Summary
This paper deals with the modelling of a plastic beam experiencing softening. This kind of behaviour is observed in steel or reinforced concrete structural members undergoing large rotation amplitudes, which may occur typically for civil engineering structures in the seismic area. The homogeneous cantilever beam loaded by a concentrated force at its extremity is considered. This simple structural problem with gradient bending moment allows an analytical treatment of the evolution problem. It is shown that a local plastic softening model makes the evolution problem ill-posed. Moreover, if we require the plastic curvature to be a continuous variable of the spatial coordinate, Wood’s paradox is encountered. A non-local gradient plastic model is developed in order to overcome this paradox. However classical gradient plastic models may not eliminate the ill-posedness since the beam response may not be continuous with respect to the loading parameter. The new gradient plastic model, presented in this paper, is similar to previous classical gradient models. The main difference is the yield moment considered as a non-local material parameter. This permits to ensure continuity between the elastic and the plastic regions during the loading process. These solutions are controlled by the ratio between the material length and the geometrical length of the beam. The new evolution problem may remain ill-posed as it possesses a finite number of solutions (which can be unique). Closed-form solutions of the unknown deflection are given.
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References
F. M. Mazzolani V. Piluso (1996) Theory and design of seismic resistant steel frames Chapman & Hall London
A. R. Kemp (1985) ArticleTitleInteraction of plastic local and lateral buckling J. Struct. Div. A.S.C.E. 111 2181–2197
A. F. Luckey P. F. Adams (1969) ArticleTitleRotation capacity of beams under moment gradient J. Struct. Div A.S.C.E. 95 1173–1188
U. Kuhlmann (1989) ArticleTitleDefinition of flange slenderness limits on the basis of rotation capacity values J. Constr. Steel Res. 14 21–40 Occurrence Handle10.1016/0143-974X(89)90068-0
G. Ballio C. A. Castiglioni (1994) ArticleTitleSeismic behavior of steel sections J. Constr. Steel Res. 29 21–24 Occurrence Handle10.1016/0143-974X(94)90055-8
R. Spangemacher G. Sedlasek (1992) ArticleTitleZum Nachweis ausreichender Rotationsfähigkeit von Fließgelenkverfahren Stahlbau 61 329–339
P. Boeraeve B. Lognard J. Janss J. C. Gérardy J. B. Schleich (1993) ArticleTitleElasto-plastic behavior of steel frame works J. Constr. Steel Res. 27 3–21 Occurrence Handle10.1016/0143-974X(93)90003-B
R. H. Wood (1968) Some controversial and curious developments in the plastic theory of structures J. Heyman F. A. Leckie (Eds) Engineering plasticity Cambridge University Press Cambridge 665–691
Jirasek, M., Bazant, Z. P.: Inelastic analysis of structures. New York: Wiley 2002.
R. Hill (1958) ArticleTitleA general theory of uniqueness and stability in elastic-plastic solids J. Mech. Phys. Solids. 6 236–249 Occurrence Handle10.1016/0022-5096(58)90029-2
G. Pijaudier-Cabot Z. P. Bazant (1987) ArticleTitleNon-local damage theory J. Eng. Mech. 113 1512–1533
A. Huerta G. Pijaudier-Cabot (1994) ArticleTitleDiscretization influence on regularization by two localization limiters J. Eng. Mech. 120 1198–1218
R. H. J. Peerlings M. G. D. Geers R. Borst Particlede W. A. M. Brekelmans (2001) ArticleTitleA critical comparison of non-local and gradient-enhanced softening continua Int. J. Solids Struct. 38 7723–7746 Occurrence Handle10.1016/S0020-7683(01)00087-7
Z. P. Bazant A. Zubelewicz (1988) ArticleTitleStrain softening bar and beam: exact nonlocal solution Int. J. Solids Struct. 24 659–673 Occurrence Handle10.1016/0020-7683(88)90015-7
H. L. Schreyer Z. Chen (1986) ArticleTitleOne-dimensional softening with localization J. Appl. Mech. 53 791–797
R. Borst Particlede H. B. Mühlhaus (1992) ArticleTitleGradient-dependent plasticity: formulation and algorithmic aspects Int. J. Num. Meth. Engng. 35 521–539
Nilsson, C.: Non-local strain softening bar revisited. Int. J. Solids Struct. 34, 33–34, 4399–4419 (1997).
C. Polizotto G. Borino (1998) ArticleTitleA thermodynamics-based formulation of gradient-dependent plasticity Eur. J. Mech. A/Solids 17 741–761 Occurrence Handle10.1016/S0997-7538(98)80003-X
H. B. Mühlhaus E. C. Aifantis (1991) ArticleTitleA variational principle for gradient plasticity Int. J. Solids Struct. 28 845–857 Occurrence Handle10.1016/0020-7683(91)90004-Y
Zbib, H., Aifantis, E.C.: On the localization and post-localization behavior of plastic deformation, I, II, III. Res. Mech. 23, 261–277, 279–292, 293–305 (1988).
Galilée: Discorsi e Dimonstrazioni Matematiche, intorno á due nuove Scienze, 1638. In: Sur les épaules des géants –les plus grands textes de physique et d’astronomie (Hawking, S., ed.), pp. 154– 182. Paris: Dunod 2002
Salençon, J. (1990). An introduction to the yield design theory and its application to soil mechanics. Eur J. Mech. A/Solids 9, 5, 477–500
G. Maier V. Carvelli G. Cocchetti (2000) ArticleTitleOn direct methods for shakedown and limit analysis Eur. J Mech. A/Solids 19 79–100
B. Halphen Q. S. Nguyen (1975) ArticleTitleSur les matériaux standards généralisés J. Mécanique 14 39–63
N. Challamel (2003) ArticleTitleUne approche de plasticité au gradient en construction métallique C. R. Acad. Sci 331 647–654
G. I. Barenblatt (1987) Dimensional analysis Gordon and Breach London
Z. P. Bazant (1976) ArticleTitleInstability, ductility and size effect in strain softening concrete J. Engng. Mech. 102 331–344
R. Chambon D. Caillerie N. El Hassan (1998) ArticleTitleOne-dimensional localization studied with a second grade model Eur. J. Mech. A/Solids 17 637–656 Occurrence Handle10.1016/S0997-7538(99)80026-6
R. Chambon J. C. Moullet (2004) ArticleTitleUniqueness studies in boundary value problems involving some second gradient models Comput. Meth. Appl. Mech. Engng. 193 2771–2796 Occurrence Handle10.1016/j.cma.2003.10.017
G. Engeln-Müllges F. Uhlig (1996) Numerical algorithms with Fortran Springer Berlin
Massonnet, C., Save, M.: Calcul plastique des constructions. Centre Belgo-Luxembourgeois d’information de l’acier. Bruxelles (1961).
Salençon, J.: Calcul á la rupture et analyse limite. Presses de l’Ecole Nationale des Ponts et Chaussées (1983).
G. Royer-Carfagni (2001) ArticleTitleCan a moment-curvature relationship describe the flexion of softening beams? Eur. J. Mech. A/Solids 20 253–276 Occurrence Handle10.1016/S0997-7538(00)01128-1
G. Maier A. Zavelani J. C. Dotreppe (1973) ArticleTitleEquilibrium branching due to flexural softening J. Engng. Mech. 99 897–901
G. Cocchetti G. Maier (2003) ArticleTitleElastic-plastic and limit-state analyses of frames with softening plastichinge models by mathematical programming Int. J. Solids Struct. 40 7219–7244 Occurrence Handle10.1016/S0020-7683(03)00363-9 Occurrence HandleMR2042403
M. Jirasek (1997) ArticleTitleAnalytical and numerical solutions for frames with softening hinges J. Engng. Mech 123 8–14 Occurrence Handle10.1061/(ASCE)0733-9399(1997)123:1(8)
M. A. Marante R. Picon J. Florez-Lopez (2004) ArticleTitleAnalysis of localization in frame members with plastic hinges Int. J. Solids Struct. 41 3961–3975 Occurrence Handle10.1016/j.ijsolstr.2004.02.014
Z. P. Bazant (2001) ArticleTitleScaling of failure of beams, frames and plates with softening hinges Meccanica 36 67–77 Occurrence Handle10.1023/A:1011913403727
P. L. Darvall (1984) ArticleTitleCritical softening of hinges in portal frames J. Struct. Engng. 110 157–162
P. L. Darvall P. A. Mendis (1985) ArticleTitleElastic-plastic-softening analysis of plane frames J. Struct. Engng. 111 871–888
A. Hillerborg M. Modeer P. E. Petersson (1976) ArticleTitleAnalysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements Cement Concrete Res. 6 773–782 Occurrence Handle10.1016/0008-8846(76)90007-7
Z. P. Bazant G. Pijaudier-Cabot J. Pan (1987) ArticleTitleDuctility, snap-back, size effect, and redistribution in softening beams or frames J. Struct. Engng. 113 2348–2364
Z. M. Wang M. L. Peterson (2001) ArticleTitleVariational principles for softening gradient dependent plasticity Int. J. Solids Struct. 38 8685–8700 Occurrence Handle10.1016/S0020-7683(01)00053-1
C. Polizotto (2003) ArticleTitleUnified thermodynamic framework for non-local/gradient continuum theories Eur J. Mech. A/Solids 22 651–668 Occurrence Handle10.1016/S0997-7538(03)00075-5
R. Borst Particlede J. Pamin (1996) ArticleTitleGradient plasticity in numerical simulation of concrete cracking Eur. J Mech. A/Solids 15 295–320
A. N. Tikhonov V. Y. Arsenine (1977) Solutions to ill-posed problems Winston-Wiley New York
Hutchinson, J. W.: Plastic buckling. Adv. Appl. Mech., vol. 14. Academic Press, New York, pp. 67–114 (1974).
Q. S. Nguyen (2000) Stabilité et mécanique non linéaire Hermés Paris
Q. S. Nguyen (1998) ArticleTitleComparaison des énergies dépensées en flambage plastique C. R. Acad. Sci. 326 353–358
T. Svedberg K. Runesson (1997) ArticleTitleA thermodynamically consistent theory of gradient-regularized plasticity coupled to damage Int. J. Plasticity 13 669–696 Occurrence Handle10.1016/S0749-6419(97)00033-8
E. Lorentz S. Andrieux (2003) ArticleTitleAnalysis of non-local models through energetic formulations Int. J Solids Struct. 40 2905–2936 Occurrence Handle10.1016/S0020-7683(03)00110-0
J. J. Climenhaga P. P. Johnson (1972) ArticleTitleMoment-rotation curves for locally buckling beams J. Struct Div. 98 1239–54
F. M. Mazzolani V. Gioncu (2002) Ductility of seismic-resistant steel structures Spon Press London
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Challamel, N., Hjiaj, M. Non-local behavior of plastic softening beams. Acta Mechanica 178, 125–146 (2005). https://doi.org/10.1007/s00707-005-0225-3
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DOI: https://doi.org/10.1007/s00707-005-0225-3