, Règlement de la position du tube à l'aide d'un comparateur de sorte que l'axe du tube coïncidé à l'axe de rotation

, Mesure de différentes sections, chaque 2mm sur la hauteur du tube, et exploitation des données via le logiciel Agilent

, Traitement des données par MATLAB pour récupérer la géométrie réelle du tube

, Les données des mesures sont enregistrées sous forme de tensions électriques sur un PC muni d'une carte d'acquisition. Les signaux sont enregistrés avec une fréquence d'échantillonnage de 0.2s. Le défaut est déterminé par la différence entre le rayon réel et le rayon moyen selon la norme, p.4291

, On exploite ici la géométrie réelle du tube (Figure 138). La qualité du tube est évaluée selon l, Ci-dessous nous analysons les résultats des mesures de géométrie à l'aide du logiciel Matlab R2012a

. Références,

R. Watkins, Buried Pipe Encased in Concrete, Pipeline Engineering and Construction, pp.1-10

D. Berti, R. Stutzman, E. Lindquist, and M. Eshghipour, Technical Forum: Buckling of Steel Tunnel Liner Under External Pressure, Journal of Energy Engineering, vol.124, issue.3, pp.55-89, 1998.

A. Limam, Flambage de coques cylindriques sous combinaison de chargements : pression interne et compression axiale, 1991.

M. A. Crisfield, Non-linear Finite Element Analysis of Solids and Structures, vol.1, 1991.

C. Bisagni, Numerical analysis and experimental correlation of composite shell buckling and postbuckling, Composites Part B: Engineering, vol.31, issue.8, pp.655-667, 2000.

K. Jorabchi and K. Suresh, A robust continuation method to pass limit-point instability, Finite Elements in Analysis and Design, vol.47, issue.11, pp.1253-1261, 2011.

Y. Yang and M. Shieh, Solution method for nonlinear problems with multiple critical points, AIAA Journal, vol.28, issue.12, pp.2110-2116, 1990.

J. Y. Yeong-bin and Y. , Recent developments in geometrically nonlinear and postbuckling analysis of framed structures, Applied Mechanics Reviews-APPL MECH REV, vol.56, issue.4, 2003.

K. Bathe and E. N. Dvorkin, On the Automatic Solution of Nonlinear Finite Element Equations, 1983.

H. Chen and G. E. Blandford, Work-increment-control method for non-linear analysis, Int. J. Numer. Meth. Engng, vol.36, issue.6, pp.909-930, 1993.

G. A. Wempner, Discrete approximations related to nonlinear theories of solids, International Journal of Solids and Structures, vol.7, issue.11, pp.1581-1599, 1971.

E. Riks, An incremental approach to the solution of snapping and buckling problems, International Journal of Solids and Structures, vol.15, issue.7, pp.529-551, 1979.

B. Kröplin, A viscous approach to post buckling analysis, Engineering Structures, vol.3, issue.3, pp.187-189, 1981.

A. Aimi, M. Diligenti, and F. Freddi, Numerical aspects in the SGBEM solution of softening cohesive interface problems, Journal of Computational and Applied Mathematics, vol.210, issue.12, pp.22-33, 2007.

X. S. Bashir-ahmed-memon, Arc-length technique for nonlinear finite element analysis, Journal of Zhejiang University. Science, vol.5, issue.5, pp.618-646, 2004.

M. Ritto-corrêa and D. Camotim, On the arc-length and other quadratic control methods: Established, less known and new implementation procedures, Computers & Structures, vol.86, issue.11-12, pp.1353-1368, 2008.

, 12-3 Abaqus, User's manuals

M. A. Crisfield, A fast incremental/iterative solution procedure that handles 'snap-through, Computers & Structures, vol.13, issue.1-3, pp.55-62, 1981.

E. Ramm, Strategies for Tracing the Nonlinear Response Near Limit Points, Nonlinear Finite Element Analysis in Structural Mechanics, pp.63-89, 1981.

G. Powell and J. Simons, Improved iteration strategy for nonlinear structures, Int. J. Numer. Meth. Engng, vol.17, issue.10, pp.1455-1467, 1981.

S. King and T. Richards, Solving Contact Problems with Abaqus, DS UK Ltd, 2013.

J. O. Hallquist, G. L. Goudreau, and D. J. Benson, Sliding interfaces with contact-impact in large-scale Lagrangian computations, Computer Methods in Applied Mechanics and Engineering, vol.51, issue.1-3, pp.107-137, 1985.

A. B. Chaudhary and K. Bathe, A solution method for static and dynamic analysis of threedimensional contact problems with friction, Computers & Structures, vol.24, issue.6, pp.855-873, 1986.

P. Wriggers, T. Vu, E. Van, and . Stein, Finite element formulation of large deformation impactcontact problems with friction, Computers & Structures, vol.37, issue.3, pp.319-331, 1990.

A. Klarbring, A mathematical programming approach to three-dimensional contact problems with friction, Computer Methods in Applied Mechanics and Engineering, vol.58, issue.2, pp.175-200, 1986.

N. Strömberg, An implicit method for frictional contact, impact and rolling, European Journal of Mechanics-A/Solids, vol.24, issue.6, pp.1016-1029, 2005.

F. Wang, L. Wang, J. Cheng, and Z. Yao, Contact force algorithm in explicit transient analysis using finite-element method, Finite Elements in Analysis and Design, vol.43, issue.6-7, pp.580-587, 2007.

J. Oliver, A. E. Huespe, and J. C. Cante, An implicit/explicit integration scheme to increase computability of non-linear material and contact/friction problems, Computer Methods in Applied Mechanics and Engineering, vol.197, pp.1865-1889, 2008.

M. L. Wilkins, , 1969.

F. Moirot, Etude de la stabilite d'un equilibre en presence de frottement de coulomb, 1998.

É. X. , G. A. , and L. , Assessment of performance of various shell elements in tube bending and hydroforming in computational industrial environment, presented at the X International Conference on Computational Plasticity COMPLAS X, CIMNE, 2009.

D. Vasilikis and S. A. Karamanos, Stability of confined thin-walled steel cylinders under external pressure, International Journal of Mechanical Sciences, vol.51, issue.1, pp.21-32, 2009.

D. Vasilikis and S. A. Karamanos, Buckling Design of Confined Steel Cylinders Under External Pressure, J. Pressure Vessel Technol, vol.133, issue.1, pp.11205-011205, 2010.

Y. Bai, S. Yuan, P. Cheng, P. Han, W. Ruan et al., Confined collapse of unbonded multilayer pipe subjected to external pressure, Composite Structures, vol.158, pp.1-10, 2016.

J. Jeyapalan and R. Watkins, Modulus of Soil Reaction (E?) Values for Pipeline Design, Journal of Transportation Engineering, vol.130, issue.1, pp.43-48, 2004.

A. Omara, L. Guice, W. Straughan, and F. Akl, Buckling Models of Thin Circular Pipes Encased in Rigid Cavity, Journal of Engineering Mechanics, vol.123, issue.12, pp.1294-1301, 1997.

D. Vasilikis and S. A. Karamanos, Stability of confined thin-walled steel cylinders under external pressure, International Journal of Mechanical Sciences, vol.51, issue.1, pp.21-32, 2009.

Z. Li, L. Wang, Z. Guo, and H. Shu, Elastic buckling of cylindrical pipe linings with variable thickness encased in rigid host pipes, Thin-Walled Structures, vol.51, pp.10-19, 2012.

V. M. , D. Kritisch-aussendruck-zylindrischer-rohre, and . Vdi-z, , vol.58, pp.750-755, 1914.

L. H. Donnell, Beams, plates and shells, 1976.

S. B. Batdorf, A Simplified Method of Elastic-stability Analysis for Thin Cylindrical Shells: Donnell's equation. National Advisory Committee for Aeronautics, 1947.

D. O. Brush and B. O. Almroth, Buckling of bars, plates, and shells, 1975.

S. S. Council and B. G. Johnson, Guide to Stability Design Criteria for Metal Structures, 1976.

, Stabilité des coques cylindriques et sphériques, Séminaire Flambement des structures I.T.B.T.P, Saint Rémy Les Chevreuses, Editions du Bâtiment et Travaux publics (Fr), pp.22-1980, 1981.

D. F. Windenburg, Collapse by Instability of Thin Cylindrical Shells Under External Pressure, 2010.

R. G. Sturm, A Study of the collapsing pressure of thin-walled cylinders, 1941.

G. Gerard, H. Becker, and N. Y. University, Handbook of structural stability: Strength of thinwing construction. National Aeronautics and Space Administration, 1959.

B. and A. Jc, Initial post-buckling behaviour of cylindrical shells under external pressure, J Math Phys, 1968.

C. De-paor, D. Kelliher, K. Cronin, W. M. Wright, and S. G. Mcsweeney, Prediction of vacuum-induced buckling pressures of thin-walled cylinders, Thin-Walled Structures, vol.55, pp.1-10, 2012.

J. H. Wang and A. Koizumi, Buckling of cylindrical shells with longitudinal joints under external pressure, Thin-Walled Structures, vol.48, issue.12, pp.897-904, 2010.

M. Lévy, Mémoire sur un nouveau cas intégrable du problème de l'élastique et l'une des ses applications, Journal de Mathématiques Pures et Appliquées, pp.5-42, 1884.

, Timoshenko-Theory of Elastic Stability, 2014.

S. Kyriakides and E. Corona, Mechanics of Offshore Pipelines: Volume 1 Buckling and Collapse. Amsterdam, 2007.

T. , Working stresses for columns and thin-walled structures, ASME Journal of Applied Mechanics, vol.1, pp.173-183, 1933.

W. Flügge and H. Jacot, Traduit par H. Henri Jacot, Statique et dynamique des coques : Par Wilhelm Flügge, 1960.

M. K. Yeh and S. Kyriakides, On the Collapse of Inelastic Thick-Walled Tubes Under External Pressure, J. Energy Resour. Technol, vol.108, issue.1, pp.35-47, 1986.

S. J. , .. W. , and A. , Stability of shells structures: Buckling and collapse, 2007.

N. Yamaki, Elastic stability of circular cylindrical shells, 1984.
DOI : 10.1115/1.3169089

URL : http://appliedmechanics.asmedigitalcollection.asme.org/data/journals/jamcav/26253/501_2.pdf

V. I. Seide, THE DEVELOPMENT OF DESIGN CRITERIA FOR ELASTIC STABILITY OF THIN SHELL STRUCTURES, p.218, 1960.

U. Schulz, , 1981.

L. H. Sobel, Effects of boundary conditions on the stability of cylinders subject to lateral and axial pressures, AIAA Journal, vol.2, issue.8, pp.1437-1440, 1964.

L. El-bayoumy, Buckling of a Circular Elastic Ring Confined to a Uniformly Contracting Circular Boundary, Journal of Applied Mechanics, vol.39, issue.3, pp.758-766, 1972.

K. W. , A consistent first approximation in the general theory of thin elastic shells, Proc. 1st IUTAM Symposium on Shell Theory, pp.12-33, 1960.

Y. , Post buckling and imperfection sensitivity of circular cylindrical shells under compression, Theoretical and Applied Mechanics, W.T. KOITER, 1977.

, Plastic Stability Theory of Thin Shells, Journal of the Aeronautical Sciences (Institute of the Aeronautical Sciences), vol.24, issue.4, pp.269-274, 1957.

T. A. Netto, U. S. Ferraz, and A. Botto, On the effect of corrosion defects on the collapse pressure of pipelines, International Journal of Solids and Structures, vol.44, pp.7597-7614, 2007.

P. T. Hsu, J. Elkon, and T. H. Pian, Note on the Instability of Circular Rings Confined to a Rigid Boundary, Journal of Applied Mechanics, vol.31, issue.3, pp.559-562, 1964.

R. Chicurel, Shrink Buckling of Thin Circular Rings, Journal of Applied Mechanics, vol.35, issue.3, pp.608-610, 1968.
DOI : 10.1115/1.3601259

J. Bucciarelli, L. L. , and T. H. Pian, Effect of Initial Imperfections on the Instability of a Ring Confined in an Imperfect Rigid Boundary, Journal of Applied Mechanics, vol.34, issue.4, pp.979-984, 1967.

T. C. Soong and I. Choi, Buckling of an Elastic Elliptical Ring Inside a Rigid Boundary, Journal of Applied Mechanics, vol.52, issue.3, pp.523-528, 1985.
DOI : 10.1115/1.3169094

W. J. Bottega, On the behavior of an elastic ring within a contracting cavity, International Journal of Mechanical Sciences, vol.31, issue.5, pp.349-357, 1989.

A. Omara, L. K. Guice, W. T. Straughan, and F. Akl, INSTABILITY OF THIN PIPES ENCASED IN OVAL RIGID CAVITY, Journal of Engineering Mechanics, vol.126, issue.4, 2000.
DOI : 10.1061/(asce)0733-9399(2000)126:4(381)

J. A. Cheney, Pressure Buckling of Ring Encased in Cavity, Journal of the Engineering Mechanics Division, vol.97, issue.2, pp.333-343, 1971.

I. D. Khaled-el-sawy, Stability of Loosely Fitted Liners Used to Rehabilitate Rigid Pipes, Journal of Structural Engineering-asce-J STRUCT ENG-ASCE, vol.124, issue.11, 1998.

O. Thepot, A new design method for non-circular sewer linings, Tunneling and Underground Space Technology, vol.15, issue.4, pp.25-41, 2000.

R. Montel and &. Béton, La Houille Blanche, issue.5, pp.560-568, 2011.

S. C. Aggarwal and M. Cooper, External Pressure Testing of Insituform Linings, Internal Report, 1984.

, Long-Term Structural Behavior of Pipeline Rehabilitation Systems, Journal of Infrastructure Systems, vol.1, issue.4, pp.214-220, 1995.

, Collapse Resistance of Cured in Place Pipes | NASTT-North American Society For Trenchless Technology, 2013.

K. El-sawy and I. Moore, Stability of Loosely Fitted Liners Used to Rehabilitate Rigid Pipes, Journal of Structural Engineering, vol.124, issue.11, pp.1350-1357, 1998.
DOI : 10.1061/(asce)0733-9445(1998)124:11(1350)

J. C. Boot, Elastic buckling of cylindrical pipe linings with small imperfections subject to external pressure, Tunnelling and Underground Space Technology, vol.12, pp.3-15, 1997.
DOI : 10.1016/s0886-7798(98)00018-2

K. M. El-sawy and A. M. Sweedan, Elastic stability analysis of loosely fitted thin liners-A proposed simplified procedure and evaluation of existing solutions, Tunnelling and Underground Space Technology, vol.25, issue.6, pp.689-701, 2010.

C. Madryas and A. Szot, Structural sensitivity of circular sewer liners to geometrical imperfections, Tunnelling and Underground Space Technology, vol.18, issue.4, pp.421-434, 2003.
DOI : 10.1016/s0886-7798(03)00065-8

J. Gumbel, New Approach to Design of Circular Liner Pipe to Resist External Hydrostatic Pressure, pp.1-18, 2001.
DOI : 10.1061/40574(2001)77

S. Kyriakides and S. Youn, On the collapse of circular confined rings under external pressure, International Journal of Solids and Structures, vol.20, issue.7, pp.699-713, 1984.

S. Kyriakides, Propagating buckles in long confined cylindrical shells, International Journal of Solids and Structures, vol.22, issue.12, pp.1579-1597, 1986.
DOI : 10.1016/0020-7683(86)90064-8

, Timoshenko-Theory of Elastic Stability, 2014.

R. Troisvallets, R. Montel, and &. Extérieure, La Houille Blanche, issue.5, pp.551-559, 2011.

E. C. and C. Steelwork, European Recommendations for Steel Construction. London, 1981.

J. L. Valdeolivas and J. C. Mosquera, A Full 3D Finite Element Model for Buckling Analysis of Stiffened Steel Liners in Hydroelectric Pressure Tunnels, J. Pressure Vessel Technol, vol.135, issue.6, pp.61205-061205, 2013.

W. J. Bottega, On the separation of concentric elastic rings, International Journal of Mechanical Sciences, vol.35, issue.10, pp.851-866, 1993.

F. Li and S. Kyriakjdes, On the response and stability of two concentric, contacting rings under external pressure, International Journal of Solids and Structures, vol.27, issue.1, pp.1-14, 1991.

S. Kyriakides and L. Lee, Buckle propagation in confined steel tubes, International Journal of Mechanical Sciences, vol.47, issue.4-5, pp.603-620, 2005.
DOI : 10.1016/j.ijmecsci.2004.11.009

F. S. Li and S. Kyriakides, On the Propagation Pressure of Buckles in Cylindrical Confined Shells, Journal of Applied Mechanics, vol.57, issue.4, pp.1091-1094, 1990.

K. El-sawy and I. D. Moore, Parametric Study for Buckling of Liners: Effect of Liner Geometry and Imperfections, pp.416-423

J. C. Boot and J. E. Gumbel, Structural analysis of sewer linings by B. Falter (Trenchless, Tunnelling and Underground Space Technology, vol.11, issue.2, pp.49-52, 1996.

K. M. El-sawy, Inelastic stability of liners of cylindrical conduits with local imperfection under external pressure, Tunnelling and Underground Space Technology, vol.33, pp.98-110, 2013.

M. H. Jawad, Buckling of Shells of Revolution, Theory and Design of Plate and Shell Structures, pp.338-349, 1994.

M. H. Jawad, Buckling of Cylindrical Shells, Theory and Design of Plate and Shell Structures, pp.300-337, 1994.

C. T. Ross, A Proposed Design Chart to Predict the Inelastic Buckling Pressures for Conical Shells Under Uniform External Pressure, Marine Technology, vol.44, issue.2, pp.77-81, 2007.

E. , working group 8.4, Buckling of Steel Shells: European Design Recommendations. ECCS, 2008.

, Theory and design of plate and shell structures-relié-Maan Jawad-Livre-Soldes, 2015.

I. , Methods for the assessement of departure from roundnessMeasurement of variations in radius. Multiple. Distributed through American National Standards Institute, vol.213, 1985.

, Eurocode 3-Calcul des structures en acier-Partie 1-6 : Résistance et stabilité des structures en coque, Annexe nationale à la NF EN, 1993.

G. , Überkritisches Verhalten eines Starr Ummantelten Kreisrohres bei Wasserdrunck von Aussen und Temperaturdehnung.' (Post-Critical Behavior of a Rigidly Encased Circular Pipe Subject to External Water Pressure and Thermal Extension), Der Stahlbau, vol.7, pp.212-217, 1977.

A. Omara, L. Guice, W. Straughan, and F. Akl, Buckling Models of Thin Circular Pipes Encased in Rigid Cavity, Journal of Engineering Mechanics, vol.123, issue.12, pp.1294-1301, 1997.

S. C. Aggarwal and M. J. Cooper, External pressure testing of Insituform lining, Internal Rep, 1984.

W. T. Straughan, L. K. Guice, and C. Mal-duraipandian, Long-Term Structural Behavior of Pipeline Rehabilitation Systems, Journal of Infrastructure Systems, vol.1, issue.4, pp.214-220, 1995.