, Greek indices ?, ?, ? = 1, 2 denote in-section dimensions and Latin indices i, j, k, l = 1, 2, 3, all three dimensions

M. Abambres, D. Camotim, N. Silvestre, and K. J. Rasmussen, Gbt-based structural analysis of elastic-plastic thin-walled members, Computers & Structures, vol.136, pp.1-23, 2014.

, Building Code Requirements for Structural Concrete and Commentary, ACI, vol.124, 2014.

R. L. Actis, B. A. Szabo, and C. Schwab, Hierarchic models for laminated plates and shells, Comput. Methods Appl. Mech. Eng, vol.172, issue.1-4, pp.79-107, 1999.

R. K. Al-rub and G. Z. Voyiadjis, Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro-and nanoindentation experiments, Recent Advances in Multiscale Modeling of Plasticity, vol.20, p.41, 2004.

C. Amrouche, P. G. Ciarlet, L. Gratie, and S. Kesavan, On the characterizations of matrix fields as linearized strain tensor fields, Journal de Mathématiques Pures et Appliquées, vol.86, issue.2, pp.116-132, 2006.

A. Anthoine, J. Guedes, and P. Pegon, Non-linear behaviour of reinforced concrete beams: From 3d continuum to 1d member modelling, Computers & Structures, vol.65, issue.6, pp.949-963, 1997.

J. Argyris, B. Boni, U. Hindenlang, and M. Kleiber, Finite element analysis of twoand three-dimensional elasto-plastic frames-the natural approach, Computer Methods in Applied Mechanics and Engineering, vol.35, issue.2, pp.221-248, 1982.

S. Baba and T. Kajita, Plastic analysis of torsion of a prismatic beam, International Journal for Numerical Methods in Engineering, vol.18, issue.6, p.86, 1982.

K. Bathe and A. Chaudhary, On the displacement formulation of torsion of shafts with rectangular cross-sections, International Journal for Numerical Methods in Engineering, vol.18, issue.10, p.85, 1982.

K. Bathe and P. M. Wiener, On elastic-plastic analysis of i-beams in bending and torsion, Computers and Structures, vol.17, issue.5, pp.711-718, 1983.

S. Benscoter, A theory of torsion bending for multicell beams, vol.20, p.62, 1954.

D. Bigoni and A. Piccolroaz, Yield criteria for quasibrittle and frictional materials, International Journal of Solids and Structures, vol.41, issue.11, 2004.

J. Bleyer, Méthodes numériques pour le calcul à la rupture des structures de génie civil, 2015.

J. Bleyer and P. De-buhan, Yield surface approximation for lower and upper bound yield design of 3d composite frame structures, Computers and Structures, vol.129, p.85, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00870789

B. Bognet, F. Bordeu, F. Chinesta, A. Leygue, and A. Poitou, Advanced simulation of models defined in plate geometries: 3D solutions with 2D computational complexity, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01462825

, Comput. Methods Appl. Mech. Eng, vol.84, pp.1-12

N. Boulton, Plastic twisting and bending of an i-beam in which the warp is restricted, International Journal of Mechanical Sciences, vol.4, issue.6, pp.491-502, 1962.

M. Brünig and S. Ricci, Nonlocal continuum theory of anisotropically damaged metals, International Journal of Plasticity, vol.21, issue.7, 2005.

N. Buannic and P. Cartraud, Higher-order effective modeling of periodic heterogeneous beams. I. Asymptotic expansion method, Int. J. Solids Struct, vol.38, p.53, 2001.

N. Buannic and P. Cartraud, Higher-order effective modeling of periodic heterogeneous beams. II. Derivation of the proper boundary conditions for the interior asymptotic solution, Int. J. Solids Struct, vol.38, pp.40-41, 2001.

N. N. Bui, M. Ngo, M. Nikolic, D. Brancherie, and A. Ibrahimbegovic, Enriched timoshenko beam finite element for modeling bending and shear failure of reinforced concrete frames, Computers and Structures, vol.42, p.126, 2014.
URL : https://hal.archives-ouvertes.fr/hal-02002879

I. Carol, E. Rizzi, and K. Willam, On the formulation of anisotropic elastic degradation.: Ii. generalized pseudo-rankine model for tensile damage, International Journal of Solids and Structures, vol.38, issue.4, p.41, 2001.

E. Carrera and G. Giunta, Refined beam theories based on a unified formulation, International Journal of Applied Mechanics, vol.34, issue.1, pp.117-143, 2010.

E. Carrera, G. Giunta, and M. Petrolo, Beam Structures, vol.48, p.86, 2011.

P. Ceresa, L. Petrini, and R. Pinho, Flexure-shear fiber beam-column elements for modeling frame structures under seismic loading -state of the art, Journal of Earthquake Engineering, vol.11, issue.sup1, pp.46-88, 2007.

A. Chatterjee, An introduction to the proper orthogonal decomposition, Current Science, vol.78, issue.7, pp.808-817, 2000.

W. Chen, Plasticity in Reinforced Concrete, p.124, 2007.

F. Chinesta, P. Lavevèze, and E. Cueto, A short review in model order reduction based on proper generalized decomposition, Archives od computational methods in engineering, vol.18, issue.4, p.84, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01004940

D. Christopherson, A theoretical investigation of plastic torsion in an i-beam, Journal of Applied Mechanics, vol.7, p.85, 1940.

P. G. Ciarlet and P. Ciarlet, Another approach to linearized elasticity and Korn's inequality, Comptes Rendus Mathematique, vol.339, issue.4, pp.307-312, 2004.

U. Cicekli, G. Z. Voyiadjis, A. , and R. K. , A plasticity and anisotropic damage model for plain concrete, International Journal of Plasticity, vol.23, issue.10, p.41, 2007.

G. Corre, A. Lebée, K. Sab, M. K. Ferradi, and X. Cespedes, Higher-order beam model with eigenstrains, vol.100, p.165, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01709881

G. Corre, A. Lebée, K. Sab, M. K. Ferradi, and X. Cespedes, Higher-order elastoplastic beam model, vol.142, p.165, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01691226

. De-saint and A. J. Venant, De la torsion des prismes: avec des considérations sur leur flexion ainsi que sur l'équilibre des solides élastiques en flexion. Imprimerie Imperiale, vol.15, p.48, 1855.

F. M. De-sciarra, Hardening plasticity with nonlocal strain damage, International Journal of Plasticity, vol.34, p.41, 2012.

T. De-soza, Eléments MEMBRANE et GRILLE_MEMBRANE, 2015.

J. De-vree, W. Brekelmans, and M. Van-gils, Comparison of nonlocal approaches in continuum damage mechanics, Computers & Structures, vol.55, issue.4, pp.581-588, 1995.

X. Du, D. Lu, Q. Gong, and M. Zhao, Nonlinear unified strength criterion for concrete under three-dimensional stress states, Journal of Engineering Mechanics, vol.40, 2010.

R. Echter and M. Bischoff, Numerical efficiency, locking and unlocking of {NURBS} finite elements, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.5-8, p.65, 2010.

R. El-fatmi, A refined 1D beam theory built on 3D Saint-Venant's solution to compute homogeneous and composite beams, J. Mech. Mater. Struct, vol.11, issue.4, p.62, 2016.

P. H. Feenstra and R. D. Borst, A composite plasticity model for concrete, International Journal of Solids and Structures, vol.33, issue.5, 1996.

M. K. Ferradi and X. Cespedes, A new beam element with transversal and warping eigenmodes, Comput. & Struct, vol.131, issue.7, pp.12-33, 2014.

M. K. Ferradi, X. Cespedes, and M. Arquier, A higher order beam finite element with warping eigenmodes, Eng. Struct, vol.46, issue.7, pp.748-762, 2013.

M. K. Ferradi, A. Lebée, A. Fliscounakis, X. Cespedes, and K. Sab, A model reduction technique for beam analysis with the asymptotic expansion method, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01691129

. Struct, , vol.172, p.165

J. Flejou, Réponse statique d'une poutre béton armé (section en T) à comportement linéaire, 2011.

A. Gendy and A. Saleeb, Generalized yield surface representations in the elastoplastic three-dimensional analysis of frames, Computers and Structures, vol.49, issue.2, p.85, 1993.

G. Giunta, G. De-pietro, H. Nasser, S. Belouettar, E. Carrera et al., A thermal stress finite element analysis of beam structures by hierarchical modelling, Composites Part B: Engineering, vol.95, pp.179-195, 2016.

M. Godio, I. Stefanou, K. Sab, and J. Sulem, Multisurface plasticity for cossear materials: Plate element implementation and validation, International Journal for Numerical Methods in Engineering, p.127, 2016.

R. Gonçalves, M. Ritto-corrêa, and D. Camotim, A new approach to the calculation of cross-section deformation modes in the framework of generalized beam theory, Computational Mechanics, vol.46, issue.5, pp.759-781, 2010.

S. Govindjee, G. J. Kay, and J. C. Simo, Anisotropic modelling and numerical simulation of brittle damage in concrete, International Journal for Numerical Methods in Engineering, vol.38, issue.21, p.124, 1995.

P. Grassl and M. Jirásek, Damage-plastic model for concrete failure, International Journal of Solids and Structures, vol.43, issue.22, p.124, 2006.

P. Grassl, K. Lundgren, and K. Gylltoft, Concrete in compression: a plasticity theory with a novel hardening law, International Journal of Solids and Structures, vol.39, issue.20, p.124, 2002.

E. Hansen, K. Willam, C. , and I. , A two-surface anisotropic damage/plasticity model for plain concrete, p.41, 2000.

R. Hill and M. Siebel, Lxxiii. on combined bending and twisting of thin tubes in the plastic range. The London, Edinburgh, and Dublin Philosophical Magazine, Journal of Science, vol.42, issue.330, pp.722-733, 1951.

R. Hill and M. P. Siebel, On the plastic distortion of solid bars by combined bending and twisting, Journal of the Mechanics and Physics of Solids, vol.1, issue.3, p.85, 1953.

D. H. Hodges, Nonlinear Composite Beam Theory, p.49, 2006.

S. Hsieh, E. Ting, C. , and W. , Applications of a plastic-fracture model to concrete structures, Computers & Structures, vol.28, issue.3, pp.373-393, 1988.

T. J. Hughes and R. L. Taylor, Unconditionally stable algorithms for quasi-static elasto/visco-plastic finite element analysis, Computers & Structures, vol.8, issue.2, pp.169-173, 1978.

A. Ibrahimbegovi? and F. Frey, An efficient implementation of stress resultant plasticity in analysis of Reissner-Mindlin plates, Int. J. Numer. Methods Eng, vol.36, issue.2, 1993.

. Icab, Calcul des structures en béton, vol.124, 2005.

D. Iesan, Saint-Venant's problem for inhomogeneous and anisotropic elastic bodies, J. Elast, vol.6, issue.3, p.62, 1976.

D. I. Jouravskii, Remarques sur la résistance d'un corps prismatique et d'une pièce composée en bois ou en tôle de fer à une force perpendiculaire à leur longueur, vol.12, p.60, 1856.

M. Juki?, B. Bostjan, and A. Ibrahimbegovic, Failure analysis of reinforced concrete frames by beam finite element that combines damage, plasticity and embedded discontinuity. Engineering Structures, vol.43, p.126, 2014.

J. S. Kim, M. Cho, and E. C. Smith, An asymptotic analysis of composite beams with kinematically corrected end effects, Int. J. Solids Struct, vol.45, issue.7-8, pp.1954-1977, 2008.

J. Kim and K. Wang, On the asymptotic boundary conditions of an anisotropic beam via virtual work principle, Int. J. Solids Struct, vol.48, pp.2422-2431, 2011.

A. G. Kolpakov, Problem of the theory of beams with initial stresses, Journal of Applied Mechanics and Technical Physics, vol.33, issue.6, pp.897-902, 1993.

A. G. Kolpakov, Application of homogenization method to justification of 1-D model for beam of periodic structure having initial stresses, International Journal of Solids and Structures, vol.35, issue.22, pp.2847-2859, 1998.

A. G. Kolpakov, Stressed Composite Structures: Homogenized Models for Thin-Walled Nonhomogeneous Structures with Initial Stresses. Foundations of Engineering Mechanics, p.50, 2012.

R. Krieg and S. Key, Implementation of a time dependent plasticity theory into structural computer programs, Constitutive equations in viscoplasticity: Computational and engineering aspects, vol.90, pp.125-137, 1976.

H. Kupfer, H. K. Hilsdorf, and H. Rusch, Behavior of concrete under biaxial stresses, ACI Journal Proceedings, vol.66, p.131, 1969.

L. Borderie and C. , Phénomènes unilatéraux dans un matériau endommageable: Modéli-sation et application à l'analyse des structures en béton, p.40, 1991.

P. Ladeveze, Nonlinear Computational Structural Mechanics -new approaches and non-incremental methods of calculation, Mechanical Engineering Series, vol.36, p.84, 1999.

M. Lahmar, F. Naccache, and R. Fatmi, Thermo-mechanical analysis of composite beams, Compos. Struct, vol.162, 2017.

. Leroux, Modèle multiaxial d'endommagement anisotrope: gestion numérique de la rupture et application à la ruine de structures en béton armé sous impacts, p.40, 2012.

J. L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lecture Notes in Mathematics, issue.23, 1973.

J. Lubliner, J. Oliver, S. Oller, and E. Oñate, A plastic-damage model for concrete, International Journal of Solids and Structures, vol.25, pp.299-326, 1988.

A. Marini and E. Spacone, Analysis of reinforced concrete elements including shear effects, ACI Structural Journal, vol.103, pp.645-655, 2006.

J. Mazars, A description of micro-and macroscale damage of concrete structures, Engineering Fracture Mechanics, vol.25, issue.5, p.124, 1986.

J. Mazars, P. Kotronis, F. Ragueneau, and G. Casaux, Using multifiber beams to account for shear and torsion. applications to concrete structural elements, Computational Mechanics Applied to Mechanical Engineering, vol.195, p.126, 2006.
URL : https://hal.archives-ouvertes.fr/hal-01079701

G. Meschke, R. Lackner, and A. Mang, An anisotropic elastoplastic-damage model for plain concrete, International Journal for Numerical Methods in Engineering, vol.42, pp.703-727, 1998.

B. Miara and L. Trabucho, A Galerkin spectral approximation in linearized beam theory, Math. Model. Numer. Anal, vol.26, issue.3, p.84, 1992.

J. Michel and P. Suquet, Nonuniform transformation field analysis, Int. J. Solids Struct, vol.40, issue.25, p.86, 2003.
URL : https://hal.archives-ouvertes.fr/hal-00088331

J. Michel and P. Suquet, Computational analysis of nonlinear composite structures using the nonuniform transformation field analysis, Advances in Computational Plasticity, vol.193, pp.5477-5502, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00088245

B. Miled, D. Ryckelynck, and S. Cantournet, A priori hyper-reduction method for coupled viscoelastic-viscoplastic composites, Computers and Structures, vol.119, pp.95-103, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00790175

R. V. Mises, Mechanik der festen körper im plastisch-deformablen zustand. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, vol.89, pp.582-592, 1913.

S. Moulin, Elément de poutre multifibre (droite), 2010.

A. Nadai, Der beginn des fliessvorganges in einem tordierten stab, Zeitschrift für angewande Mathematik und Mechanik, vol.29, 1923.

A. Nadai, Plasticity. McGraw-Hill, vol.29, p.85, 1931.

S. Oller and A. H. Barbat, Moment-curvature damage model for bridges subjected to seismic loads, Computational Methods Applied to Mechanical Engineering, vol.124, p.125, 2005.

P. C. Olsen, Rigid plastic analysis of plane frame structures, Computer Methods in Applied Mechanics and Engineering, vol.179, issue.1, 1999.

N. S. Ottosen, A failure criterion for concrete, Engineering Mechanics Division. Journal, vol.103, issue.4, pp.527-535, 1977.

D. R. Owen and E. Hinton, Finite elements in plasticity, vol.271, p.125, 1980.

M. Papadrakakis and V. Papadopoulos, A computationally efficient method for the limit elasto plastic analysis of space frames, Computational Mechanics, vol.16, issue.2, pp.132-141, 1995.

H. Park and J. Kim, Plasticity model using multiple failure criteria for concrete in compression, International Journal of Solids and Structures, vol.42, issue.8, 2005.

B. H. Pham, D. Brancherie, L. Davenne, and A. Ibrahimbegovic, Stress-resultant models for ultimate load design of reinforced concrete frames and multi-scale parameter estimates, Computational Mechanics, vol.42, p.126, 2012.
URL : https://hal.archives-ouvertes.fr/hal-02002878

F. Poltronieri, A. Piccolroaz, D. Bigoni, and S. R. Baivier, A simple and robust elastoplastic constitutive model for concrete, vol.60, p.41, 2014.

L. Prandtl, Zur torsion von prismatischen stäben, Physikalische Zeitschrift, vol.4, p.29, 1903.

S. Radfar, Modélisation d'éléments de structure en béton armé renforcées par collage de PRF: application à la rupture de type peeling, 2013.

S. Roussette, J. Michel, and P. Suquet, Nonuniform transformation field analysis of elastic-viscoplastic composites, Composites Science and Technology, vol.69, issue.1, p.86, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00499000

M. A. Sadowsky, An extension of the sand-heap analogy in plastic torsion applicable to cross sections having one or more holes, Journal of Applied Mechanics, vol.8, pp.166-168, 1941.

M. M. Saeed, Torsion, shear and bending in reinforced concrete beams, p.16, 1962.

M. R. Salari, S. Saeb, K. J. Willam, S. J. Patchet, and R. C. Carrasco, A coupled elastoplastic damage model for geomaterials, Computer Methods in Applied Mechanics and Engineering, vol.193, issue.27, p.124, 2004.

E. Sanchez-palencia, Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics, vol.127, p.53, 1980.

F. S. Shaw, The torsion of solid and hollow prisms in the elastic and plastic range by relaxation methods. Australian council for Aeronautics, p.29, 1944.

M. Silhavy, The Mechanics and Thermodynamics of Continuous Media. Theoretical and Mathematical Physics, p.135, 2013.

J. Simo, A finite strain beam formulation. The three-dimensional dynamic problem. Part I, Comput. Methods Appl. Mech. Eng, vol.49, issue.1, 1985.

J. Simo and T. Hughes, Computational Inelasticity. Interdisciplinary applied mathematics, vol.131, p.134, 1998.

J. Simo and R. Taylor, Consistent tangent operators for rate-independent elastoplasticity, Computer Methods in Applied Mechanics and Engineering, vol.48, issue.1, pp.101-118, 1985.

J. Smith and O. Sidebottom, Inelastic behaviour of load-carrying members, vol.29, p.85, 1965.

W. Sokolovsky, Theory of plasticity, vol.29, p.85, 1946.

E. Spacone, F. C. Filippou, and F. F. Taucer, Fibre beam-column model for nonlinear analysis of r/c frames: Part i. formulation. Earthquake Engineering and Structural Dynamics, vol.25, p.125, 1996.

E. Spacone and S. Limkatanyu, Response of rc members including bond-slip effects, ACI Structural Journal, vol.43, p.125, 2000.

, Note interne, STRAINS, p.142, 2016.

, CEB-FIP model code 90, Comité Euro International du Béton, p.160, 1993.

S. Timoshenko, On the transverse vibrations of bars of uniform cross-section, Philos. Mag. Ser, vol.6, issue.253, p.48, 1922.

B. ?tok and M. Halilovi?, Analytical solutions in elasto-plastic bending of beams with rectangular cross section, Applied Mathematical Modelling, vol.33, issue.3, pp.1749-1760, 2009.

L. Trabucho and J. Viaño, Derivation of generalized models for linear elastic beams by asymptotic expansion methods. Applications of multiple scaling in mechanics, Proc. Int. Conf, vol.4, p.24, 1986.

L. Trabucho and J. Viaño, A new approach of timoshenko's BEAM theory by asymptotic expansion method, Math. Model. Numer. Anal, vol.24, issue.5, pp.651-680, 1990.

L. Trabucho and J. Viaño, Mathematical modelling of rods, Handb. Numer. Anal, vol.4, p.53, 1996.

L. Trabucho and J. M. Viano, A derivation of generalized saint venant's torsion theory from three dimensional elasticity by asymptotic expansion methods, Journal of Applicable Analysis, vol.31, pp.129-148, 1988.

L. Trabucho and J. M. Viano, Existence and characterization of higher-order terms in an asymptotic expansion method for linearized elastic beams, Asymptotic Analysis, vol.2, pp.223-255, 1989.

G. C. Tsiatas and N. G. Babouskos, Elastic-plastic analysis of functionally graded bars under torsional loading, Composite Structures, vol.176, pp.254-267, 2017.

P. Vidal, L. Gallimard, and O. Polit, Composite beam finite element based on the proper generalized decomposition, Computers & Structures, vol.36, pp.102-103, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01366924

V. Z. Vlasov, Thin-walled elastic beams. National Science Foundation and Department of Commerce, vol.18, p.49, 1961.

M. Vogelius and I. Babuska, On a Dimensional Reduction Method I. The Optimal Selection of Basis Functions, Math. Comput, vol.37, issue.155, 1981.

M. Vogelius and I. Babuska, On a Dimensional Reduction Method II. Some Approximation-Theoretic Results, Math. Comput, vol.37, issue.155, 1981.

M. L. Wilkins, Methods in Computational Physics, chapter Calculation of ElasticPlastic Flow, p.90, 1964.

K. Willam and E. Warnke, Constitutive model for the triaxial behavior of concrete, Proceedings seminar on concrete structures subjected to triaxial stresses, p.40, 1974.

J. Y. Wu, J. Li, and R. Faria, An energy release rate-based plastic-damage model for concrete, International Journal of Solids and Structures, vol.43, issue.3, p.41, 2006.

W. Yu and D. H. Hodges, Elasticity Solutions Versus Asymptotic Sectional Analysis of Homogeneous, Isotropic, Prismatic Beams, J. Appl. Mech, vol.71, pp.15-23, 2004.

W. Yu, D. H. Hodges, and J. C. Ho, Variational asymptotic beam sectional analysis -An updated version, Int. J. Eng. Sci, vol.59, pp.40-64, 2012.

W. Yu, V. V. Volovoi, D. H. Hodges, and X. Hong, Validation of the Variational Asymptotic Beam Sectional Analysis, AIAA J, vol.40, issue.10, pp.2105-2112, 2002.

Q. Zhao, P. Cartraud, and P. Kotronis, Justification of the asymptotic expansion method for homogeneous isotropic beams by comparison with the Saint-Venant solution, J. Elast, vol.53, p.56, 2015.

O. C. Zienkiewicz, The finite element method. McGraw-Hill London, p.107, 1977.