REF] Mohri | Vibration analysis of buckled thin-walled beams with open sections[END_REF] proposed a FEM solution to the linear free vibration analysis of pre-and post-buckled open thin-walled cross section beams subjected to special boundary conditions, neglecting warping inertia. In all of these research efforts the angle of twist per unit length is considered as a warping parameter with the exception of the aforementioned research effort of Simo and Vu-Quoc [START_REF] Simo | A Geometrically-exact rod model incorporating shear and torsion-warping deformation[END_REF] who employed an independent one.

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of bars of arbitrary doubly symmetric constant cross section, taking into account the effects of geometrical nonlinearity (finite displacement -small strain theory) and secondary twisting moment deformation. The bar is subjected to arbitrarily distributed or concentrated conservative dynamic twisting and warping moments along its length, while its edges are subjected to the most general axial and torsional (twisting and warping) boundary conditions. The resulting coupling effect between twisting and axial displacement components is also considered and a constant along the bar compressive axial load is induced so as to investigate the dynamic response at the (torsional) post-buckled state. The bar is assumed to be adequately laterally supported so that it does not exhibit any flexural or flexural -torsional behaviour. A coupled nonlinear initial boundary value problem with respect to the variable along the bar angle of twist and to an independent warping parameter is formulated. The resulting equations are further combined to yield a single partial differential equation with respect to the angle of twist. The problem is numerically solved employing the Analog Equation Method [START_REF] Katsikadelis | The analog equation method. A boundary-only integral equation method for nonlinear static and dynamic problems in general bodies[END_REF], a BEM based method, leading to a system of nonlinear Differential -Algebraic Equations (DAE). The essential features and novel aspects of the present formulation compared with previous ones are summarized as follows.

i. The cross section is an arbitrarily shaped doubly symmetric thin or thick walled one. The formulation does not stand on the assumptions of a thin-walled structure and therefore the cross section's torsional and warping rigidities are evaluated "exactly" in a numerical sense.

ii. The beam is supported by the most general boundary conditions including elastic support or restraint.

iii. The trigonometric terms of the cross sectional twisting rotations appearing in the displacement field are retained without employing any simplifying approximations.

iv. The present investigation focuses to torsional vibrations and provides a unified framework for the theoretical statement and numerical comparison between shear deformable and shear undeformable bars undergoing linear or nonlinear, free of forced vibrations. v. Nonlinear free vibrations at torsional postbuckling configurations and primary resonance excitations are numerically examined (in the second part of this contribution) revealing several aspects of nontrivial nonlinear phenomena. vi. The adopted numerical procedure can efficiently analyze torsional vibrations of bars at postbuckling configurations without employing any assumptions on the form of the modeshapes of deformation.

vii. The proposed method employs a BEM approach (requiring boundary discretization exclusively for the cross sectional analysis) resulting in line or parabolic elements instead of area elements of the FEM solutions (requiring the whole cross section to be discretized into triangular or quadrilateral area elements), while a small number of line elements are required to achieve high accuracy.

STATEMENT OF THE PROBLEM

Displacements, strains, stresses

Let us consider a prismatic bar of length l (Fig. 1), of constant arbitrary doubly symmetric cross-section of area A . The homogeneous isotropic and linearly elastic material of the beam cross-section, with modulus of elasticity E , shear modulus G and mass density ρ occupies the two dimensional multiply connected region Ω of the y,z plane and is bounded by the ( )

j j 1,

( )

w w m m x,t = moments acting in the x direction (Fig. 1b). The bar is assumed to be adequately laterally supported so that it does not exhibit any flexural or flexural -torsional behaviour Under the aforementioned loading, the displacement field of the bar accounting for large twisting rotations is assumed to be given as

( ) ( ) ( ) ( ) P m x S u x, y,z,t u x,t x,t y,z η φ = + (1a) ( ) ( ) ( ) ( ) x x v x, y,z,t z sin x,t y 1 cos x,t θ θ = - - - (1b) ( ) ( ) ( ) ( ) x x w x,y,z,t y sin x,t z 1 cos x,t θ θ = - - (1c)
where u , v , w are the axial and transverse bar displacement components with respect to the Syz system of axes [43];

( )

x x,t θ is the angle of twist; P S φ is the primary warping function with respect to the shear center S [1]; ( )

x x,t η and ( ) m u x,t denote an independent warping parameter [START_REF] El Fatmi | Non-uniform warping including the effects of torsion and shear forces. Part I: A general beam theory[END_REF] and an "average" axial displacement of the bar's cross section, respectively, that will be later discussed. By dropping the displacement and rotation components related to flexure of the displacement field adopted in [START_REF] Simo | A Geometrically-exact rod model incorporating shear and torsion-warping deformation[END_REF][START_REF] Machado | Lateral buckling of thin-walled composite bisymmetric beams with prebuckling and shear deformation[END_REF],

eqns (1) are precisely recovered. Moreover, by neglecting the time variation of the defined quantities, adopting the approximations of the linear theory of torsion

( x x sinθ θ ≈ ,
x cos 1 θ ≈ ) and having in mind that only bisymmetrical cross sections of bars under axial and torsional loading conditions are considered, the displacement field of the linear nonuniform warping beam theory can be obtained [START_REF] El Fatmi | Non-uniform warping including the effects of torsion and shear forces. Part I: A general beam theory[END_REF].

Substituting eqns (1) in the three-dimensional nonlinear strain-displacement relations, the non vanishing strain resultants are obtained, after neglecting the nonlinear terms of the axial displacement component [45][START_REF] Rothert | Nichtlineare Stabstatik[END_REF][START_REF] Brush | Buckling of bars, plates and shells[END_REF], as

( ) ( ) 2 P 2 2 xx m x S x 1 u y z 2 ε ηφ θ ′ ′ ′ = + + + (2a) P S xy x x z y φ γ η θ ∂ ′ = - ∂ P S xz x x y z φ γ η θ ∂ ′ = + ∂ (2b,c)
where the second-order geometrically nonlinear term in the right hand side of eqn (2a) is often described as the "Wagner strain" [START_REF] Trahair | Nonlinear Elastic Nonuniform Torsion[END_REF]. From eqns (2b,c) it is observed that shear strains are linear. However, if flexural effects were considered, both the normal and shear strains would be enriched with a series of nonlinear terms [START_REF] Machado | Lateral buckling of thin-walled composite bisymmetric beams with prebuckling and shear deformation[END_REF].

Considering strains to be small, employing the second Piola -Kirchhoff stress tensor, the Hooke's stress-strain law [START_REF] Vlasov | Thin-walled elastic beams, Israel Program for Scientific Translations[END_REF][START_REF] Armenakas | Advanced Mechanics of Materials and Applied Elasticity[END_REF] and eqns (2), the work contributing stress components are given as where

( ) ( ) 2 P 2 2 xx m x S x 1 S E u y z 2 η φ θ ⎡ ⎤ ′ ′ ′ = + + + ⎢ ⎥ ⎣ ⎦ (3a)
P P S xy x S G z y φ θ ⎛ ⎞ ∂ ′ = - ⎜ ⎟ ⎜ ⎟ ∂ ⎝ ⎠ P P S xz x S G y z φ θ ⎛ ⎞ ∂ ′ = + ⎜ ⎟ ⎜ ⎟ ∂ ⎝ ⎠ (4a,b) ( ) P S S xy x x S G y φ η θ ∂ ′ = - ∂ ( ) P S S xz x x S G z φ η θ ∂ ′ = - ∂ (5a,b)
denote the well known primary (St. Venant) shear stress distribution accounting for uniform torsion [START_REF] Sapountzakis | Torsional vibrations of composite bars by BEM[END_REF] and the secondary (warping) shear stress distribution accounting for nonuniform torsion, respectively.

Primary warping function

∂ ∂ ∂ + + - = ∂ ∂ ∂ in Ω , ( ) x 0,l ∀ ∈ (7)
it is easily concluded that this equation can not be satisfied. The same conclusion also holds for the associated boundary condition of this equation, as this is thoroughly discussed in [START_REF] El Fatmi | Non-uniform warping including the effects of torsion and shear forces. Part I: A general beam theory[END_REF] for the linear static case. Observing eqns (5a,b), it comes up that adopting the displacement field given in eqns (1), warping shear stresses follow the distribution of the derivatives of P S φ . Moreover, a warping shear stress distribution including a secondary warping function S S φ is proved not to violate both the aforementioned equation of motion [START_REF] Sapountzakis | Secondary torsional moment deformation effect by BEM[END_REF] and the associated boundary condition, as

proposed in [START_REF] Sapountzakis | Warping shear stresses in nonlinear nonuniform torsional vibrations of bars by BEM[END_REF]. Therefore, employing eqns [START_REF] Vlasov | Thin-walled elastic beams, Israel Program for Scientific Translations[END_REF] to obtain accurate values of warping shear stresses is of doubtful validity, especially near the boundary of the cross section.

Nevertheless, the present formulation depending on an independent warping parameter shear stresses in global equilibrium, which has not been achieved in previous research efforts [START_REF] Sapountzakis | Warping shear stresses in nonlinear nonuniform torsional vibrations of bars by BEM[END_REF] for the case of closed shaped cross sections.

Equations of motion

In order to establish the equations of motion, the principle of virtual work under a Total Lagrangian formulation is employed as this is accomplished in [START_REF] Sapountzakis | Nonlinear nonuniform torsional vibrations of bars by the boundary element method[END_REF]. Performing the decomposition of shear strains into primary and secondary parts, as it is described for shear stresses in eqns (3b-c, 4, 5), the contribution of shear stresses in the virtual work of internal forces can be written as

Carrying out suitable integration by parts and exploiting both eqns (6a) and (6b), it is easily proven that the underlined terms in the above equation vanish. The remaining terms can be written more conveniently as ( )

l P S 1 t x t x x x 0 I M M d x δθ δη δθ = ⎡ ⎤ ′ ′ = - - ⎣ ⎦ ∫ where P t M , S t
M are the primary and secondary twisting moments, respectively [START_REF] Sapountzakis | Warping shear stresses in nonuniform torsion by BEM[END_REF], defined here as d

P P P P P S S t x y x z M S z S y y z Ω φ φ Ω ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ∂ ∂ = -+ + ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ∫ (9a)
d P P S S S S S t x y x z M S S y z Ω φ φ Ω ⎛ ⎞ ∂ ∂ = - + ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ∫ (9b)
Substituting eqns (4-5) into eqns (9), the above stress resultants are given (with respect to the kinematical components) as

P t t x M GI θ ′ = , ( ) S S t t x x M GI η θ ′ = - - where t I , S t I
are the primary (St. Venant) [START_REF] Sapountzakis | Solution of non-uniform torsion of bars by an integral equation method[END_REF] and secondary [START_REF] Murín | An effective finite element for torsion of constant crosssections including warping with secondary torsion moment deformation effect[END_REF] torsion constants, respectively given as

2 2 d P P S S t I y z y z z y Ω φ φ Ω ⎛ ⎞ ∂ ∂ = + + - ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ∫ d P P S S S t I A y z z y θ Ω φ φ Ω ⎛ ⎞ ∂ ∂ = - + ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ∫ (10a,b)
with A θ defined as the "effective shear area due to the restrained torsional warping" [START_REF] Kim | Exact dynamic/static stiffness matrices of non-symmetric thinwalled beams considering coupled shear deformation effects[END_REF]. The required algebra to reach the above expressions starting from those of eqns ( 9) is presented in Appendix A. Kim & Kim in [10] propose an expression to evaluate A θ of thin-walled cross sections within the context of the stress resultants defined in this paper, while Rubin [START_REF] Rubin | Wölbkrafttorsion von Durchlaufträgern mit konstantem Querschnitt unter Berücksichtigung sekundärer Schubverformungen[END_REF] presents expressions to evaluate the secondary torsion constant S t I of thin-walled cross sections. Numerical evaluation of A θ of arbitrarily shaped cross sections is proposed in the recent developments of Kraus [START_REF] Kraus | Computerorientierte Bestimmung der Schubkorrekturfaktoren gewalzter I-Profile[END_REF] employing the FEM and of Sapountzakis and Mokos [START_REF] Sapountzakis | Secondary torsional moment deformation effect by BEM[END_REF] employing the BEM, within the context of linear static torsional loading conditions. Throughout the present work it is assumed (unless otherwise mentioned) that A 1 θ = , which evidently leads to the relation

S t P t
I I I =with P I denoting the polar moment of inertia. It is worth here mentioning that the above relations absolutely conform to the geometrically exact beam theory of Simo and Vu-Quok [START_REF] Simo | A Geometrically-exact rod model incorporating shear and torsion-warping deformation[END_REF] and the nonuniform warping beam theory of El Fatmi [START_REF] El Fatmi | Non-uniform warping including the effects of torsion and shear forces. Part I: A general beam theory[END_REF].

It is also convenient to define the axial stress resultant N and the warping moment w M arising from normal stresses as in [START_REF] Sapountzakis | Nonlinear nonuniform torsional vibrations of bars by the boundary element method[END_REF]. Substituting the stress components given in eqns (3), the strain ones given in eqns (2) and the displacement ones given in eqns (1) to the principle of virtual work, the governing partial differential equations of motion of the bar are obtained after some algebra as ( )

m m P x x Au EAu EI n x,t ρ θ θ ′′ ′ ′′ - - = (11a) ( ) ( ) ( ) 2 S S P x t t x t x PP x x P m x P m x t 3 I G I I GI EI EI u EI u m x,t 2 ρ θ θ η θ θ θ θ ′′ ′ ′ ′′ ′ ′′ ′′ ′ - + + - - - = (11b) ( ) ( ) S S x S x t x x w C EC GI m x,t ρ η η η θ ′′ ′ - + - =- (11c)
subjected to the initial conditions ( ( )

x 0,l ∈ ) ( ) ( ) m m 0 u x,0 u x = ( ) ( ) m m 0 u x,0 u x = (12a,b) ( ) ( ) x x 0 x,0 x θ θ = ( ) ( ) x x 0 x,0 x θ θ = (12c,d) ( ) ( ) x x 0 x,0 x η η = ( ) ( ) x x 0 x,0 x η η = (12e,f)
together with the boundary conditions at the bar ends x 0,l =

1 2 m 3 N u α α α + = 1 t 2 x 3 M β β θ β + = 1 w 2 x 3 M β β η β + = (13a,b,c)
where N , t M , w M are the axial force, twisting and warping moments at the bar ends, respectively given as

( ) 2 m P x 1 N EAu EI 2 θ ′ ′ = + (14a) ( ) ( ) 3 S S t t t x t x P m x P P x 1 M G I I GI EI u EI 2 θ η θ θ ′ ′ ′ ′ = + - + + w S x M EC η′ = - (14b,c) while i α , i β , i β = ( i 1,2,3
) are time dependent functions specified at the boundary of the bar. The geometric constants PP I , S C appearing in eqns [START_REF] Back | A shear-flexible element with warping for thin-walled open beams[END_REF][START_REF] Saadé | Non-uniform torsional behavior and stability of thin-walled elastic beams with arbitrary cross sections[END_REF] are given in [START_REF] Sapountzakis | Nonlinear nonuniform torsional vibrations of bars by the boundary element method[END_REF],

while the externally applied loads are related to the components of the traction vector with respect to the undeformed surface of the bar x t , y t , z t as ( )

d x n x,t t Ω Ω = ∫ (15a) ( ) ( ) ( ) d t y x x z x x m x,t t zcos ysin t ycos z sin Ω θ θ θ θ Ω = - - + - ∫ (15b) ( ) d P w xS m x,t t Ω φ Ω = -∫ (15c)
The boundary conditions ( 13) are the most general boundary conditions for the problem at hand, including also the elastic support. It is worth here noting that the terms of the boundary conditions of eqns ( 13) are nonlinear by virtue of eqns ( 14), while linear viscous damping could also be included in eqns [START_REF] Back | A shear-flexible element with warping for thin-walled open beams[END_REF] without any special difficulty.

Moreover, from eqns [START_REF] Back | A shear-flexible element with warping for thin-walled open beams[END_REF] it is concluded that all of the inertia terms are linear.

However, if flexural effects were considered, nonlinear inertia terms would arise in the governing equations as well [START_REF] Machado | Lateral buckling of thin-walled composite bisymmetric beams with prebuckling and shear deformation[END_REF].

It is worth here noting that all the relations established so far are completely analogous to those of the Timoshenko beam theory, modeling the shear -bending loading conditions of bars. The analogy of all the kinematical and stress components with the ones of the Timoshenko beam theory is thoroughly presented in [START_REF] Murín | An effective finite element for torsion of constant crosssections including warping with secondary torsion moment deformation effect[END_REF] for the linear static case.

The established initial boundary value problem is a coupled and nonlinear one. A significant reduction on the set of differential equations can be achieved by neglecting the axial inertia term m Au ρ of eqn (11a), a common assumption among dynamic beam formulations. Ignoring this term, two partial differential equations with respect to two unknown displacement components ( ( )

x x,t θ , ( ) x x,t η
) can be obtained. These equations can be further combined by performing similar algebraic manipulations as the ones described in [START_REF] Sapountzakis | Flexural-torsional buckling analysis of composite beams by BEM including shear deformation effect[END_REF][START_REF] Sapountzakis | Shear deformation effect in flexuraltorsional vibrations of beams by BEM[END_REF][START_REF] Sapountzakis | Nonlinear dynamic analysis of Timoshenko beams by BEM. Part I: Theory and numerical implementation[END_REF][START_REF] Prokic | On fivefold coupled vibrations of Timoshenko thin-walled beams[END_REF] so as to formulate a single partial differential equation with respect to ( )

x x,t θ . This equation can then be directly compared with the corresponding one presented in [START_REF] Sapountzakis | Nonlinear nonuniform torsional vibrations of bars by the boundary element method[END_REF] that does not account for secondary twisting moment deformation effect (STMDE). In what follows, these procedures are described in detail for the case of constant along the bar axial load which is of great practical interest, especially in torsional postbuckling analysis of bars.

Reduced initial boundary value problem for constant along the bar axial load

For the case of constant along the bar axial load, the axial boundary conditions (13a) are written as

( ) m u 0,t 0 = ( ) ( ) N l,t N l,t = (16a,b)
Employing the aforementioned simplifications, eqn (11a) gives

P m x x I u A θ θ ′′ ′ ′ ′ = - , ( )
x 0,l ∀ ∈ which after subsequent integration and utilization of eqns ( 16) yields

( ) 2 P m x I 1 N u 2 A EA θ ′ ′ = - + , [ ] x 0,l ∀ ∈ . Substituting
I 3 I GI GI N GI EI m x,t A 2 ρ θ θ η θ θ ⎛ ⎞ ′′ ′ ′ ′′ - + + + - = ⎜ ⎟ ⎝ ⎠ (17)
along with the governing eqn (11c), the pertinent initial conditions (12c,d,e,f) and boundary conditions (13b, c). It is worth here noting that eqns (14a,c) hold, whereas eqn (14b) is modified as

( ) 3 S S P t t t x t x n x I 1 M GI GI N GI EI A 2 θ η θ ⎛ ⎞ ′ ′ = + + - + ⎜ ⎟ ⎝ ⎠ (18)
where n I is a nonnegative geometric cross sectional property related to the geometrical nonlinearity expressed as

2 P n PP I I I A = - .
The equations of motion ( 17), (11c) can be further combined by performing similar algebraic manipulations with those presented in [START_REF] Sapountzakis | Flexural-torsional buckling analysis of composite beams by BEM including shear deformation effect[END_REF][START_REF] Sapountzakis | Shear deformation effect in flexuraltorsional vibrations of beams by BEM[END_REF][START_REF] Sapountzakis | Nonlinear dynamic analysis of Timoshenko beams by BEM. Part I: Theory and numerical implementation[END_REF]

I I 1 3 3 E C N G I N E I A 2 GI A EC EI κ θ θ θ ⎛ ⎞ ⎡ ′′ ′ ′′ ′ ′′ ′ ′ ′ ′′ - + + - + + ⎜ ⎟ ⎢ ⎜ ⎟ ⎣ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ ⎤ ′ ′ ′′′′ ′′ ′ ′′ + + + - + - + ⎜ ⎟ ⎜ ⎟ ⎥ ⎜ ⎟ ⎦ ⎝ ⎠ ⎝ ⎠ ′′ ′ ′′ ′ + + ( ) 2 S S x x x t w t t S S t t C EC 3 m m m m 2 GI GI ρ θ θ θ ⎡ ⎤ ′′ ′ ′′′′ ′ ′′ + = + + - ⎢ ⎥ ⎣ ⎦ (19)
after neglecting the higher order term ( )

S S P t x C I GI ρ ρ θ ⎡ ⎤ ⎣ ⎦ . Eqn. ( 19
)
I 1 3 N 3 2 GI GI A GI GI η θ θ θθ θ θ κ ⎛ ⎞ ⎡ ⎤ ′ ′′′ ′ ′′ ′ ′′′ = + + + + ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠ (20a) ( ) ( ) ( ) 3 P P t t x S x n x S t 2 2 n S x x x x S t I I 1 1 M GI N EC N EI A 2 GI A EI 3 -EC 3 2 GI θ θ θ κ θ θ θ θ ⎛ ⎞ ⎛ ⎞ ′ ′′′ ′ = + - + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎡ ⎤ ′ ′′ ′ ′′′ + ⎢ ⎥ ⎣ ⎦ (20b) ( ) 2 n P w S x S x x S S t t EI I 1 3 M EC N EC 2 GI A GI θ θ θ κ ⎛ ⎞ ⎡ ⎤ ′′ ′ ′′ = + + ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠ (20c)
In the above equations κ is an auxiliary geometric constant related to the secondary twisting moment deformation effect given as ( )

S S t t t I I I κ = + .
Comparing the formulated reduced initial boundary value problem and the one presented in [START_REF] Sapountzakis | Nonlinear nonuniform torsional vibrations of bars by the boundary element method[END_REF] where the STMDE is not taken into account, it is concluded that this effect alters the expressions of warping inertia, warping stiffness and external loading and induces higher order nonlinear inertia and stiffness terms in the governing partial differential equation. Some nonlinear stiffness terms are also induced in the kinematical and stress components at the bar ends. If the STMDE is neglected, the displacement field (eqns (1)) must be updated by substituting x x η θ ′ = (and all the subsequent equations accordingly) as well as eqns [START_REF] Kollár | Flexural-torsional vibration of open section composite beams with shear deformation[END_REF][START_REF] Park | A finite element analysis of discontinuous thinwalled beams considering nonuniform shear warping deformation[END_REF] by substituting 1 κ = and dropping all the terms related to S t I .

INTEGRAL REPRESENTATIONS -NUMERICAL SOLUTION

For the angle of twist x θ

According to the precedent analysis, the nonlinear nonuniform torsional vibration problem of shear deformable bars reduces to establishing the displacement component ( )

x x,t θ having continuous partial derivatives up to the fourth order with respect to x and up to the second order with respect to t , satisfying the nonlinear initial boundary value problem described by the partial differential equation ( 19), the initial conditions (12c,d) along the bar and the boundary conditions (13b,c) at the bar ends x 0,l = . This problem is solved using the Analog Equation Method [START_REF] Katsikadelis | The analog equation method. A boundary-only integral equation method for nonlinear static and dynamic problems in general bodies[END_REF]. Employing this method as this is presented in [START_REF] Sapountzakis | Nonlinear nonuniform torsional vibrations of bars by the boundary element method[END_REF] and applying eqn [START_REF] Kollár | Flexural-torsional vibration of open section composite beams with shear deformation[END_REF] to L collocation points, L semidiscretized nonlinear equations of motion are formulated as

{ } { } ( ) ( ) , , , , , , = nl nl 1 2 1 2 3 + + + M d K d m H H d d k H H H d f ( 21
)
where d is a L 8 + generalized unknown vector, nl m , nl k are nonlinear generalized mass vector and stiffness vector respectively and , , M K f are generalized mass matrix, stiffness matrix and force vector respectively, given as ( ) t m x,t , respectively at the collocation points by using appropriate finite differences [START_REF] Sapountzakis | Nonuniform torsion of bars of variable cross section[END_REF]. Eqns [START_REF] Cortinez | Vibration and buckling of composite thin-walled beams with shear deformability[END_REF] along with 8 algebraic equations constitute a system of simultaneous L 8 + Differential -Algebraic equations (DAE). The details of the derivation as well as the solution algorithm of this system of equations can be retrieved in [START_REF] Sapountzakis | Nonlinear nonuniform torsional vibrations of bars by the boundary element method[END_REF]. simply supported according to its torsional boundary conditions, while the left end is immovable and the right end is subjected to a compressive axial load according to its axial boundary conditions.

{ } { } { } ( ) { } ( ){ } ( ){ } ( ) { } ( ){ } ( ){ } ( ) { } ( ){ } 2 nl n S 1 2 1 2 1 i i i i i S i t 2 1 1 2 1 1 i i i i i EI C 3 6 GI 3 3 ρ ⎡ ⎡ ⎤ = - + ⎣ ⎦ ⎢ ⎣ ⎤ ⎡ ⎤ + + ⎣ ⎦ ⎥ ⎦ m H d H d H d H d H d H d H d H d H d H d (22a) { } { } { } ( ) { } { } ( ){ } ( ) { } ( ) { } { } 2 3 nl S n 1 2 n 2 i i i S i t 2 1 2 3 1 i i i i i EC 3 EI EI 3 2 GI 3 9 2 ⎡ ⎡ ⎤ ⎡ ⎤ = - + ⎣ ⎦ ⎣ ⎦ ⎢ ⎣ ⎤ ⎡ ⎤ + + ⎥ ⎣ ⎦ ⎦ k H d H d H d H d H d H d H d d (22b)
P P P 0 S 2 S S t t EI I 1 I C N GI GI A ρ ρ κ ⎡ ⎤ ⎛ ⎞ = - + + ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ M H H ( 22c
) P P t 2 S S t I I 1 GI N EC N A GI A κ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ = - + + + ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ K H I 0 (22d) S S S S t t C E C GI GI ρ ⎧ ⎫ ⎪ ⎪ ′ ′ ′ = + + - ⎨ ⎬ ⎪ ⎪ ⎩ ⎭ t w t t f m m m m (

For the primary warping function

In the first example, for comparison reasons, the axial load -torsional fundamental natural frequency relation of the aforementioned bar has been investigated.

Mohri et. al. [START_REF] Mohri | Vibration analysis of buckled thin-walled beams with open sections[END_REF], without considering the secondary twisting moment deformation effect, proposed analytical load -frequency relations by dropping the higher order warping inertia term and assuming that (i) the fundamental modeshape of vibration follows a sinusoidal form

x sin l π ⎛ ⎞ ⎜ ⎟ ⎝ ⎠
both in the pre-and postbuckling region, (ii) the bar vibrates harmonically, (iii) vibrations of small amplitude around a static equilibrium state are performed. These load ( N ) -frequency ( f ω ) relations for the pre-and postbuckling regions are given, respectively, as

2 2 f cr , N 1 N θ θ ω ω ⎛ ⎞ = - ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 2 2 f cr , N 2 1 N θ θ ω ω ⎛ ⎞ = - ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ (23a,b)
where θ ω is the natural frequency for vanishing axial load and cr , N θ is the torsional buckling load of the bar, respectively given as [41] [START_REF] Mohri | Vibration analysis of buckled thin-walled beams with open sections[END_REF][START_REF] Mohri | Flexural-torsional post-buckling analysis of thin-walled elements with open sections[END_REF] assuming that the twisting deformation mode follows a sinusoidal form

x sin l π ⎛ ⎞ ⎜ ⎟ ⎝ ⎠
proposed the following analytical expression to evaluate ( )

0 l / 2 θ ( ) ( ) , prebuckling region
, postbuckling region

2 0 P cr , 2 n 0 l / 2 I 8l N N EAI 3 θ θ π ⎧ ⎪ = ⎨ ± - ⎪ ⎩ (25)
In Fig. 2, the load -frequency relations of eqns [START_REF] Minghini | Vibration analysis with second-order effects of pultruded FRP frames using locking-free elements[END_REF] θ ω relation is rather complex and differs significantly from the one of the prebuckling region. In the pre-buckled state, the increase of initial amplitude of vibration always leads to an increase of the bar's stiffness and eventually an increase of f ω (see also [START_REF] Sapountzakis | Nonlinear nonuniform torsional vibrations of bars by the boundary element method[END_REF], and [START_REF] Rozmarynowski | Non-linear free torsional vibrations of thinwalled beams with bisymmetric cross-section[END_REF] for the case of axially immovable ends and

n I 0 = ).
In Figs. 5,6 the time histories of the angle of twist ( ) In Figs.

x0 x 0 θ = , ( )
x0 x 0 θ = ).
f ,lin ω is the fundamental natural frequency of the bar undergoing linear torsional vibrations, ignoring STMDE and is numerically evaluated by performing modal analysis as this is presented in [START_REF] Sapountzakis | Dynamic analysis of 3-D beam elements including warping and shear deformation effects[END_REF].

In Figs. 9, 10, the time histories of the angle of twist ( )

x l / 2,t θ at the midpoint of the bar are presented (with or without STMDE) for two cases of analysis, namely ignoring or considering geometrical nonlinearity, respectively. The linear analyses are carried out by dropping all the nonlinear terms of eqns [START_REF] Kollár | Flexural-torsional vibration of open section composite beams with shear deformation[END_REF], [START_REF] Park | A finite element analysis of discontinuous thinwalled beams considering nonuniform shear warping deformation[END_REF]. As expected, only in the linear cases (Fig. 9) deformations continue to increase with time. The beating phenomenon observed in the nonlinear response (Fig. 10) is explained from the fact that large twisting rotations increase the bar's fundamental natural frequency f ω (by increasing the stiffness of the bar), thereby causing a detuning of f ω with the frequency of the external loading ( f ,lin ω

). After the angle of twist reaches its maximum value, the amplitude of twisting deformations decreases, leading to the reversal of the previously mentioned effects. Moreover, in Fig. 11, the time histories of the primary twisting moment

Example 4 -Linear forced vibrations, closed shaped cross section

In the last example, the (geometrically) linear response of a bar Murín and Kutiš [START_REF] Murín | An effective finite element for torsion of constant crosssections including warping with secondary torsion moment deformation effect[END_REF]. Throughout the entire numerical example the geometric constants of the bar are assumed to take the values presented in [START_REF] Murín | An effective finite element for torsion of constant crosssections including warping with secondary torsion moment deformation effect[END_REF] (

2 A 0,240m = , 4 t I 0,089824m = , S 4 t I 0,001107m = , 6 S C 0,000193m =
), while the results have been 29

obtained employing 31 nodal points (longitudinal discretization). The bar's left end is clamped, while its right end is free and subjected to zero axial load, zero warping moment and prescribed twisting moment ( ( )

N l,t 0 = , ( ) w M l,t 0 = , ( ) t t M l,t M = ).
In the beginning, for comparison purposes, the linear static loading conditions are investigated by dropping all the inertia and nonlinear terms of eqns [START_REF] Kollár | Flexural-torsional vibration of open section composite beams with shear deformation[END_REF], [START_REF] Park | A finite element analysis of discontinuous thinwalled beams considering nonuniform shear warping deformation[END_REF]. A concentrated twisting moment t M 32kNm = is applied at the bar's right end and the computed results of several kinematical and stress components taking into account or ignoring the STMDE are presented in Table 6 as compared wherever possible with those obtained from a FEM solution [START_REF] Murín | An effective finite element for torsion of constant crosssections including warping with secondary torsion moment deformation effect[END_REF]. The coincidence of the results between the proposed and the FE methods is remarked. Moreover, from the comparison of the results obtained considering or ignoring the STMDE, it is concluded that STMDE influences decisively the stress components, while it exhibits a less pronounced effect on the kinematical components. Thus, the significance of STMDE in linear static analysis of bars of closed shaped cross sections, already reported in the literature [START_REF] Murín | An effective finite element for torsion of constant crosssections including warping with secondary torsion moment deformation effect[END_REF], is verified.

Following the linear static loading conditions, the linear forced vibrations are investigated by dropping all the nonlinear terms of eqns [START_REF] Kollár | Flexural-torsional vibration of open section composite beams with shear deformation[END_REF], [START_REF] Park | A finite element analysis of discontinuous thinwalled beams considering nonuniform shear warping deformation[END_REF]. More specifically, the resonance of the examined bar is studied by applying a distributed twisting moment 6 concerning static analysis can be extended in linear dynamic analysis of bars of closed shaped cross sections.

CONCLUDING REMARKS

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of simply or multiply connected cylindrical bars of arbitrary doubly symmetric cross section, taking into account the effects of geometrical nonlinearity and secondary twisting moment deformation. The displacement components are expressed so as to be valid for large twisting rotations (finite displacement -small strain theory), while the use of an independent warping parameter makes it possible to account for warping shear stresses in global equilibrium of the bar.

The main conclusions that can be drawn from this investigation are The developed procedure retains most of the advantages of a BEM solution over a FEM approach, although it requires longitudinal domain discretization.

APPENDIX A

Expression of the primary torsion constant t I : Substituting eqns (4) into eqn (9a), the primary torsion constant of eqn (10a) is obtained as

2 2 P P S S t 1 2 I z y d I I y z Ω φ φ Ω ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ∂ ∂ ⎢ ⎥ = - + + = + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ∫ (A.1)
where

P P 2 2 S S 1 I y z y z d z y Ω φ φ Ω ⎛ ⎞ ∂ ∂ = + + - ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ∫ (A.2a) P P P P S S S S 2 I z y d y y z z Ω φ φ φ φ Ω ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ∂ ∂ ∂ ∂ = -+ + ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ∂ ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ∫ (A.2b)
Carrying suitable integration by parts with respect to y for the first term of eqn (A.2b) and with respect to z for the second term of eqn (A.2b) and exploiting both eqns (6a) and (6b), 2 I is proved to vanish. Consequently, t I is proven to be given from expression (10a).

Expression of the secondary torsion constant S t I : Substituting eqns (5) into eqn (9b), the secondary torsion constant of eqn (10b) is obtained as

2 2 P P S S S t I A d y z θ Ω φ φ Ω ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ∂ ∂ ⎢ ⎥ = + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ∫ (A.3)
Carrying suitable integration by parts with respect to y for the first term of eqn (A. Using the Gauss divergence theorem, eqn (A.5) yields the exact formula of eqn (10b).

Finally it is worth here noting that the alternative definitions of P t M , S t M as ( )

d P P P t x y x z M S z S y Ω Ω ⎡ ⎤ = -+ ⎣ ⎦ ∫ ( ) d S S S t x y x z M S z S y Ω Ω ⎡ ⎤ = -+ ⎣ ⎦ ∫ (A.6a,b)
can be easily proven to be identical to the ones presented in eqns (9a,b).

n t y z 1 Γ 2 Γ Κ Γ 1 C≡S (Ω) s ω r P q = - P α q (a)
x,u [START_REF] Minghini | Vibration analysis with second-order effects of pultruded FRP frames using locking-free elements[END_REF] [START_REF] Mohri | Vibration analysis of buckled thin-walled beams with open sections[END_REF] along with pairs of values ( f ω , N ) obtained from present study (example 1).

u. 2 . 3

 23 The primary warping function is evaluated independently by exploiting local equilibrium considerations along the longitudinal x axis (after ignoring the inertia and the nonuniform torsion theory terms) from the solution of the following boundary value problem ∂ + ∂ ∂ is the Laplace operator and / n ∂ ∂ denotes the directional derivative normal to the boundary Γ . Since the problem at hand has Neumann type boundary condition, the evaluated warping function contains an integration constant (parallel displacement of the cross section along the bar axis),which is evaluated following the procedure presented in Sapountzakis[START_REF] Sapountzakis | Solution of non-uniform torsion of bars by an integral equation method[END_REF]. The meaning of the "average" axial displacement of the cross section of the bar can now be explained as eqn (1a) can be written as d d Warping shear stress distribution, independent warping parameter x η By substituting eqns (1a,3) on the differential equation describing local equilibrium along the longitudinal axis x[START_REF] Sapountzakis | Nonlinear nonuniform torsional vibrations of bars by the boundary element method[END_REF]

xη

  and the angle of twist x θ , makes it possible to accurately analyze bars of either closed or open shaped cross sections. This formulation can also account for warping

  22e) In the above equations, { } i ⋅ denotes the (arbitrary) i-th row .with I and 0 being the L L × identity matrix and the L 8 × rectangular matrix with zero elements, respectively, m are vectors containing the values of the dynamic external loading at the L nodal points. The elements of ′ w m , ′′ t m can be written with respect to the values of ( ) w m x,t ,

φExample 1 -

 1 The integral representations and the numerical solution for the evaluation of the angle of twist x θ presented in the previous section assume that the warping S C and the torsion t I , S t I constants are already established. The expressions of these constants indicate that their evaluation presumes that P S φ at any interior point of the domain Ω of the cross section of the bar is known. Once P S φ is established, S C and t I constants are evaluated by converting the domain integrals into line integrals along the boundary using the corresponding relations presented in Sapountzakis and Mokos [1], while the line integral expression of On the basis of the analytical and numerical procedures presented in the previous sections, a computer program has been written and representative examples have been studied to demonstrate the validation, efficiency and the range of applications of the developed method. It is noted that the probability of the bar's exhibiting chaotic motion in its nonlinear response is not investigated within the present study. Small amplitude free vibrations, open shaped cross section In the first three examples, an open thin-walled I-shaped cross section bar ( been studied, while the numerical results have been obtained employing 21 nodal points (longitudinal discretization) and 400 boundary elements (cross section discretization). The geometric constants of the bar are computed as

2 θ

 2 results, the nonlinear initial boundary value problem (eqns[START_REF] Kollár | Flexural-torsional vibration of open section composite beams with shear deformation[END_REF], (12c,d), (13b,c)) has been solved for several values of constant axial load N to obtain the response of the bar in the time domain, taking into account or neglecting the STMDE. The fundamental natural frequency f ω is then computed by exploiting the first few full cycles of vibration. The computation is based on the Fast Fourier Transform (FFT) of the time history, while a Hanning data window is employed[START_REF] Brigham | Fast Fourier transform and its applications[END_REF].The free vibrations are initiated by dropping all the torsional loading terms and by employing the linear fundamental modeshape of the angle of twist as initial twisting rotations presented in[START_REF] Sapountzakis | Nonlinear nonuniform torsional vibrations of bars by the boundary element method[END_REF] and corresponds to the sinusoidal form assumed by Mohri et. al.[START_REF] Mohri | Vibration analysis of buckled thin-walled beams with open sections[END_REF]. In order to perform vibrations of small amplitude, the initial midpoint angle of twist amplitude corresponding to the static equilibrium state. Mohri et. al.

Example 2 - 2 θ

 22 are presented along with pairs of values ( f ω , N ) obtained from the proposed method, ignoring the STMDE. The pairs ( f ω , N ) are computed by using the values of initial midpoint angle of twist amplitude of bars undergoing small amplitude torsional vibrations is concluded. The higher discrepancies between the two sets of solutions in the postbuckling region as compared to the ones in the prebuckling region are attributed to the fact that the previously mentioned assumptions (i), (ii) are valid only in the prebuckling region. To illustrate this point better, in Figs. 3a,b the obtained time histories of the angle of twist are presented, respectively, demonstrating that only the buckled bar undergoes multifrequency vibrations. For comparison purposes, in Fig.3b the time history of ( ) x l / 2,t θ employing the nonlinear fundamental modeshape of the angle of twist as initial twisting rotations that the initiation of free vibrations with the nonlinear modeshape does not induce higher harmonics in the response of the bar. Finally, in Table 2 the obtained results of pairs of values ( f ω , N ) are presented taking into account or ignoring the STMDE and employing either the linear or the nonlinear fundamental modeshape to initiate the free vibrations, showing that in both pre-and post-buckling regions, the STMDE does not affect the fundamental natural frequency of bars of open thin-walled cross sections undergoing small amplitude vibrations. The insignificance of the STMDE for linear static loading conditions of bars of such cross sections has already been reported in the literature [6]. Nonlinear free vibrations, open shaped cross section In the second example, the effect of geometrical nonlinearity on the dynamic characteristics of the bar under examination (bar of example 1) at a post-buckled state is investigated. The nonlinear initial boundary value problem (eqns (19), (12c,d), (13b,c)) has been solved for N 5500kN to obtain the response of the bar in the time domain, taking into account or ignoring the STMDE. The free vibrations are initiated by dropping all the torsional loading terms and by considering the linear fundamental modeshape of the angle of twist as initial twisting rotations STMDE [35]. The fundamental natural frequency f ω and the position around which vibrations are performed are then computed by exploiting the first few full cycles of vibration.In Fig.4, the variation of the torsional fundamental natural frequency f ω with respect to the initial midpoint angle of twist amplitude geometrical nonlinearity has a profound effect οn both the natural frequency of the bar and the positions around which vibrations are performed. From Table3, it is observed that these positions are non constant ones and depend on the initial amplitude of vibration, while for large values of ( ) x0 l / 2 θ they coincide with the static equilibrium position of the prebuckling region. From Fig.4it is also concluded that the

  t at the bar's right end, respectively, for two cases of the initial midpoint angle of twist amplitude ( taking into account or ignoring the STMDE. From these figures it is verified that especially for the case of axial displacement's response is twice as much as the one of the twisting rotation's one. This effect has been already documented in the literature for nonlinear torsional vibrations of bars at prebuckling configurations[START_REF] Sapountzakis | Nonlinear nonuniform torsional vibrations of bars by the boundary element method[END_REF] and nonlinear flexural vibrations of bars at prebuckling configurations with axially immovable ends[START_REF] Bhashyam | Galerkin finite element method for non-linear beam vibrations[END_REF]. Apparently, this is no longer valid when vibrations at postbuckling configurations are performed around positions with Figs.5, 6, it is also observed that the STMDE does not influence the kinematical components of buckled bars of open thin-walled cross sections, undergoing large amplitude vibrations. This conclusion does not depend on the magnitude of the initial midpoint angle of twist amplitude.

(

  shaped cross section is studied. The cross section is of total height h 1,64m = , total width b 1,05m = , horizontal walls thickness h t 0,04m = and vertical walls thickness v t 0,05m = and is identical to the box-shaped cross section studied by

  f ,lin ω is the fundamental natural frequency of the bar undergoing linear torsional vibrations, ignoring STMDE and is numerically evaluated by performing modal analysis as this is presented in [63]. In Fig. 12 the time histories of the secondary twisting moment t at the bar's left end taking into account or ignoring (scaled quantity are presented demonstrating the decisive influence of the aforementioned effect to this stress resultant. Finally, in Fig. 13 the time histories of the warping moment ( ) w M 0,t at the bar's left end are depicted taking into account or ignoring the STMDE. From Figs. 12-13, the conclusions already drawn from Table

  a. The numerical technique presented in this investigation is well suited for computer aided analysis of cylindrical bars of arbitrarily shaped doubly symmetric cross section, supported by the most general twisting and warping boundary conditions and subjected to the combined action of arbitrarily distributed or concentrated time dependent conservative axial and torsional loading.b. The geometrical nonlinearity leads to coupling between the torsional and axial equations of motion and alters the modeshapes of vibration. Consequently, the initiation of small amplitude free vibrations of buckled bars with the linear fundamental modeshape as initial twisting rotations induces higher harmonics in the bar's response, but its fundamental natural frequency is only slightly affected. c. Large twisting rotations have a profound effect on both the positions around which vibrations are performed and the fundamental natural frequency of buckled bars undergoing large amplitude free vibrations. The computation of dynamic characteristics of buckled bars differs from the one of bars at a pre-buckled state where the increase of initial amplitude of vibration always leads to an increase of the fundamental natural frequency. d. The natural frequency of the axial displacement's response of buckled bars undergoing large amplitude free vibrations is twice as much as the one of the twisting rotation's one, when vibrations are performed around the static equilibrium position of the prebuckling configuration. e. As expected, geometrical nonlinearity bounds the (twisting) deformations of bars at a pre-buckled state subjected to primary resonance excitations. A beating phenomenon is observed in the time histories of kinematical and stress components. f. The secondary twisting moment deformation affects negligibly the kinematical and stress components of bars of open shaped thin-walled cross sections undergoing free or forced vibrations of small or large amplitude. g. The secondary twisting moment deformation affects the kinematical components of bars of closed shaped cross sections undergoing linear vibrations. Its effect is much more pronounced on stress components, concluding that it cannot be neglected in linear dynamic analysis of bars of such cross sections.

3 )

 3 and with respect to z for the second term of eqn (A.3) and exploiting eqn (8a),

Fig. 1 .

 1 Fig. 1. Prismatic bar of an arbitrarily shaped doubly symmetric constant cross section occupying region Ω (a) subjected to axial and torsional loading (b).

Fig. 2 .

 2 Fig.2. Axial load N -torsional fundamental natural frequency f ω relation of eqns

Fig. 3 .Fig. 4 .3Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .

 345678 Fig.3. Time histories of the angle of twist at the midpoint of the bar of example 1, for a prebuckling ( N 1000kN = -) (a) and a postbuckling ( N 5000kN = -) (b) axial loading.

Fig. 9 .

 9 Fig.9. Time histories of the angle of twist at the midpoint of the bar of example 3 taking into account or ignoring secondary twisting moment deformation effect-Geometrically linear case.

Fig. 10 .Fig. 11 .Fig. 12 .Fig. 13 .

 10111213 Fig.10. Time histories of the angle of twist at the midpoint of the bar of example 3 taking into account or ignoring secondary twisting moment deformation effect -Geometrically nonlinear case.

Fig. 8 .Fig. 9 .

 89 Fig.8. Time histories of various twisting moment components at the left end of the free vibrating bar of example 2 taking into account the secondary twisting moment deformation effect for initial midpoint angle of twist amplitude ( ) x0 l / 2 2,50 rad θ

Fig. 10 .

 10 Fig.10. Time histories of the angle of twist at the midpoint of the bar of example 3 taking into account or ignoring secondary twisting moment deformation effect -Geometrically nonlinear case.

Fig. 11 .

 11 Fig.11. Time histories of various twisting moment components at the left end of the bar of example 3 taking into account the secondary twisting moment deformation effect (geometrically nonlinear analysis).

Fig. 12 .

 12 Fig.12. Time histories of the secondary twisting moment

Fig. 13 .

 13 Fig.13. Time histories of the warping moment

Primary resonance, geometrical nonlinearity, open shaped cross section

  [START_REF] Sapountzakis | Secondary torsional moment deformation effect by BEM[END_REF][START_REF] El Fatmi | Non-uniform warping including the effects of torsion and shear forces. Part I: A general beam theory[END_REF], the time histories of the primary twisting moment

	extreme values of the kinematical and stress components depicted in Figs. 5-8, obtained
	for the time interval	( ) 0,00 t 0,12 sec ≤ ≤	are presented taking into account or ignoring
	the STMDE, for	θ	( ) x0 l / 2 2,20 rad =	. From this table it is deduced that the STMDE
	influences negligibly both kinematical and stress components of buckled bars of open
	thin-walled cross sections undergoing large amplitude vibrations.
	Example 3 -In the third example, the forced vibrations of the bar under examination (bar of
	example 1) at a pre-buckled state ( N 0 = ) are investigated to determine the effects of
	geometrical nonlinearity and secondary twisting moment deformation. More
	specifically, the primary resonance of the bar is studied by applying a concentrated
	twisting moment	M	t ,ext	at its midpoint given as	M	t ,ext	( ) t M sin t0 =	(	ω	f ,lin	t	)	, where
	M	t0	=	5kNm	and	ω	f ,lin 214,23 sec =	1 -	(initial conditions	( )
														P t M (first two
	terms of eqn (20b) including the twisting moment caused by the axial load N ), the
	linear secondary twisting moment	M	S t ,lin	(third and fourth terms of eqn (20b)), the
	nonlinear twisting moment	M	t,nl	(rest nonlinear terms of eqn (20b)) and the total
	twisting moment	t M M =	P t	+	M	S t , l i n	+	M	t, n l	at the bar's left end are presented taking
	into account the STMDE for two cases of the initial midpoint angle of twist amplitude
	(	θ	( ) x0 l / 2 2,20 rad =	,	θ	( ) x0 l / 2 2,50 rad =	), respectively. Finally, in Table 4 the

  From Fig.10and from Table5it is deduced that the STMDE influences negligibly both kinematical and stress components of bars of open thinwalled cross sections undergoing forced vibrations.

										P t M (first two terms of eqn (20b)), the linear secondary twisting
	moment	M	S t ,lin	(third and fourth terms of eqn (20b)), the nonlinear twisting moment
	M	t ,nl	(rest nonlinear terms of eqn (20b)) and the total twisting moment
	t M M =	P t	+	M	S t , l i n	+	M	t, n l	at the bar's left end are presented taking into account the
	STMDE and performing geometrically nonlinear analysis. Finally, in Table 5 the
	extreme values of the kinematical and stress components depicted in Figs. 10-11,
	obtained for the time interval	( ) 0,00 t 0,30 sec ≤ ≤	are presented taking into account or
	ignoring the STMDE.

Table 1 .

 1 Axial load N , midpoint angle of twist amplitude

	θ	( ) 0 l / 2	corresponding to

Table 3 .

 3 Initial midpoint angle of twist amplitudes

						θ	( ) x0 l / 2	and positions	θ	( ) x,m l / 2
	around which vibrations are performed of the free vibrating bar of example
	2 ( N	5500kN = -	).
		θ	( )( ) x0 l / 2 rad	θ	( )( ) x,m l / 2 rad
			1,60		1,60
			1,75		1,60
			1,85		1,58
			1,95		1,55
			2,00		1,53
			2,05		1,50
			2,15		1,41
			2,20		1,34
			2,22		1,23
			2,25		0,00
			2,35		0,00
			2,50		0,00
			3,50		0,00
			4,50		0,00

Table 4 .

 4 Extreme values of various kinematical and stress components of the free

	vibrating bar of example 2, obtained for	( ) 0,00 t 0,12 sec ≤ ≤
	(	θ	( ) x0 l / 2 2,20 rad =	).
							with STMDE	without STMDE
			max	θ	x	( )( ) l / 2 rad	2,307	2,304
			min	θ	x	( )( ) l / 2 rad	0,393	0,373
			( )( ) max u l m m	-0,019	-0,019
			( )( ) min u l m m	-0,046	-0,046
			( )( max M 0 kNm P t	)	-7,178	-6,796
			( )( ) min M 0 kNm P t	-59,927	-59,927
			max M	S t ,lin	( )( ) 0 kNm	29,277	28,775
			min M		S t ,lin	( )( 0 kNm	)	-16,208	-15,697
			max M	t ,nl	( )( 0 kNm	)	89,261	88,321
			min M	t ,nl	( )( ) 0 kNm	0,144	0,129
			( )( max M 0 kNm t	)	55,993	55,266
			( )( ) min M 0 kNm t	-27,665	-26,828

Table 5 .

 5 Extreme values of various kinematical and stress components of the bar of

	example 3, obtained for	( ) 0,00 t 0,30 sec ≤ ≤	.
							with STMDE	without STMDE
	max	θ	x	( )( ) l / 2 rad	1,378	1,376
	min	θ	x	( )( ) l / 2 rad	-1,393	-1,388
	( )( ) max M 0 kNm P t	17,407	17,359
	( )( ) min M 0 kNm P t	-17,555	-17,537
	max M	S t ,lin	( )( ) 0 kNm	12,464	12,205
	min M		S t ,lin	( )( ) 0 kNm	-12,318	-12,491
	max M	t ,nl	( )( 0 kNm	)	19,199	18,887
	min M	t ,nl	( )( ) 0 kNm	-19,690	-19,474
	( )( max M 0 kNm t	)	49,070	48,427
	( )( min M 0 kNm t	)	-49,545	-49,499

Table 6 .

 6 Kinematical and stress components for linear static loading conditions of the bar of example 4, taking into account or ignoring the secondary twisting moment deformation effect.

			with STMDE		without STMDE
		x 0 =	x l =		x 0 =	x l =
			Present		Present
		FEM [6]	study -	FEM [6]	study -	Present study -AEM
			AEM		AEM
	( ) M Nm P t	31610,4 31610,2	31999,5	31999,5	0	32000,1
	( ) M Nm S t	389,6	389,6	0.5	0.5	32000,0	-0,1
	( ) 2 M Nm w	-263,95 -264,14	0	0	-2397,12	0

Comment [Authors2]: 2nd remark

Comment [Authors3]: 2nd remark

Comment [Authors4]: 2nd remarkComment [Authors5]: 2nd remark

Comment [Authors6]: 2nd & 3rd remarks

Comment [Authors7]: 5th remark

Acknowledgements

The authors would like to thank the Senator Committee of Basic Research of the National Technical University of Athens, Programme "PEVE-2008", R.C. No: 65 for the financial support of this work.


Pierre Hyvernat
email: pierre.hyvernat@univ-savoie.fr

FROM COHERENT TO FINITENESS SPACES

Keywords: 1998 ACM Subject Classification: F.4.1, F.3.2. Key words and phrases: Linear Logic, Coherent Space, Finiteness Space, Ramsey Theorem

This short note presents a new relation between coherent spaces and finiteness spaces. This takes the form of a functor from Coh to Fin commuting with the additive and multiplicative structure of linear logic. What makes this correspondence possible and conceptually interesting is the use of the infinite Ramsey theorem. Along the way, the question of the cardinality of the collection of finiteness spaces on N is answered. Basic knowledge about coherent spaces and finiteness spaces is assumed.

Introduction

The category of coherent spaces was the first denotational model for linear logic [START_REF] Girard | Linear logic[END_REF]: the basic objects are countable reflexive non-oriented graphs, and we are more specifically interested by their cliques (complete subgraphs). If C is such a graph, we write C(C) for the collection of its cliques. Coherent spaces enjoy a rich algebraic structure where the important operations are:

• the (reflexive closure of the) complement, written C ⊥ 1 ; • the product, written C 1 ⊗ C 2 ;

• the disjoint union, written C 1 ⊕ C 2 .

If one forgets about edges and only looks at vertices, the corresponding operations are simply the identity, the usual cartesian product "×" and the disjoint union "⊕".

More recently, T. Ehrhard introduced the notion of finiteness space [START_REF] Ehrhard | Finiteness spaces[END_REF] to give a model for the differential λ-calculus [START_REF] Ehrhard | The differential lambda-calculus[END_REF], which can be seen as an enrichment of linear logic. The point that interests us most here is that the collection of finitary sets of a finiteness space is closed under finite unions. This is definitely not the case with the cliques of a coherent space. This property is crucial for the interpretation of non-deterministic sums of terms [START_REF] Ehrhard | Finiteness spaces[END_REF][START_REF] Vaux | The algebraic lambda calculus[END_REF], which correspond in the models to linear combinations of simple terms. In this note, we only look at a qualitative version, where coefficients play no role. In other words, coefficients live in the rig (ring without negatives) {0, 1} with 1 + 1 = 1.

Very briefly, a finiteness space is given by a countable set |F|, called its web, and a collection F of subsets of |F|, called the finitary sets, which satisfies

F ⊥ ⊥⊥ ⊥ = F with D ⊥ ⊥ def = x ∀y ∈ D, #(x ∩ y) < ℵ 0 where #(x)
is the cardinal of x, and ℵ 0 is the least infinite cardinal. In natural language, the crucial property is that whenever x ∈ F and y ∈ F ⊥ ⊥ , the intersection x ∩ y is finite. Algebraic constructions similar to the ones for coherent spaces can be defined on finiteness spaces, and they are characterized by:

• the dual F ⊥ ⊥ ;

• the tensor

F 1 ⊗ F 2 def = {r | π 1 (r) ∈ F 1 , π 2 (r) ∈ F 2 }; • the coproduct F 1 ⊕ F 2 def = {x 1 ⊎ x 2 | x 1 ∈ F 1 , x 2 ∈ F 2 }.
Here again, if one forgets about finitary sets and only looks at the webs of finiteness spaces, the corresponding operations are just the identity, the usual cartesian product and the disjoint union.

Remarks.

(1) The operations on finiteness spaces are actually defined in a way that makes it clear that they yield finiteness spaces. They are then proved to be equivalent to the definitions given above [START_REF] Ehrhard | Finiteness spaces[END_REF]. (2) Any operator of the form X → X * = {y | ∀x ∈ X , R(x, y)} is contravariant with respect to inclusion and yields a closure operator when applied twice. In particular, any set of the form Y = X * satisfies Y = Y * * . (3) There is another presentation of coherent spaces that closely matches the definition of finiteness spaces: a coherent space is given by a collection C of subsets of |C| which satisfy C * * = C, where D * = {x | ∀y ∈ D, #(x ∩ y) ≤ 1}.

From Coherence to Finiteness

The idea is rather simple: we would like to close the collection of cliques of a coherent space under finite unions. Unfortunately (but unsurprisingly), the notion of "finite unions of cliques" is not very well behaved, especially with respect to the dual. Recall that an anticlique of C (also called independent, or stable sets) is a clique in C ⊥ . We consider the following notion:

Definition 1.1.
If C is a coherent space, we call a subset of |C| finitely incoherent if it doesn't contain infinite anticliques. We write F (C) for the collection of all finitely incoherent subsets of C.

The next lemma follows directly from the definition.

Lemma 1.2.

(1) Any finite subset of |C| is finitely incoherent;

(2) a subset of a finitely incoherent subset is finitely incoherent;

(3) finitely incoherent subsets are closed under finite unions;

(4) any clique is finitely incoherent.

Note however that a finitely incoherent set needs not be a finite union of cliques: take for example the graph composed of the disjoint union of all the complete graphs K n for n ≥ 1. This graph doesn't contain an infinite clique, but it is not a finite union of anticliques; so, its dual is finitely incoherent but is not a finite union of cliques.

The next lemma is more interesting as it implies that the collection of finitely incoherent subsets forms a finiteness space.

Lemma 1.3. If C is a coherent space, we have:

C(C) ⊥ ⊥ = F(C ⊥ ) .
Proof.

(⊆) Let x be in C(C) ⊥ ⊥ , and suppose, by contradiction, that x is not in F(C ⊥ ), i.e., x contains an infinite anticlique y of C ⊥ . This set y is a clique in C, i.e., y ∈ C(C). Since x ∩ y = y is infinite, this contradicts the hypothesis that x ∈ C(C) ⊥ ⊥ .

(⊇) Let x be finitely incoherent in C ⊥ , i.e., x doesn't contain an infinite clique of C; let y be in C(C). Since x ∩ y ∈ C(C) and x ∩ y is contained in x, it cannot be infinite. This shows that x ∈ C(C) ⊥ ⊥ .

By remark 2 on page 2, we thus get the expected corollary:

Corollary 1.4. If C is a coherent space, then F(C) is a finiteness space.
What was unexpected is the following:

Lemma 1.5. If C is a coherent space, then:

F(C ⊥ ) = F (C) ⊥ ⊥ .
Proof. Because of the previous lemma, and because _ ⊥ ⊥ is contravariant with respect to inclusion, we only need to show that C(C) ⊥ ⊥ ⊆ F (C) ⊥ ⊥ . Suppose that x ∈ C(C) ⊥ ⊥ , and let y ∈ F (C); we need to show that x ∩ y is finite.

• Since x ∩ y ⊆ y ∈ F(C), x ∩ y cannot contain an infinite anticlique;

• since x ∩ y ⊆ x ∈ C(C) ⊥ ⊥ , x ∩ y cannot contain an infinite clique. Those two points imply, by the infinite Ramsey theorem, 1 that x ∩ y is finite.

The other linear connectives are similarly behaved with respect to the notion of finitely incoherent sets. We have: Lemma 1.6. If C 1 and C 2 are coherent spaces, then we have both

F(C 1 ⊕ C 2 ) = F(C 1 ) ⊕ F (C 2 ) , and F(C 1 ⊗ C 2 ) = F(C 1 ) ⊗ F(C 2 )
where the connectives on the left are the coherent spaces' ones, and the connectives on the right are the finiteness spaces' ones. 1 Infinite Ramsey theorem: suppose G is a countably infinite set, then, for every assignment of c colors to the subsets of G of cardinality n, there is an infinite I ⊆ G s.t. all subsets of I of cardinality n have the same color. (Refer to [START_REF] Plumpton | On a problem of formal logic[END_REF] or one of the many textbooks on combinatorics covering it.) For n = 2 and c = 2, it amounts to "each countably infinite graph has an infinite clique or an infinite anticlique".

Proof. The ⊕ part is direct; for the ⊗ part, recall that r ∈ F (C 1 ) ⊗ F(C 2 ) is equivalent to π 1 (r) ∈ F(C 1 ) and π 2 (r) ∈ F(C 2 ).

(⊆) Suppose r doesn't contain an infinite anticlique; neither π 1 (r) nor π 2 (r) can contain an infinite anticlique, as it would imply the existence of an infinite anticlique in r.

(⊇) Suppose that r ∈ F(C 1 ) ⊗ F (C 2 ) contains an infinite anticlique r ′ of C 1 ⊗ C 2 . At least one of π 1 (r ′ ) or π 2 (r ′ ) must be infinite, otherwise, r ′ itself would be finite. Suppose π 1 (r ′ ) is infinite; because π 1 (r ′ ) ⊆ π 1 (r), it cannot contain an infinite anticlique. By the infinite Ramsey theorem, it thus contains an infinite clique x.

For each a ∈ x, chose one element b inside the fiber r

′ (a) = {b | (a, b) ∈ r ′ }.
Two such b's cannot be coherent as it would contradict the fact that r ′ is an anticlique.

In particular, all such b's are distinct. We have constructed an infinite anticlique in

π 2 (r ′ ) ⊆ π 2 (r) ∈ F (C 2 ). Contradiction!
For finiteness spaces, the operation ⊕ coincides with its dual [START_REF] Ehrhard | Finiteness spaces[END_REF]. In particular, for finiteness spaces coming from from coherent spaces, we have

F(C 1 ) ⊕ F (C 2 ) ⊥ ⊥ = F(C 1 ) ⊥ ⊥ ⊕ F(C 2 ) ⊥ ⊥ .
Lemmas 1.5 and 1.6 thus imply that

F (C 1 ⊕ C 2 ) ⊥ = F C ⊥ 1 ⊕ C ⊥ 2 .
The direct proof of this equality is also quite easy.

Both coherent spaces and finiteness spaces form categories, where:

• a morphism from C to D in Coh is a clique in (C ⊗ D ⊥ ) ⊥ ,
• a morphism from F to G in Fin is a finitary set in (F ⊗ G ⊥ ⊥ ) ⊥ ⊥ . In both cases, morphisms are special relations between webs and composition is the usual composition of relations:

r • s def = (a, c) ∃b (a, b) ∈ s and (b, c) ∈ r .
From all the above, we can conclude that:

Proposition 1.7. The operation F(_) can be lifted to a functor from Coh to Fin:

(1) it sends C to F(C) (2) and r ∈ Coh[C, D] to r ∈ Fin[F (C), F (D)]. Moreover, this functor commutes with _ ⊥ , _ ⊗ _ and _ ⊕ _.
This functor is faithful (but not full), and it is not injective on objects as adding or removing any finite number of edges to a coherent space doesn't change its image via F(_). Moreover, this functor commutes with the forgetful functors from Coh and Fin to Rel, the category of sets and relations.

In a sense, coherent spaces allow one to define a collection of simple finiteness spaces. An informal argument regarding this simplicity can be found in the following remark: the logical complexity of the formula expressing "x ∈ A ⊥ ⊥⊥ ⊥ ", i.e., "x is finitary with respect to A" changes when A comes from a coherent space. If we write y ∞ ⊂ x for "y is an infinite subset of x", we have [START_REF] Ehrhard | Finiteness spaces[END_REF]:

x ∈ A ⊥ ⊥ ⇐⇒ ∀y ∞ ⊂ x y / ∈ A
whenever A is downward closed. Thus:

x ∈ A ⊥ ⊥⊥ ⊥ ⇐⇒ ∀y ∞ ⊂ x ∃z ∞ ⊂ y z ∈ A .
Because y ∞ ⊂ x is a Σ 1 1 -formula (no universal second-order quantifiers), "x ∈ A ⊥ ⊥⊥ ⊥ " is a Π 1 2 -formula (second-order quantifiers are ∀∃). Note that even if A isn't downward closed, the formula expressing "x ∈ A ⊥ ⊥⊥ ⊥ " is still a Π 1 2 -formula. For the particular case when A is the set of cliques of a coherent spaces C, we obtain

x ∈ A ⊥ ⊥⊥ ⊥ ⇐⇒ x ∈ C(C ⊥ ) ⊥ ⊥ ⇐⇒ ∀y ∞ ⊂ x ∃ a, b ∈ y (a, b) ∈ C
which is only a Π 1 1 -formula.

Cardinality of Finiteness Spaces

So, coherent spaces can be used to define a collection of finiteness spaces closed under the linear operations (_ ⊥ , _ ⊗ _ and _ ⊕ _). It is natural to ask whether all finiteness spaces can be obtained in this way. The previous informal remark about the logical complexity of coherence versus finiteness points toward a negative answer. Here is a formal proof which also answers a question raised by T. Ehrhard:

Proposition 2.1. If A is infinite countable,

the cardinality of finiteness spaces on A is exactly that of P(P(A)). The cardinality doesn't change if we consider finiteness spaces up-to isomorphisms, i.e., up-to permutations of A.

Since the cardinality of coherent spaces on A is the same as that of P(A × A) ≃ P(A), we can conclude that:

Corollary 2.2. If A is infinite countable, there are strictly more finiteness spaces on A than coherent spaces on A.

Proof. Let A be infinite countable; up-to isomorphism, we can assume that A = B <ω , the set of finite sequences of bits. If x is an infinite sequence of bits, write x ↓ for the set of finite approximations of x; and if X is a set of such "real numbers", write X ↓ for the set {x ↓ | x ∈ X}. We have X ↓ ⊂ P(A) for any such set X. Suppose now that X = X ′ with, for example, x ∈ X but x / ∈ X ′ . Since x ↓ is infinite and x ↓ ∈ X ↓ , we have x ↓ / ∈ X ↓⊥ ⊥ . However since two different reals must differ on some finite approximation, we have that x ↓ ∈ X ′↓⊥ ⊥ . Thus, the finiteness spaces (A, X ↓⊥ ⊥ ) and (A, X ′↓⊥ ⊥ ) differ.

This defines an injective map X → (A, X ↓⊥ ⊥ ) from arbitrary sets of reals to finiteness spaces on A. This shows that finiteness spaces on A have at least the same cardinality as P(R) ≃ P(P(A)). Since it cannot be more than that, we have equality.

An isomorphism in the category Fin is a particular relation with a left and right inverse. This implies that it is in fact the graph of a bijection, and thus, two finiteness spaces F 1 and F 2 on a set A are isomorphic if and only if there is a bijection σ : A → A such that

F 2 = σ • F 1 def = σ(x)
x ∈ F 1 where σ(x) when x ⊆ A is simply the direct image of the set x. Because the cardinality of each equivalence class is at most # P(A) (this is the cardinality of permutations on A), and since κ × # P(A) = max κ, #(P(a)) , there must be at least # P(P(A)) such equivalence classes to cover the whole collection of finiteness spaces.

The cardinality of finiteness spaces on A up-to isomorphism is thus the same as that of finiteness spaces on A up-to plain equality: # P(P(A)) .

It is slightly interesting to note that the same reasoning doesn't apply to higher cardinalities since #(A <ω ) = #(A) if A is uncountable.

Conclusion

The situation with respect to full linear logic isn't totally clear. We have a base category FinCoh with:

• coherent spaces as objects • and finitely incoherent linear maps as morphisms:

FinCoh[C, D] = F (C ⊗D ⊥ ) ⊥ .
This category is a linear, full subcategory of the category Fin of finiteness spaces.

Lifting the usual set-based notion of exponentials for coherent spaces to this category is impossible: because the web of !C is the collection of finite cliques of C (uniformity), this construction isn't even functorial. Take for example K n and K ⊥ n : since their sets of vertices have the same finite cardinality, they are isomorphic in FinCoh. However !K n and !(K ⊥ n ) have sets of vertices of different cardinalities, namely 2 n and n + 1: they cannot be isomorphic in FinCoh. 2 The multiset-based notion of exponentials, where one defines the web |!C| to be the collection of finite multisets whose support is a clique, doesn't seem to help. It is not functorial in any canonical way, as shown by the same example of K n ≃ K ⊥ n : the corresponding sets of vertices for !K n and K ⊥ n are M f {1, . . . , n} and M f {1} ∪ • • • ∪ M f {n}. Note that in both cases, the collection of finitely incoherent sets consists of all the subsets of the web, so that a non-canonical isomorphism is still possible. However, if such an isomorphism exists, it doesn't commute to the forgetful functors to the category of sets and relations.

Using the non-uniform coherent spaces [START_REF] Bucciarelli | On phase semantics and denotational semantics: the exponentials[END_REF] isn't a solution either as it introduces a third relation: neutrality. A non-uniform coherent space is thus a non-oriented graph with two kinds of edges: strict coherence edges and neutral edges. Neutral edges are left unchanged by duality and we take the complement of the rest. In the usual coherent spaces, the only neutral edges are the loops around vertices. The natural notion is to define a clique as a set of vertices that are pairwise coherent or neutral, but one could also consider "strict" cliques. Thus, there are two possible definitions of C(C): other doesn't hold. With the strict versions of C(C) and F(C), we go as far as the proof of Lemma 1.5. However, • x ∩ y doesn't contain an infinite strict anticlique • and x ∩ y doesn't contain an infinite strict clique only implies, by the infinite Ramsey theorem for three colors, that x ∩ y is finite or contains an infinite set of pairwise neutral vertices.

Finding an appropriate notion of exponential to extend this category to a model of the algebraic λ-calculus, or better yet, of the differential λ-calculus is thus left open at this point.

C

  coherence or neutrality) strict cliques (pairwise strict coherence) , and similarly, there are two possible definitions of F(C):F (C) def =x's that do not contain anticliques x's that do not contain strict anticliques .None of the four possibilities enjoys the adequate properties. When using cliques and anticliques, point 4 of Lemma 1.2 fails: we do not have C(G) ⊆ F (C). The cliques/strict anticliques and strict cliques/anticliques versions fail at Lemma 1.3: one inclusion or the2 An isomorphism in FinCoh is in particular an isomorphism in Rel which is an isomorphism in Set.

This work has been partially funded by the French project choco (ANR-07-BLAN-0324).