C Mercier

A Khelil

F Al Mahmoud
email: firas.al-mahmoud@univ-lorraine.fr

J.-L Blin-Lacroix
email: jean-luc.blin-lacroix@univ-lorraine.fr

A Pamies

EXPERIMENTAL INVESTIGATIONS OF BUCKLING BEHAVIOUR OF STEEL SCAFFOLDS

Keywords: Scaffold, Buckling behaviour, Buckling curve, Initial imperfections, Circular tubular section

The analysis and design of structures sensitive to second order effects require the consideration of initial imperfections. Scaffold structures, due to their low stiffness, are very sensitive to second order effects. The accuracy of the design of these structures relies in part on the estimation of initial imperfections affecting the structure. In the 1960s, the C.E.C.M., the European Convention of Steel Construction, carried out a series of experimental tests to determine the degree of initial imperfections of each type of cross-section. Five buckling curves were established. The European design code for steel structures, Eurocode 3 Part 1-1, classifies hot-rolled hollow tubular sections as profiles with few imperfections, i.e. buckling curve 'a', and cold-formed sections as sections with many imperfections, i.e. buckling curve 'c'. A test campaign was conducted to analyse the experimental buckling behaviour of hot-rolled and cold-formed hollow sections of scaffold structures. Twenty-nine buckling tests on simply supported members and nine buckling tests on scaffolding ladders, from various European suppliers, were carried out. Statistical studies of the experimental results point out that the estimate of the initial imperfection currently defined for the hot-rolled and cold-formed tubular sections is far too penalizing for sections used in scaffolding.

Introduction

The scaffold structures present a low rigidity, with regard to those of buildings, and a high slenderness ratio. Under axial compression, cross-section resistance does not govern the behaviour of such structures. Indeed, second order effects appear and the failure of these structures is caused by divergence of equilibrium (Figure 1). Amplitude of these second order effects is influenced by the presence of initial imperfections. It is therefore necessary to include them in the structures' design.

The structures design is performed in two steps: (i) the structures' analysis, known as the global analysis; (ii) the resistance and stability verifications. The more the global analysis is carried out in depth, the more verification criteria, which result from it, are simple, and vice versa. The European design code for steel structures, Eurocode 3 Part 1-1 [START_REF]NF EN 1993-1-1: Eurocode 3: Design of steel structures. Part 1-1: General rules and rules for buildings[END_REF], allow to take them into account in two different ways:

-The global analysis includes initial imperfections and second order effects. The structures verification is therefore based on cross-section criteria.

-The global analysis does not include initial imperfections and second order effects.

The structures verification is therefore done using so-called instability criteria.

However, whatever the method used, the precision of the verification relies on the assessment of the initial imperfections affecting the structures. Indeed, if the initial imperfections of a structure are overestimated, the internal stresses will in turn be overestimated; this will lead to the oversizing of the structure and therefore an additional cost for the manufacturers. On the contrary, if the initial imperfections are underestimated, the internal stresses will also be underestimated and the structure will be undersized, which will lead to safety problems. An accurate estimation of the initial imperfections affecting scaffolding structures is therefore crucial for the optimal design of these structures.

In the 1960s, the European Convention of Steel Construction (C.E.C.M.) [START_REF] Sfintesco | Fondement expérimental des courbes européennes de flambement[END_REF] led a test campaign to assess the degree of initial imperfections in each type of cross-section. This led, in Eurocode 3 Part 1-1, to the creation of five buckling curves, from 'a0' to 'd', corresponding to almost perfect profiles to profiles with a significant degree of initial imperfection, respectively. Each of these buckling curves corresponds to an imperfection factor α, enabling to take into account the degree of initial imperfections in the structure design. Steel scaffold structures are mostly designed in cold-formed tubular sections, but can equally be made with hot-rolled sections. The imperfection factor is therefore set at 0.49 in the first case and 0.21 in the second.

Scaffolding structures have initial imperfections that affect the stability and load-bearing capacity of the structure. The study of the stability of scaffolds has been the subject of many researches over the last fifty years, because of the risk of collapse. Milojkovic et al. [START_REF] Milojkovic | Factors of safety for standard scaffold structures[END_REF][START_REF] Milojkovic | Determination of the factors of safety of standard scaffold structures[END_REF] presented a report on an investigation conducted by the Health and Safety Executive [START_REF] Pallet | Investigation into aspects of falsework[END_REF] in the United Kingdom into inherent defects in scaffold structures. They showed that these defects can reduce the capacity of a scaffold to less than 10% of its design capacity. Among the defects, they identified lack of anchorage to the façade, verticality defects of the structure, straightness defects of the scaffold standards and inadequate foundations. On the other hand, Lew [START_REF] Lew | CONSTRUCTION FAILURES AND THEIR LESSONS[END_REF] showed that errors in falsework design are also one of the main sources of scaffold collapses. Many researchers have thus devoted themselves to the study of the stability of scaffolds and to the development of structural models allowing their load capacities to be estimated.

Peng et al. [START_REF] Peng | Stability of independent heavy-duty scaffolds: an experimental study[END_REF] carried out experimental studies to determine the influence on the load capacity of heavy duty scaffolds of different parameters such as: (i) the different numbers of stories, (ii) the presence or absence of top and base screw jacks, (iii) the presence or absence of horizontal braces. They showed that the number of stories did not significantly reduce the load capacity of a scaffold. They also showed that the clevises and screw jack feet enhance the load capacity of a scaffold because they provide bending moment stiffness. The tests on scaffolds without horizontal braces showed that the horizontal braces provided strength to the structure.

Chandrangsu and Rasmussen [START_REF] Chandrangsu | Structural modelling of support scaffold systems[END_REF][START_REF] Chandrangsu | Structural modelling of support scaffold systems[END_REF] developed three-dimensional analysis models to predict the ultimate load of scaffold structures. The results obtained with the numerical model are compared with the results of a full-scale test carried out on a three-by-three bay formwork system, to ensure the reliability of the numerical model. They thus showed that the ultimate load from the numerical analysis is close to the failure loads obtained with the experimental test.

Peng et al. [START_REF] Peng | Simplified models for analysis and design of modular falsework[END_REF][START_REF] Peng | High clearance scaffold systems during construction -I. Structural modelling and modes of failure[END_REF] examined the behaviour of two-storey steel scaffolds. They developed simplified theoretical analysis models in two-dimensional to predict the load capacity of scaffold structures. They showed [START_REF] Peng | Structural modeling and analysis of modular falsework systems[END_REF] that a three-dimensional scaffold with regular configuration could be analysed more simply as a two-dimensional scaffold while being accurate and safe. Indeed, the cross-braces provide stiff lateral support in the three-dimensional scaffold.

Cimellaro and Domaneschi [START_REF] Cimellaro | Stability analysis of different types of steel scaffolds[END_REF] investigated diverse type of steel scaffolds, namely: (i) joint tube scaffold, (ii) multidirectional scaffold and (iii) prefabricated scaffold. Finite element models are developed to identify the parameters that affect the load capacity of the scaffolds. They therefore showed that the presence of manufacturing imperfections, the eccentricities of the applied vertical load reduce the load capacity of the structure. They proposed an empirical formula to predict the critical load of common scaffolds.

Weesner and Jones [START_REF] Weesner | Experimental and analytical capacity of frame scaffolding[END_REF] compared the load capacities of four different scaffolds obtained with experimental tests and finite element model. They concluded that numerical models provided a fair estimate of the load capacity of the scaffolds.

Peng and al. [START_REF] Peng | Experimental and numerical studies of practical system scaffolds[END_REF] studied the failure modes and load capacities of different scaffolds. Numerical analysis models supplemented these experimental tests. These models were initially calibrated using experimental tests. They were then used to deduce the load capacity of scaffolds with other conditions like additional diagonal bracings. Klasson and al. [16] studied the effects of initial imperfections on the predicted strength and stiffness or steel columns. They highlighted the difficulty of studying complex structures containing several columns with multiple bracings. This difficulty stems from the considerable number of possible combinations in the shape of the initial imperfections.

Based on a review of the extant literature, many researches have been conducted to study the stability of the scaffolds. Cai et al. [START_REF] Cai | Nonlinear stability analysis of a radially retractable suspen-dome[END_REF], Gao et al. [START_REF] Gao | Experimental study on stability of high strength steel long columns with box-sections[END_REF], Iu [START_REF] Iu | Generalised element load method with whole domain accuracy for reliable structural design[END_REF], Chan and Lo [START_REF] Chan | Stability design of system scaffold with differential settlement[END_REF], Liu et al. [START_REF] Liu | Curved-quartic-function elements with end-springs in series for direct analysis of steel frames[END_REF], Mercier et al. [START_REF] Mercier | Analysis of the global and local imperfection of structural members and frames[END_REF][START_REF] Mercier | Lateral stability of slender cold-rolled hollow tubular sections with initial imperfections[END_REF] and Yang et al. [START_REF] Yang | Flexural buckling design of fabricated austenitic and duplex stainless steel columns[END_REF] agree that initial imperfections must be taken into account in the design of structures.

Zhao and Chen [START_REF] Zhao | Analysis of door-type modular steel scaffolds based on a novel numerical method[END_REF] developed numerical models to reveal the influence of the initial imperfections on scaffolding structures from one-bay to four-bay. They were therefore able to show that for an initial imperfection lower than h/1000, where h is the height of the scaffold modular, the ultimate load of capacity of the structure is merely slightly reduced. However, no real experimentation has been carried out to thoroughly determine the initial imperfections affecting the scaffold structures.

Gao et al. [START_REF] Gao | Experimental study on stability of high strength steel long columns with box-sections[END_REF] conducted an experimental study on ten long steel columns with box-sections. They were therefore able to point out that the behaviour of this type of long column is similar to the curve 'b' in Eurocode 3.

In this article, similar experimental tests have been carried out to analyse the scaffold structures behaviour under axial compression. Simply supported members and scaffold ladders, with a tubular section, were tested. The objective is to determine which buckling curve currently defined in Eurocode 3 better reflects the real behaviour of this type of structure. This test campaign was realized in collaboration with the French Syndicate of Scaffolding, Formwork and Shoring (SFECE).

2

Experimental tests on simply supported members

Experimental procedure

Tests on simply supported members [START_REF] Khamisi | Stability of tubular steel structures: buckling and lateral torsional buckling[END_REF] constitute the first step of the test campaign led in the laboratory to analyse the buckling behaviour of hollow tubular section structures. The test device (Figure 2) enabled to test simply supported members of 2 meters length. Simple steel tubes or steel scaffold standards were tested up to failure under axial compression. A total of ten hot-rolled members and nineteen cold-formed elements were tested. The tested members were galvanized and non-galvanized steel tubes, of 48.3 mm of diameter and 2.7 mm or 3.2 mm in thickness, widely used in the domain of scaffolding. The specific yielding strength and the modulus of elasticity were 320 MPa and 210 GPa, respectively. A naming system was assumed in the study to distinguish the tested specimens: TN for the non-galvanized tubes, TG for the galvanized tubes, PGD for the non-galvanized standards and PG for the galvanized standards.

Screwdrivers supports fabricated in the laboratory were the supports of the tested elements to ensure pin-jointed supports (Figure 3). The loading frame applied a vertical load on the specimens. A sensor displacement, located at mid-height of the tested element (Figure 4) and connected to the central processing unit, measured the lateral displacement of the element and acquire the data. It enabled monitoring the deflection of the single member according to the axial load. The experimental loaddeflection responses were therefore obtained. The experimental load and failure stress, of each tested specimen, are summarized in Table 1.

Statistical study

As part of the test campaign led by the C.E.C.M. in the 1960s, Jacquet [START_REF] Jacquet | Essais de flambement et exploitation statistique[END_REF] carried out a statistical study to determine conventional values of failure load and failure stress for each type of cross-section and each slenderness ratio tested. All these values of failure stress enable to draw up the current five buckling curves.

To enable the comparison between our experimental results and the results obtained in the 1960s, the statistical study carried out here is based on that of Jacquet's one.

To carry out the statistical study, it is necessary to ensure the distribution of the results can be approximated by a statistical normal distribution (Gauss distribution). The method of the moments was therefore employed.

The parameters determined with the method of the moments are the following:

(1) average:

= ∑ x m N (1)
(2) deviation from the average: = -

i i y x m (2)
(3) values: (4) values:

2 2 = ∑ i S y (3)
2 2 1 = - S K N (6) ( )( ) 3 3 1 2 = - - NS K N N (7) ( ) ( ) ( )( )( ) 2 4 3 4 1 3 1 1 2 3 + - - = - - - N N S N S K N N N (8)
(5) mean square deviation:

( )

1 2 2 = s K (9)
(6) values:

3 1 3 2 2 = K G K (10) 4 2 2 2 = K G K ( 11
)
(7) mean square deviation s1 and s2:

( ) ( )( )( ) 1 6 1 2 1 3 - = - + + N N s N N N (12) ( ) ( )( )( )( ) 2 2 24 1 3 2 3 5 - = - - + + N N s N N N N ( 13
) (8) ratio: 1 1 1 = V G s ( 14
) 2 2 2 = V G s ( 15
)
A distribution is perfectly normal if V1 = V2 = 0. However, the normality of the distribution can be accepted as long as the values V1 and V2 remain low, i.e. less than 3. The values of V1 and V2, given by Table 2, show that the distribution of the results can be considered as normal distribution. The convention value of the failure stress is therefore done by the following equation:

, 2 = - u conv m s σ (16)
Table 3 presents the conventional values of the failure load and the failure stress obtained from the experimental tests. 7 plots the distribution of experimental results. The conventional value of the failure stress is emphasised. It can be observed that the totality of the experimental results obtained is beyond the conventional failure stress resulting from the statistical study. Eurocode [START_REF] En | Eurocode: Basis of structural design[END_REF] states the 5% fractile value as the characteristic value, for an unfavourable low value. The both conventional values of the The conventional values of failure stress can be visualized among the European buckling curves (Figure 8). The reference buckling curve of such sections is the buckling curve 'a'.

y

Figure 8 Comparison between experimental values and European buckling curves for hot-rolled elements

From the equation of the European buckling curves (equation ( 17)), drawn up by Maquoi and Rondal [START_REF] Maquoi | Mise en équation des nouvelles courbes européennes de flambement[END_REF], the values of imperfection factor α stemming from the experiment were determined (Table 8). It is apparent that the standard theoretical imperfection factor of Eurocode 3 is nowhere near the test results with respect to the hot-rolled elements. The imperfection factors determined from the statistical studies of the experimental test were around 0.05. Based on these results, it is possible to admit an imperfection factor of 0.13 would allow a better estimation of the initial imperfections affecting this type of element. This value of 0.13 could be applied to hot-rolled circular tubular steel sections, whose diameter and slenderness range are those common in the scaffold field.

σ α λ λ α λ λ λ λ       = ⋅ + - + - + - + -         2 2 2 2 2 2 2 1 1 0,04 1 0,04 4 2 (17)

Calibration of an analytical model and a finite element model

Two models have been developed: an analytical one and a finite element model.

For the analytical model, the mid-height deflection of the column must be expressed as a function of the load applied to the column. The deflection ∆ physically measured at mid-height of the single member is:

0 0 ∆ = -= ⋅ - fin cr N d e e N N (18)
where: dfin represent the total deflection of the member under axial load, e0 is the initial imperfection of the element, N is the compression load and Ncr is the critical load.

The article 5.3.2 [START_REF] Peng | High clearance scaffold systems during construction -I. Structural modelling and modes of failure[END_REF] of Eurocode 3 Part1-1 defines the initial imperfection e0 as:

( ) 0 0, 2 = ⋅ - ⋅ el W e A α λ (19)
where: α is the imperfection factor, λ is the non-dimensional slenderness, Wel is the elastic section modulus and A is the area of the cross-section.

The critical load is deduced from the equation:

2 2 = cr cr EI N L π ( 20
)
where Lcr is the buckling length of the element, E is the elastic modulus and I is the second moment of area.

The tested elements were single members with pin-jointed supports; therefore, the buckling length equals the length of the elements.

Equations [START_REF] Gao | Experimental study on stability of high strength steel long columns with box-sections[END_REF] to [START_REF] Chan | Stability design of system scaffold with differential settlement[END_REF] hence make it possible to establish the force-deflection response of a simply supported element for any value of imperfection factor α, i.e. several amplitudes of initial imperfections. It will thus be possible to compare the experimental response to different analytical responses, to obtain the analytical model that best reflects the actual behaviour of the structure. Three imperfection factor values will be compared: (i) an imperfection factor of 0.21: the conventional value for this type of cross-section established by Eurocode 3;

(ii) an imperfection factor of 0.13: the imperfection factor value as close as possible to the values obtained by the statistical study; (iii) an imperfection factor value obtained through the calibration of the analytical model.

The second model developed is a finite element model. The software suite for finite element analysis Abaqus was used to create this finite element model. The measured dimensions of the components were used for cross-sectional properties in the finite element models. The nonlinearity properties of the steel material are applied to the model. The stress-strain relationship is based on results of tensile tests carried out on specimens from tested hot-rolled elements.

The calibration was achieved by changing the amplitude of the initial imperfections of the model, to obtain the finite element model that best reflects the actual behaviour of the structure. The failure (ultimate load) in the finite element model was observed when the load limit point was reached, which was associated with large displacements. The calibration results show that an imperfection factor of 0.08 and 0.05, for 2.7 mm and 3.7 mm of thickness respectively, gives a load-deflection response similar to the experimental one. These two imperfection factor values, derived from the analytical and finite element models, are significantly close to the values derived from the statistical study of the experimental results. The comparison of the load-deflection responses obtained with the analytical model and the finite element model shows that these two models are equivalent and both accurately predicting the behaviour and failure load of hotrolled simply supported members.

The comparison of the analytical response with an imperfection factor of 0.21 to the experimental results makes it possible to affirm this response does not reflect the real behaviour of the structure. This implies that the initial imperfection degree of a buckling curve 'a', i.e. an imperfection factor of 0.21, is too important. The initial imperfections affecting these elements are actually considerably slighter, as experimentation has shown. An imperfection factor of 0.13, i.e. a buckling curve 'a0', would make it possible to get closer to the experimental response while nevertheless preserving a safety margin.

Experimentations on cold-formed hollow tubular sections

Test results

Figure 11 and Figure 12 illustrate the experimental load-deflection responses obtained from the acquisition of the measurements on hot-rolled simply supported elements. The experimental load and failure stress, of each tested specimen, are summarized in Table 5.

Statistical study

The statistical study is carried out on the similar principle as the previous one. Table 6 presents the values of V1 and V2. In view of the values of the both parameters, it is therefore satisfactory to assume the normality of the distribution of the experimental results. The convention value of the failure stress is therefore done by the equation [START_REF] Klasson | Slender steel columns: How they are affected by imperfections and bracing stiffness[END_REF]. Table 7 provides the conventional values of the failure load and the failure stress obtained from the experimental tests. The conventional values of failure stress can be visualized among the European buckling curves (Figure 14). The reference buckling curve of such sections is the buckling curve 'c'. From the equation of the European buckling curves (equation ( 17)), drawn up by Maquoi and Rondal [START_REF] Maquoi | Mise en équation des nouvelles courbes européennes de flambement[END_REF], the values of imperfection factor α stemming from the experimental were determined (Table 8). It is apparent that the standard theoretical imperfection factor of Eurocode 3 is nowhere near the test results with respect to the cold-formed elements. The imperfection factor determined from the statistical study of the experimental test was less than 0.21; so closer to the imperfection factor of hotrolled hollow tubular sections than the current imperfection factor of cold-formed ones.

These tests on simply supported members enable to assess initial imperfections of a single element. However, other initial imperfections may be present in a structure like inclinations between vertical components. Full-scale tests on scaffolding ladders were therefore carried out to determine the importance of this lack of fit at joints between scaffoldings standards and their influence on the buckling behaviour.

Calibrations of an analytical model and a finite element model

As before, these experimental results will be used to calibrate two models: the analytical model and the finite element model. Since the test configuration has not changed compared to the hot-rolled element tests, the two models, analytical and finite element, are identical. Only the calibration of these models was distinct. Indeed, the aim was to determine the degree of imperfection to better reflect the force-deflection response of the cold-formed element tests.

As regards the analytical model, three imperfection factor values will be compared: (i) an imperfection factor of 0.49: the conventional value for this type of section established by Eurocode 3; (ii) an imperfection factor of 0.21: the imperfection factor value as close as possible to the values obtained by the statistical study; (iii) an imperfection factor value obtained through the calibration of the analytical model. The calibration results show that an imperfection factor of 0.12 and 0.14, for 2.7 mm and 3.7 mm of thickness respectively, gives a load-deflection response similar to the experimental one. These two imperfection factor values, derived from the analytical and finite element models, are significantly close to the values derived from the statistical study of the experimental results. The comparison of the load-deflection responses obtained with the analytical model and the finite element model shows, once again, that these two models are equivalent. The both accurately predict the behaviour and failure load of cold-formed simply supported members.

The comparison of the analytical response with an imperfection factor of 0.49 to the experimental results makes it possible to affirm this response does not reflect the real behaviour of the structure. This implies that the initial imperfection degree of a buckling curve 'c', i.e. an imperfection factor of 0.49, is too important. The initial imperfections affecting these elements are actually considerably slighter, as experimentation has shown. An imperfection factor of 0.21, i.e. a buckling curve 'a', would make it possible to get closer to the experimental response while nevertheless preserving a safety margin as shown by Khamisi [26] in his thesis dissertation.

Experimental tests on scaffolding ladders

Experimental procedure

The second step of the test campaign launched in the laboratory consists of carrying out tests on scaffolding ladders (Figure 17) [START_REF] Mercier | Development of a model of stability for steel hollow tubular section beam-columns sensitive to second order effects[END_REF]. The test device (Figure 18) enabled to test ladders of 4 meters high and 0.7 meters wide, under axial compression. A total of nine elements was tested. An appended structure was created to ensure the assembly and restrain out-of-plane displacement of the tested structures as illustrated in Figure 19a. No load was applied on this appended structure. The testing device consists on a trimmer to transmit the load between the loading frame and the top of the structure tested (Figure 19b), and a wire rope to restrain its out-of-plane displacements (Figure 19c). The scaffolding ladders were tested under pure compression. The loads were applied by hydraulic cylinders located on the loading frames (Figure 19d) on the trimmer to transmit the load to the top of the ladder. The test was carried out with materials from several scaffolding suppliers. The cross-sectional characteristics of these scaffoldings are presented in Table 9. All the cross-sections were cold-formed hollow tubular The horizontal deflections of the ladders were measured halfway up, i.e. 2 meters from the case level, using a displacement sensor. This sensor was connected to a central processing unit to acquire the data and monitor the deflection of the ladder according to the load applied by both loading frames. Several loading and unloading cycles were performed. The experimental load-deflection responses were therefore obtained.

Test results

Figure 21 and Figure 22 illustrate the experimental load-deflection responses obtained from the acquisition of the measurements at mid-height of the scaffolding ladders. Loosenesses were observed during the first two loading cycles. These loosenesses were due to the assembly clearances of the vertical components with each other (inclinations between vertical components). The maximal assembly clearances, i.e. the play between the nominal inner diameter of the tube and the nominal outer diameter of the spigot, measured on the tested ladders did not exceed 3 mm.

Table 10 presents the failure load of each scaffolding ladder tested. These ultimate loads were obtained after several loading cycles.

Statistical study

The statistical study is conducted on the similar principle as the two previous ones. The tests on the scaffolding ladders being expensive, only one or two tests were carried out for each configuration. The statistical study was therefore unconducted on the values of failure stresses but on the values of imperfection factors resulting from the experimentation.

The imperfection factors stemming from the experimentation were determined from the equation of European buckling curve of Maquoi and Rondal [START_REF] Maquoi | Mise en équation des nouvelles courbes européennes de flambement[END_REF] and are given in the Table 11. The values of V1 and V2, given by Table 12, show that the distribution of the results of scaffolding ladder tests can be considered as normal distribution. The convention value of the imperfection factor is therefore done by the equation:

2 m s α = + (21)
The convention value of the imperfection factor for scaffolding ladder tests is: α = 0.252. Figure 23 plots the distribution of experimental results. The conventional value of the imperfection factor is emphasised. It can be observed that the totality of the experimental results obtained is below the conventional imperfection factor resulting from the statistical study. Eurocode [START_REF] En | Eurocode: Basis of structural design[END_REF] states the 95% fractile value as the characteristic value, for an unfavourable high value. The conventional imperfection factor value are therefore safe. The results of these full-scale tests highlight that the conventional imperfection factor for cold-formed hollow tubular sections, i.e. α = 0.49, is too safe and too constraining as regard of the sections used in the field of scaffolding. Based on these results, it is possible to admit an imperfection factor of 0.34 would allow a better estimation of the initial imperfections affecting this type of element while nevertheless preserving a safety margin.

Calibrations of an analytical model and a finite element model

As for simply supported members, the experimental load-deflection responses will be used to calibrate two models: an analytical model and a finite element model.

For the analytical model, the mid-height deflection of the column must be expressed as a function of the load applied to the column. The deflection ∆ physically measured at mid-height of the ladder is:

0 0 fin cr N d d d N N ∆ = -= ⋅ - (22)
where: dfin is the total displacement under axial load, d0 is the initial imperfection of the scaffolding ladder, N is the compression load and Ncr is the critical load.

The initial displacement d0 is given by the expression below:

⋅   = + = ⋅ +             (23)
where e0,sup is the initial imperfection determined on the height e 0,sup h , e 0,sup h represents the distance between the upper inflection point and the top of the scaffolding ladder (Figure 24) and e0 is the initial imperfection defined in the article 5.3.2 [START_REF] Peng | High clearance scaffold systems during construction -I. Structural modelling and modes of failure[END_REF] of Eurocode 3 Part1-1 as:

( ) 0 0, 2 el W e A α λ = ⋅ - ⋅ (24)
The critical load is obtained from the equation:

2 2 cr cr EI N L π = ( 25
)
where Lcr is the buckling length of the element. The tested elements were scaffolding ladders; the buckling length does not equal the length of the elements. Under axial compression, the scaffolding ladder was deformed as shown in Figure 24. The buckling length represents the length between two consecutive turning points. The buckling length of each configuration tested is summarized in Table 13. The second model developed is a finite element model. The software suite for finite element analysis Abaqus was used to create this finite element model. The measured dimensions of the components were used for cross-sectional properties in the finite element models. The nonlinearity properties of the steel material are applied to the model. The stress-strain relationship is based on results of tensile tests carried out on specimens from tested hot-rolled elements.

The standard/transom stiffness, given in Table 9, has been taken into account in all models. The ladders with screwdriver supports were modelled with pin-jointed supports; while the ladders with clevises and screw jack feet were modelled with supports with a stiffness of 20 kN.m/rad. The calibration was achieved by changing the amplitude of the initial imperfections of the model, to obtain the finite element model that best reflects the actual behaviour of the structure. The failure (ultimate load) in the finite element model was observed when the load limit point was reached, which was associated with significant displacements. The calibration of analytical and finite element models shows that the imperfection factor must be lower or at most equal to 0.25 to obtain a load-deflection response similar to the experimental one. The imperfection factor values obtained above are considerably lower than the conventional value of 0.49 imposed by Eurocode 3 for this type of cross-section. Moreover, when comparing the analytical response with an imperfection factor of 0.49 with the experimental responses, it can be noted that this response does not reflect the actual behaviour of scaffolding ladders. This implies that the initial imperfection degree of a buckling curve 'c', i.e. an imperfection factor of 0.49, is too important.

It can equally be sighted that the imperfection values resulting from analytical and finite element model are close to or even lower than the value of 0.252 resulting from the statistical study. This implies this value of 0.252 is well representative of the behaviour of this type of structure while being safe.

The comparison of the load-deflection responses obtained with the analytical model and the finite element model shows once again that these two models are equivalent and both accurately predicting the behaviour and failure load of scaffolding ladders.

The observation of the previous results makes it possible to affirm that an imperfection factor of 0.34, i.e. a buckling curve 'b', would make it possible to get closer to the real behaviour of this type of element while nevertheless preserving a safety margin.

Conclusion

A series of experimental tests were conducted to determine and analyse the buckling of scaffolding structures. These tests were out on the on simply supported members, hot-rolled or cold-formed, on the other hand on scaffolding ladders, which govern the entire behaviour of an entire scaffolding structure.

The experimentation revealed that for all the elements tested, the intensity of the initial imperfection required by the Eurocode 3 standard, through the buckling curves, were far too important. However, it is pertinent to remember that if the initial imperfections of a structure are overestimated, this leads to an overestimation of the evaluation of the internal stresses and therefore, in fine, to the oversizing of the structure. It is therefore crucial for all designers of scaffolding structures to have an accurate estimate of the initial imperfections affecting their structures to be able to design them optimally. Accurate estimation of initial imperfections leads to optimal designing. It was therefore shown in this article that the behaviour of the hot-rolled elements was closer to the buckling curve 'a0', i.e. a 0.13 imperfection factor, than to the buckling curve 'a', i.e. a 0.21 imperfection factor, as currently established in Eurocode 3. This result is applicable to elements whose diameter and slenderness range are those commonly found in the scaffold field.

Similarly, the tests on simply supported cold-formed elements and on scaffolding ladders highlighted that the standard theoretical buckling behaviour of these sections (buckling curve 'c', i.e. a 0.49 imperfection factor), established by Eurocode 3, does not reflect their actual behaviour and is too restrictive.

The tests on simply supported cold-formed members provided an experimental imperfection factor of 0.12. These tests enabled to assess the initial imperfections that affect only one element, i.e. geometrical imperfections and residual stresses.

For this reason, the tests on scaffolding ladders were conducted. Indeed, in the case on scaffolding ladders, initial imperfections measured are equally the geometrical imperfections and residual stresses of each constituent element but also the set of imperfections affecting the structure as a whole, i.e. any inclinations between vertical components. In this case the experimentation has shown that the initial imperfections affecting these structures lead to an imperfection factor of 0.25. This value of 0.25 is the result of tests on scaffolding ladders and therefore takes into account the lack of fit at joints between scaffolding standards, within the assembly clearance limit of 3mm (maximum value observed during tests). This play of 3 mm is ordinary for scaffolding elements. Nevertheless, for a more significant clearance between tubular standards and spigots, it would be necessary to model this inclination between vertical components, in accordance with the EN 12811-1 standard [START_REF]NF EN 12811-1: Temporary works equipment. Part 1: Scaffolds -Performance requirements and general design[END_REF], in addition to the initial imperfection advocated in this article.

To get closer to the buckling curves currently defined in Eurocode 3, it is therefore proposed to adopt an imperfection factor of 0.34 to represent the behaviour of scaffolding structures. This value of 0.34 applies for cold-formed steel circular hollow tubular sections with a slenderness ratio between 1.5 and 2.5, a cross-sectional area common in scaffolding and with assembly clearances not exceeding three millimetres. On the materials investigated, this valuation of the buckling curve (from curve 'c' to curve 'b') allows a resistance gain of 5% on average. Further studies required needed for a more universal application of the value of 0.34.

Two models have equally been developed through this article: an analytical model and a finite element one. The comparison of load-deflection responses show that these both models provide reasonable agreement compared to the experimental responses. As a result, these two models combined with the imperfection factor of 0.34 make it possible to accurately predict the behaviour and failure load of scaffolding ladders. This value can then be used for the design of scaffolding structures [START_REF] Mercier | Lateral stability of slender cold-rolled hollow tubular sections with initial imperfections[END_REF][START_REF] Mercier | Development of a model of stability for steel hollow tubular section beam-columns sensitive to second order effects[END_REF].

Figure 1

 1 Figure 1 Behaviour of column under axial compression

  due to second order effects Bifurcation point

Figure 2

 2 Figure 2 Simply supported members test device

Figure 3

 3 Figure 3 Screwdrivers supports

Figure 4 Figure 5 and

 45 Figure 4 Configuration of the experimental test device for simply supported elements

Figure 5 Figure 6

 56 Figure 5 Experimental load-deflection responses at mid-height of the 2.7 mm of thickness hot-rolled elements

Figure 7

 7 Figure 7 Distribution of experimental results for hot-rolled specimens: a) 48.3x2.7mm; b) 48.3x3.2mm

Figure 9 and

 9 Figure 9 and Figure 10 compare finite element analysis results and analytical analysis results with the experimental load-deflection responses at mid-height of hot-rolled simply supported elements.

Figure 9 Figure 10

 910 Figure 9 Calibration of load-deflection responses at mid-height of the 2.7 mm of thickness hot-rolled elements

Figure 11 Figure 12

 1112 Figure 11 Experimental load-deflection responses at mid-height of the 2.7 mm of thickness cold-formed elements

Figure 13

 13 Figure 13 Distribution of experimental results for cold-formed specimens: a) 48.3x2.7mm; b) 48.3x3.2mm

Figure 14

 14 Figure 14 Comparison between experimental values and European buckling curves for cold-formed elements

  Cold-formed hollow tubular sections -48,3x2,7 mm Conventional value -Cold-formed hollow tubular sections -48,3x3,2 mm λ

Figure 15 and

 15 Figure 15 and Figure 16 compare finite element analysis results and analytical analysis results with the experimental load-deflection responses at mid-height of cold-formed simply supported elements.

Figure 15 12 Figure 16

 151216 Figure 15 Calibration of load-deflection responses at mid-height of the 2.7 mm of thickness cold-formed elements

Figure 17 Figure 18

 1718 Figure 17The testing element: scaffolding ladder

Figure 19 a

 19 Figure 19 a) Appended structure; b) Trimmer; c) Wire rope; d) Loading framesThe ladders were tested in two support configurations: (i) screwdriver's supports (Figure20a) to behave like the pin-jointed supports; (ii) clevises and screw jack feet (Figure20b) to act as the real supports of scaffolding structures (stiffness: 20 kN.m/rad).

Figure a )

 a Figure a) Screwdriver supports; b) Clevises and feet

Figure 21 Figure 22

 2122 Figure 21 Experimental load-deflection responses at mid-height of the scaffolding ladders with screwdriver supports

Figure 12

 12 Figure 12 Comparison between experimental values and European buckling curves for scaffolding ladders

Figure 23

 23 Figure 23 Distribution of experimental results for scaffolding ladders

  Conventional value of the imperfection factor

Figure 24

 24 Figure[START_REF] Yang | Flexural buckling design of fabricated austenitic and duplex stainless steel columns[END_REF] Deformation of the scaffolding ladder under axial load Equations (26) to[START_REF] Jacquet | Essais de flambement et exploitation statistique[END_REF] hence make it possible to establish the load-deflection response of a scaffolding ladder for any value of imperfection factor α, i.e. several amplitudes of initial imperfections. It will thus be possible to compare the experimental response to different analytical responses, to obtain the analytical model that best reflects the actual behaviour of the structure. Three imperfection factor values will be compared: (i) an imperfection factor of 0.49: the conventional value for this type of cross-section established by Eurocode 3; (ii) an imperfection factor of 0.34: the imperfection factor value as close as possible to the values obtained by the statistical study; (iii) an imperfection factor value obtained through the calibration of the analytical model.

Figure 25 and 25 Figure 25 Figure 26

 25252526 Figure 25 and Figure 26 compare finite element analysis results and analytical analysis results with the experimental load-deflection responses at mid-height of hot-rolled simply supported elements.

Table 1

 1 Experimental load and failure stress on hot-rolled hollow tubular specimens

	Section	Specimen	Pr,exp (kN)	σr,exp (MPa)
		TNC06	50	129.268
		TNC07	50.7	131.078
	48.3 x 2.7 mm	TNC08	52.3	135.214
		TNC09	50.5	130.561
		TNC10	48	124.097
		TNC01	60.2	132.776
		TNC02	57.2	126.159
	48.3 x 3.2 mm	TNC03	58	127.924
		TNC04	58.5	129.027
		TNC05	58.3	128.586

Table 2

 2 Parameters of the method of the moments for tests on simply supported hot-rolled members

	Section	N	m (MPa)	s (MPa)	V1	V2
	48.3 x 3.2 mm	5	128.894	2.245	0.485	0.541
	48.3 x 2.7 mm	5	130.044	3.611	-0.303	0.246

Table 3

 3 Conventional values of the failure stress and the failure load for tests on simply supported hot-rolled members

	Section	σ	, u conv	(MPa)	, u conv P	(kN)	σ	, = u conv y f σ
	48.3 x 3.2 mm		124.404	56.404		0.518
	48.3 x 2.7 mm		122.822	47.507		0.512
	Figure

Table 4

 4 Experimental values of the imperfection factor for simply supported hot-rolled members

	Section	α
	48.3 x 3.2 mm	0.042
	48.3 x 2.7 mm	0.066

Table 5

 5 Experimental load and failure stress on cold-formed hollow tubular specimens

	Section	Specimen	Pr,exp (kN)	σr,exp (MPa)
		TN01	47	121.512
		TN02	51.4	132.888
		TN03	47.5	122.805
		TN04	51	131.854
		TN05	49.8	128.751
		TG01	50.2	129.785
	48.3 x 2.7 mm	TG02 TG03	50.7 49.8	131.078 128.751
		TG04	54.7	141.419
		TG05	50.7	131.078
		PG01	50	129.268
		PG02	52.7	136.249
		PG03	50.1	129.527
		PG04	51.3	132.629
		PGD01	56.5	124.615
		PGD02	55.7	122.851
	48.3 x 3.2 mm	PGD03	58.2	128.365
		PGD04	57	125.718
		PGD05	54.2	119.543

Table 6

 6 Parameters of the method of the moments for tests on simply supported cold-formed members

	Section	N	m (MPa)	s (MPa)	V1	V2
	48.3 x 3.2 mm	5	124.219	3.611	-0.303	0.246
	48.3 x 2.7 mm	14	130.542	4.868	0.301	0.123

Table 7

 7 Conventional values of the failure stress and the failure load for tests on simply supported membersFigure 13 plots the distribution of experimental results. The conventional value of the failure stress is emphasised. As previously, it can be observed that the totality of the experimental results obtained is beyond the conventional failure stress resulting from the statistical study. The both conventional values of the failure stress are therefore safe with regard to the Eurocode[START_REF] En | Eurocode: Basis of structural design[END_REF] specifications.

	Section	σ	, u conv	(MPa)	, u conv P	(kN)	σ	, = u conv y f σ
	48.3 x 3.2 mm		116.978	53.037		0.366
	48.3 x 2.7 mm		120.807	46.727		0.378

Table 8

 8 Experimental values of the imperfection factor for cold-formed simply supporter members

	Section	α
	48.3 x 3.2 mm	0.1312
	48.3 x 2.7 mm	0.1147

Table 9

 9 Properties of cross-sections of the ladders

	Section	Yield strength	Stiffness between standard
		(MPa)	transom (kN.m/rad)

Table 10

 10 Experimental failure load of scaffolding ladders (in kN)

		Tube A	Tube B	Tube C
	1	21.22	33.37	37.66
	2	21.88	33.57	37.52
	3	28.18	37.9	40.40

Table 11

 11 Experimental imperfection factor of scaffolding ladders

		Tube A	Tube B	Tube C
	1	0.235	0.215	0.128
	2	0.170	0.205	0.135
	3	0.145	0.105	0.090
	Figure 18 allows the comparison between the experimental results and European buckling curves. The

buckling curve 'c', relating to the cold-formed sections, is emphasised.

Table 12

 12 Parameters of the method of the moments for tests on scaffolding ladders

	N	m	s	V1	V2
	9	0.1587	0.047	0.278	2.988

Table 13

 13 Buckling length Lcr of the scaffolding ladders (in meter)

		Screwdriver supports	Clevises and screw jack feet
	Tube A	2.961	2.605
	Tube B	2.340	2.326
	Tube C	2.342	2.282


Sybille Caillot-Ranjeva
email: sybille.caillot-ranjeva@u-bordeaux.fr

Samuel Gourlain
email: samuel.gourlain@u-bordeaux.fr

Hélène Amieva
email: helene.amieva@u-bordeaux.fr

Catherine Helmer
email: catherine.helmer@u-bordeaux.fr

Valérie Bergua
email: valerie.bergua@u-bordeaux.fr

Study of mutual influence between trait anxiety and risk of depression among older couples facing cancer

Keywords: Adjustment, anxiety, cancer, depression, older couples, mutual influence, oncology

Introduction:

The present study aims to examine the process of mutual influence in older couples with cancer diagnosis by studying their risk of depression.

Materials and Methods: 282 couples with one spouse diagnosed with cancer were selected from the Three-City cohort study. Dyadic analyses were used to determine whether trait anxiety affects the risk of depression and whether a mutual influence process occurs prior and post cancer diagnosis. Cross-sectional analyses were performed at two time-points: before and after receiving the diagnosis.

Results: A higher level of anxiety among cancer patients resulted in a decreased risk of depression among spousal caregivers. Moreover, a higher anxiety among spousal caregivers increased their own risk of depression, but it didn't influence depression risk among cancer patients. While there is an intra-individual relationship between a higher level of trait anxiety and a greater risk of depression prior to cancer diagnosis, there is no cross-influence between spouses.

Discussion:

The study findings indicate that a dyadic psychological adjustment process might help older adults to cope with cancer by limiting the risk of depression among spousal caregivers.

Introduction

Depression among patients with cancer is a major concern, as patients with cancer of all ages show a prevalence of depression at around 25% [START_REF] Cardoso | Depression and anxiety symptoms following cancer diagnosis: A cross-sectional study[END_REF], at least twice as high as the corresponding number in the general population [START_REF] Caruso | Depressive spectrum disorders in cancer: prevalence, risk factors and screening for depression: a critical review[END_REF]. Research on older patients with cancer is scarce [START_REF] Singer | Prevalence of mental health conditions in cancer patients in acute care-a meta-analysis[END_REF], and the results are less clear. A decrease in the prevalence depression in older adults with cancer is reported [START_REF] Hinz | Anxiety and depression in cancer patients compared with the general population[END_REF]. However, this decreased rate seems to be partly due to underestimation linked to diagnostic difficulties [START_REF] Saracino | Towards a new conceptualization of depression in older adult cancer patients: a review of the literature[END_REF], and depression remains a major concern as it is the most common psychopathological comorbidity among older adults with cancer [START_REF] Parpa | Elderly cancer patients' psychopathology: A systematic review Aging and mental health[END_REF].

In addition, depression and anxiety appear to be related among older people with cancer [START_REF] Lenze | Comorbidity of depression and anxiety disorders in later life[END_REF][START_REF] Wiesel | The relationship between age, anxiety, and depression in older adults with cancer[END_REF]. In a study of 500 patients aged 65 and over, Wiesel et al. reported prevalence rates of 20.9% for anxiety symptomatology and 12.6% for depression [START_REF] Wiesel | The relationship between age, anxiety, and depression in older adults with cancer[END_REF]. In older adults, anxiety and depression are two distinct psychological disorders often considered on the same continuum as the level of anxiety could be a prodromal stage of depression or a vulnerability to depression [START_REF] Potvin | Anxiety and 10-year risk of incident and recurrent depressive symptomatology in older adults[END_REF]. Trait anxiety defined as "a person's disposition to respond with anxiety when faced with situations perceived as threatening" [START_REF] Spielberger | Manual for the State-Trait Anxiety Inventory (Form Y1 -Y2)[END_REF] is an independent predictor of incident and recurrent depressive symptomatology in older adults. While some studies found similar results on younger patients with breast cancer [START_REF] Vries | Trait anxiety determines depressive symptoms and fatigue in women with an abnormality in the breast[END_REF][START_REF] Lockefeer | What is the relationship between trait anxiety and depressive symptoms, fatigue, and low sleep quality following breast cancer surgery?[END_REF], to the best of our knowledge, no studies have been done on older cancer persons. Yet, when studying the risk of depression in older adults with cancer it is critical to consider trait anxiety as well, beyond other known sociodemographic and clinical factors associated with depression [START_REF] Spoletini | Depression and cancer: An unexplored and unresolved emergent issue in elderly patients[END_REF][START_REF] Stommel | A Longitudinal Analysis of the Course of Depressive Symptomatology in Geriatric Patients With Cancer of the Breast, Colon, Lung, or Prostate[END_REF].

Detecting and treating depression is crucial as depression has harmful consequences for the patient including increased mortality risk [START_REF] Pinquart | Depression and cancer mortality: a meta-analysis[END_REF], increase in the number of hospital days [START_REF] Koenig | Survival and health care utilization in elderly medical inpatients with major depression[END_REF] and lower effectiveness of the treatment [START_REF] Spoletini | Depression and cancer: An unexplored and unresolved emergent issue in elderly patients[END_REF]. Managing depression is also important from a dyadic point of view since the level of depression of patients not only impacts their well-being and their chances of recovery but also their spouse's psychological well-being. There is indeed a strong relational closeness between spouses, especially important as patients get older, as advanced age increases in the importance of intimate relationships [START_REF] Carstensen | Affect in intimate relationships[END_REF]. The influence of close family (e.g., spouses) is important when evaluating mental health among older people [START_REF] English | Selective narrowing of social networks across adulthood is associated with improved emotional experience in daily life[END_REF][START_REF] Hoppmann | Spousal interrelations in old age -A mini-review[END_REF]. Emotional proximity among the spouses influences their mental health by facilitating the processes of mutual influence, especially with regard to fluctuations of positive and negative emotions during the time spent together [START_REF] Goodman | Is it contagious? Affect similarity among spouses[END_REF][START_REF] Hoppmann | Spousal interrelations in happiness in the Seattle Longitudinal Study: Considerable similarities in levels and change over time[END_REF][START_REF] Meyler | Health concordance within couples: A systematic review[END_REF][START_REF] Michalowski | Associations Between Perceived Support in Older Adult Marriages and Dyadic Covariations in Momentary Affect and Aches[END_REF]. Mutual influence is understood here as a multi-determined process underpinned by the psychological interdependence inherent in the relational proximity between spouses [START_REF] Berscheid | Emotional Experience in Close Relationships[END_REF]. Emotional and psychological contagion processes predict the emotional distress and risk of depression in couples facing cancer [START_REF] Hagedoorn | Distress in couples coping with cancer: A meta-analysis and critical review of role and gender effects[END_REF][START_REF] Lo | Couples facing advanced cancer: Examination of an interdependent relational system[END_REF][START_REF] Moser | Higher emotional distress in female partners of cancer patients: prevalence and patient-partner interdependencies in a 3year cohort[END_REF][START_REF] Segrin | Psychological and physical distress are interdependent in breast cancer survivors and their partners[END_REF][START_REF] Segrin | Interdependent psychological quality of life in dyads adjusting to prostate cancer[END_REF]. In addition, some studies suggest that being single and having low levels of social support significantly increase the risk of depression among older patients with cancer, while being married seems to have an opposite effect [START_REF] Schapmire | Depression symptoms in older adults with cancer: A multilevel longitudinal study[END_REF]. To date, the underlying process of such a protective effect is poorly understood as the processes of mutual influence is rarely studied in older couples facing cancer.

Hagedoorn et al. suggest that couples react to the disease as a dyadic emotional system. Moreover, it seems that the distress is lower when the patient has a partner. One explanation could be that the marital relationship, through dyadic rather than individual emotional mechanism, acts as a protective factor [START_REF] Hagedoorn | Distress in couples coping with cancer: A meta-analysis and critical review of role and gender effects[END_REF]. Spousal caregivers tend to have lower levels of anxiety, depression and higher levels of hope than patients with cancer [START_REF] Goldzweig | A Dialogue of Depression and Hope: Elderly Patients Diagnosed with Cancer and Their Spousal Caregivers[END_REF]. In other words, the spousal caregiver can provide significant support to the patient. To better understand this protective effect of the couple, the present study focuses on the process of mutual influence. Based on a sample of 282 couples from the Three-City (3C) cohort, this study aims firstly to determine whether the level of trait anxiety predicts the risk of depression in older spouses facing cancer. It is hypothesized that their higher level of trait anxiety will increase their risk of depression. The second goal is to examine spousal caregivers' response to the psychological vulnerability of the cancer patient. It is hypothesized that a high level of trait anxiety among patients with cancer will decrease the risk of depression among spousal caregivers. Finally, it is hypothesized that the association between the spousal caregiver's depression risk and the spouse's anxiety level depend on the timing of the cancer diagnosis. Specifically, prior to cancer diagnosis the link between the spouse's anxiety level and their partner's risk of depression is minimal. To assess this hypothesis, we will study the process of mutual influence between the spouses during pre and post-cancer diagnosis stages.

Material and Methods

Study population

The present study relies on a subset sample from the 3C study which is a multicenter epidemiological cohort study aiming to assess the relationship between vascular diseases, dementia and cognitive impairment among people aged 65 and over, initially living at home.

The study sample consisted in 9,294 people among which 1,817 couples. The participants were randomly selected from the voters' register of three French cities: Bordeaux, Montpellier and Dijon. The initial evaluation was conducted between 1999 and 2001 and is repeated approximately every two years. Trained nurses and psychologists collected a wide range of data including socio-demographics, educational level, cognitive assessment, psychological health, food habits, alcohol and tobacco consumption, lifestyle, family history of cardiovascular diseases, exhaustive medical history, medication use, disabilities, clinical measurements (blood pressure, height, weight…) and biological parameters, using a standardized questionnaire common to the three cities.

The ethics committee of the University Hospital of Kremlin-Bicêtre approved the 3C Study.

For a detailed report of the 3C Study data collection methodology, refer to [START_REF] Study | Vascular factors and risk of dementia: Design of the Three-City Study and baseline characteristics of the study population[END_REF].

Sample

Among the 1,817 couples included in the 3C cohort Study, 282 spousal dyads were selected according to the following inclusion criteria: couples living together, legally married or partners, and one of the spouses diagnosed with cancer within two years prior to the cohort follow-up. All the follow-up visits were considered in the present study. The exclusion criteria were: one of the spouses diagnosed with dementia during the same follow-up the cancer diagnosis was reported and couples diagnosed with cancer at the same follow-up as the presence of dementia in one partner or cancer in both partners could significantly alter the process of mutual influence between the spouses ; widowed couples and couples with missing data on anxiety or depression,. Our final sample included 282 couples (see flowchart, Figure1).

Insert Figure 1 2.3. Assessment Outcome: In order to consider the presence of a risk of depressive state related to cancer, we constructed a single outcome by taking into account both current depressive symptomatology and the use of antidepressants. Participants were considered to have depression when their CESD score was elevated and/or when antidepressant treatment was prescribed. The binary score was coded with 0 = absence of depressive symptomatology and no antidepressant treatment; 1 = presence of depressive symptomatology or antidepressant treatment.

The 20-item Center for Epidemiologic Studies Depression Scale [START_REF] Radloff | The CES-D scale: A self-report depression scale for research in the general population[END_REF] was used to assess the frequency of depressive symptoms on a four-point scale (0=rarely or never; 1=occasionally; 2=often; 3=most or always). The overall score (0 to 60) was dichotomized on the basis of cutoff points fixed at 17 for men and 23 for women, as previously validated in older population [START_REF] Fuhrer | La version française de l'échelle CES-D (Center for Epidemiologic Studies-Depression Scale). Description et traduction de l'échelle d'autoévaluation. The french version of the Center for Epidemiologic Studies. Depression Scale[END_REF], with 1=high depressive symptomatology and 0=low depressive symptomatology.

The use of antidepressants was reported, as were all other treatments used by the patient at the time of the follow-up.

Predictor: Trait anxiety was evaluated during the initial visit with the 20-item Spielberger

State-Trait Anxiety Inventory (STAI; 32), the scores ranging from 20 to 80, higher scores indicating higher anxiety. Couples in which one spouse had more than 2 missing items were excluded. When there was less than 3 missing items, we divided the total score by the number of items to which participants responded to obtain a score ranging from one to four, then we multiplied it by 20 as recommended by Spielberger.

Covariates. The covariates were selected from the literature based on their impact on the risk of depression in older patients with cancer. These variables were collected from both patients and their spousal caregivers.

-Sociodemographic variables included age, sex, and educational level classified in three categories (0=primary school education evidenced by a diploma, 1=short or long secondary education validated by a diploma, 2=higher educational level).

-To assess the specific impact of the cancer diagnosis on the risk of depression, depressive symptomatology and use of antidepressants were controlled at the prior follow-up visit to which the cancer diagnosis was reported. To take into account the effect of pre-existing mental health concerns beyond depression, use of anxiolytics and antipsychotic drugs at the prior follow-up visit to which the cancer diagnosis was reported were also controlled.

-To consider the effect of general health status beyond cancer, other health conditions were included. These conditions, considered as binary variables, included myocardial infarction, angina, stroke, dyspnea, heart failure, diabetes, and disability. The binary scores were coded with 0 = no medical history; 1 = medical history.

-To consider the effect of being a couple, we identify the 2 members of the same couple by using the couple's ID number, which is identical for each of the two partners.

Statistical analysis

In order to take into account the process of mutual influence between spouses the statistical methodology is based on Actor Partner Interdependence Model, APIM [START_REF] Kenny | Dyadic data analysis[END_REF] which permits to distinguish: 1. the effect of each spouse's trait anxiety level on his/her own risk of depression (actor effect); 2: the effect of each spouse's trait anxiety level on each partner's risk of depression (partner effect) (for APIM design, see Figure2). In this study, the distinguishing variable is the fact of being the participant with cancer or the spousal caregiver.

Insert Figure 2

The statistical methodology relied on several steps. In a first step, the descriptive analyses were conducted at the follow-up at which the cancer diagnosis was reported (see APIM design, Figure 2). Means and standard deviations were reported for continuous variables. In a second step, the initial data set was reorganized to allow for dyadic mixed model analyses, with the couple as the unit of analysis. To do so, we referred to the APIM methodology [START_REF] Kenny | Dyadic data analysis[END_REF]. We performed two different models over two-time windows (at pre and post cancer diagnosis).

In both Model 1 and Model 2 the level of trait anxiety, considered as a global and stable measure of anxiety, was collected at the baseline of the study as were the sociodemographic variables.

Model 1 was performed on data collected at the follow-up at which the cancer diagnosis was reported. Mixed variables include the trait anxiety score as fixed effect predictor, and comorbidities (myocardial infarction, angina, stroke, dyspnea, heart failure, diabetes, and disability) as fixed effect covariables. Individual intercept and couple I.D. were included as a random effect. Depressive symptomatology, use of antidepressants, use of anxiolytics and antipsychotic drugs were controlled at the prior follow-up visit to which the cancer diagnosis was reported.

Model 2 was performed on data collected at least two years prior to cancer diagnosis. Such a time window was used in order to study the process of mutual influence between spouses before the couple was faced with a cancer diagnosis or its first signs. Mixed variables include trait anxiety score as fixed effect predictor, and comorbidities as fixed effect covariate. Individual intercept and couple I.D. were included as random effects.

In a third step, in order to determine the most relevant adjustment variables for each model, we performed univariate logistical mixed regressions models of both own and partner healthrelated variables on own risk of depression. We kept all adjustment variables with p≤. [START_REF] Stommel | A Longitudinal Analysis of the Course of Depressive Symptomatology in Geriatric Patients With Cancer of the Breast, Colon, Lung, or Prostate[END_REF].

Model 1 and model 2 were adjusted on socio-demographic variables (age, gender, education) as well as on retained health-related variables after a backward stepwise strategy. This backward stepwise strategy consisted in dropping the variable with the highest p-value (higher than 5%) out of the model to check the relative variation of the regression parameters (for anxiety, cancer and interactions). When the variation was smaller than 10%, we considered that the variable was not a confounding factor and we dropped it out of the model.

We performed this backward stepwise strategy for each variable with p-value higher than 5%, until the model contains only variables with p-value under 5% as well as the variables that were confounding factors.

In a fourth step, we performed APIM analysis using two interactions between the distinguishing variable (cancer) and the predictors (actor and partner trait anxiety) to distinguish actor and partner effects.

Taking together the results of the two models will highlight the change in the process of mutual influence between spouses before and after facing a cancer diagnosis.

Analyses were performed using the Statistical Analysis System (SAS), software version 9.3 (SAS Institute, Inc., Cary, NC, USA) and R version 3.2.3.

Results

Sample characteristics

The descriptive statistics for all study variables are presented in Table1. Among the 282 participants with cancer, the level of depressive symptomatology does not differ before and after cancer diagnosis (p=.162) but the use of antidepressants is significantly higher after the diagnosis (p=.020). If we consider the combined score of high depressive symptomatology and use of antidepressants, it is not significantly higher after the diagnosis than before (p=.068).

Among the 282 spouses of participants with cancer, neither the level of depression (p=.336) nor the use of antidepressants (p= .083) or the combined score (p=.095) significantly differ before and after cancer diagnosis. presents the results of the dyadic mixed effects analysis in Model 2 (for confidence intervals, see Table 2 in supplemental online materials).

Insert Table1

Dyadic analysis

Insert Figure 3 and Figure 4 This study first aimed to address the effect of the trait anxiety level of participants with cancer and spousal caregivers on their own risk of depression (actor effect). Regarding model 1, after the cancer diagnosis, the spousal caregiver's level of trait anxiety is positively associated with his/her own risk of depression (OR=1.09; p<.001). There is no association between cancer participant's level of trait anxiety and his/her own risk of depression.

Regarding model 2, at least two years prior to cancer diagnosis, the future cancer participant's level of trait anxiety is positively associated with his/her own risk of depression (OR=1.09; p=.004). His/her spouse's level of trait anxiety is positively associated with his/her own risk of depression (OR=1.12; p<.001).

This study secondly aimed to address the mutual effect of each spouse's trait anxiety level on his/her partner's risk of depression at the post cancer diagnosis stage (partner effect).

Regarding model 1, after the cancer diagnosis, the cancer patient's level of trait anxiety is negatively associated with his/her spousal caregiver's risk of depression (OR=0.94; p=0.034) while there is no association between spouses' level of trait anxiety and cancer participant's risk of depression.

Regarding model 2, at least two years prior to cancer diagnosis, there is no partner effect.

Discussion

The purpose of this study was to examine the process of mutual influence in older couples facing a cancer diagnosis. We examined 1) the effect of the trait anxiety level of participants with cancer and spousal caregivers on their own risk of depression (actor effect) and, 2) the mutual effect of each spouse's trait anxiety level on his/her partner's risk of depression after a cancer diagnosis (partner effect). Accordingly, the impact of trait anxiety on the risk of depression was considered both at the individual level and between spouses, at two points of time (prior and during cancer diagnosis).

At the individual level (actor effect), the level of trait anxiety predicts the risk of depression prior to cancer diagnosis for both spouses. This finding is consistent with previous studies [START_REF] Potvin | Anxiety and 10-year risk of incident and recurrent depressive symptomatology in older adults[END_REF]. The same holds true for spousal caregivers during the post-diagnosis stage. The absence of a link between anxiety and the risk of depression in older adults recently diagnosed with cancer could be explained by the fact that the risk of depression is mainly driven by cancer itself. Another possible explanation is that older adults with cancer perceive their marital relationship as reassuring support, thus limiting the negative impact of anxiety, while spousal caregivers perceive less marital support and rely on their own resources to face the situation [START_REF] Hagedoorn | Distress in couples coping with cancer: A meta-analysis and critical review of role and gender effects[END_REF]. This account should be tested in a future study by comparing the relationship between trait anxiety and the risk of depression across married and unmarried patients. It would also be worthwhile to assess whether this relationship is partly mediated by the quality of care and support provided by the spouse. Moreover, patients who perceive that the spousal support is maladaptive or lacking are more likely to have a poor psychological adjustment to the disease [START_REF] Manne | Unsupportive partner behaviors, social-cognitive processing, and psychological outcomes in couples coping with early stage breast cancer[END_REF]. Therefore, as suggested by [START_REF] Berg | A developmental-contextual model of couples coping with chronic illness across the adult life span[END_REF] the dyadic functioning between spouses might drive their adaptation to the disease [START_REF] Berg | A developmental-contextual model of couples coping with chronic illness across the adult life span[END_REF].

At the dyadic level (partner effect), while there is no mutual influence between spouses prior to cancer diagnosis, our findings show that when the participant with cancer has a high level of trait anxiety, the spousal caregiver is less likely to be depressed. This finding is consistent with our hypothesis predicting the spousal caregiver's response to the psychological vulnerability of the participant with cancer. Older adults with cancer experiencing anxiety are likely to require the most emotional support from their partners, and to benefit from such support [START_REF] Hagedoorn | Marital satisfaction in patients with cancer: does support from intimate partners benefit those who need it the most?[END_REF]. The notion of spousal involvement facing cancer is important. The cancer patient expects the caregiver to take an active role in discussions, problem solving, and solution seeking, all of which are related to psychological adjustment [START_REF] Berg | A developmental-contextual model of couples coping with chronic illness across the adult life span[END_REF][START_REF] Hagedoorn | Marital satisfaction in patients with cancer: does support from intimate partners benefit those who need it the most?[END_REF]. This type of engagement requires substantial availability of the spousal caregiver.

Therefore, in a process of dyadic influence, spousal caregivers tend to take it upon themselves. By limiting the impact of cancer on their own risk of depression, they could improve their ability to effectively support the patient. Furthermore, the absence of a partner effect between spousal caregiver and participant with cancer before and after the cancer diagnosis supports the hypothesis of the dyadic adjustment process. In other words, the mutual influence is not a passive process of emotional contagion from one spouse to the other, but an active process of adaptation of the spouse helping to meet the psychological needs of his or her partner with cancer.

Prior research shows that an active investment in discussion and collaboration with the cancer patient can lead to emotional contagion with a potential transmission of daily negative emotions [START_REF] Berg | Collaborative coping and daily mood in couples dealing with prostate cancer[END_REF][START_REF] Berg | Affect Covariation in Marital Couples Dealing with Stressors Surrounding Prostate Cancer[END_REF][START_REF] Hagedoorn | Is self-disclosure in couples coping with cancer associated with improvement in depressive symptoms?[END_REF]. Thus, it would worthwhile to examine the mental health of spousal caregivers and to study the process of emotional contagion in daily life. It is also important to examine whether the relationship the cancer patient's anxiety and the caregiver's risk of depression persists over time. Such a relationship might be limited to a time close to the diagnosis. There might be a rebound effect with a subsequent risk of depression over longer periods of time following the diagnosis. The risk of depression in spousal caregivers should also be studied after remission and recovery, as the need for supportive behaviors is lessened.

A dyadic study of the risk of depression over time based on the trait anxiety levels of the two parties is warranted. However, this would require the use of repeated measures over time, particularly after the cancer diagnosis. Unfortunately, due to attrition related to the number of deaths and the lack of participant's follow-up, we do not have the necessary data to conduct such analyses.

Study limitations

This study has several limitations. First of all, the exact date of diagnosis for all participants is not available. Only information about whether the cancer diagnosis occurred since the last follow-up visit was available. Therefore, participants may not be at the same post-diagnosis period (diagnosis, treatment or end of treatment) which could influence their interactions and with spouses. Another potential limitation is the lack of information on cancer stage. It has been shown that the stage of the disease can influence the spouse's psychological state [START_REF] Lyons | The role of patient pain and physical function on depressive symptoms in couples with lung cancer: A longitudinal dyadic analysis[END_REF].

However, we did control for the level of disability. In addition, in order to observe different dyadic relationships between trait anxiety and the risk of depression in the context of cancer and without major health events, we used data collected after the diagnosis and prior to the diagnosis. In this way, we were able to ascertain whether there was a difference in the relationship between anxiety and the risk of depression before and after the onset of the disease.

Finally, since the analyses were done on data from a cohort study, the study was not designed specifically to examine psychological distress in patients with cancer and their spouses. As a result, we did not have access to information that would have allowed us to better explain how this relationship might constitute a dyadic adjustment process to the disease. An assessment of dyadic coping or marital adjustment would have been of great value for this. Beyond these limitations, the appropriate tools were used to assess depression in older adults with cancer (39).

Clinical implications

As pointed out by Hagedoorn et al. (26) this study supports the view that couples react to the disease as a dyadic emotional system. Thus, the questions of the risk of depression in older adults with cancer needs to consider the patient's marital support. In a similar vein, the risk of depression in spousal caregivers can only be understood in the context of their interactions with the patient. From a clinical point of view, clinicians should first assess more systematically the tendency to anxiety of older adults with cancer, their risk of depression and their spouse's risk of depression. As our findings indicate that the spouse can be an important source of support for anxious patients with cancer, it is necessary to take into account the spousal relationship. To the best of our knowledge, there are no studies on psychological interventions for older couples facing cancer. However, prior research suggests that interventions to support communication between the spouses have a positive impact on the couple's psychological balance and mutual influence processes [START_REF] Li | A literature review on the mutual impact of the spousal caregivercancer patients dyads: 'Communication', 'reciprocal influence', and 'caregiver-patient congruence[END_REF]. Thus, rather than considering the patient as an isolated unit, systematic interventions involving spouses and relying on dyadic psychological resources should be encouraged.

In conclusion, this study encourages both researchers and clinicians to consider the mental health of older spouses facing cancer through a dyadic approach, and to account not only for the experiences of each of the spouses, but also for the interdependence that exists between them.

Conflicts of interest

None declared

Author

Frequencies

  and percentages were reported for categorical variables. Differences between spouses were assessed using Student tests for continuous variables and Chi2 tests or Fisher's exact test for categorical variables. Categorical variables before and after cancer diagnosis were compared using Chi2 McNemar tests.

Figure 3 .

 3 Figure 3. presents the results of the dyadic mixed effects analysis in Model 1 and Figure 4.

Figure 1 . Title Figure 1

 11 Figures captions

Figure 2

 2 Figure 2. Title

Figure 2 .Figure 3 .Figure 3 .

 233 Figure 2. Captions

Figure 3 .

 3 Figure 3. Captions

Figure 4 .Figure 4 .

 44 Figure 4. Title Figure 4.

Figure 4 .

 4 Figure 4. Captions

Figure 1

 1 Figure 1.

Figure 2

 2 Figure 2.

  *p<.01(for CI95% see table 2 in supplemental materials) Adjusted on age, education, sex, P's use of antipsychotics, both P and SC's use of anxiolytics, P's disability, SC's myocardial infarction, SC's angina, SC's diabetes

	Tables
	Table1
	Sample characteristics of participants with cancer and their spousal caregivers (N=282
	couples), 3C study
		Participants	Spousal caregivers	Comparisons p-value
	Age, mean (SD)	74.8 (4.8)	75.0 (4.9) .674
	Men, n (%)	158 (56.0)	124 (44.0) .004
	Education, n (%)			.877
	primary school	78 (27.7)	78 (27.7)
	short or long secondary education	89 (31.6)	92 (32.6)
	higher education	115 (40.8)	112 (39.7)
	Trait anxiety, mean (SD)	38.3 (9.0)	39.5 (9.7) .127
	Before the cancer diagnosis
	High depressive symptomatology, n (%)	23 (8.6)	18 (6.7)	.403
	Use of antidepressants, n (%)	21 (7.4)	19 (6.7)	.743
	After the cancer diagnosis
	High depressive symptomatology, n (%)	26 (9.2)	32 (11.4)	.406
	Use of antidepressants, n(%)	29 (10.3)	21 (7.4)	.236
	Dyspnea, n (%)	6 (2.2)	5 (1.8)	.734
	Stroke, n (%)	4 (1.4)	6 (2.1)	.752
	Angina, n (%)	20 (7.2)	16 (5.7)	.477
	Myocardial infarction, n (%)	4 (1.4)	2 (0.7)	.686
	Heart failure, n (%)	9 (3.4)	10 (3.6)	.854
	Diabetes, n (%)	29 (10.3)	25 (8.9)	.586
	Disability, n (%)	30 (11.2)	30 (11.1)	.951

Acknowledgements 16 This work was supported by the Ligue Contre le Cancer as well as by the funders of the Three City study: National supports: Fondation pour la Recherche Médicale, Sanofi-aventis, Plan Alzheimer, Fondation de Coopération Scientifique sur la maladie d'Alzheimer et les maladies apparentées, Caisse Nationale d'Assurance Maladie des Travailleurs Salariés, Direction

Data references

The availability of data from the Three City Cohort Study is subject to the agreement of the 3 City Cohort Study Scientific Committee [START_REF]The Three-City Study[END_REF]. Investigators unaffiliated with any of the 3C-Study founding teams are welcome to propose new studies or new analyses (termed ancillary studies). 3C Study investigators are encouraged to work in collaboration with a senior investigator from one of the 3C-Study founding teams (listed in: Investigators, Study Sites, Collaborating Centres and Working Groups). The 3C-Study correspondent investigator can either be contacted directly by an investigator submitting an ancillary study or can be designated by the 3C-Study Steering Committee. Ancillary study proposals must be sent to the Coordinating Centre E3C.CoordinatingCenter@gmail.com