Refined Gradient Inelastic Flexibility-Based Formulation for Members Subjected to Arbitrary Loading
Publication: Journal of Engineering Mechanics
Volume 143, Issue 9
Abstract
This paper advances the gradient inelastic (GI) flexibility-based (FB) frame element formulation, which focused on monotonic loading conditions, to capture member responses to arbitrary (nonmonotonic) loading conditions. The GI formulation is a generalization of the strain gradient elasticity theory to inelastic continua. Contrary to nonlocal/gradient damage or plasticity models, in the GI theory, nonlocality is strictly decoupled from the constitutive relations; as a result, the GI theory can incorporate any material constitutive law (plastic, hardening, or softening), whereas for linear elastic materials, the GI theory reduces to the strain gradient elasticity theory. In the GI theory, alleviation of strain localization and response objectivity (i.e., convergence with mesh refinements) are achieved through a localization condition applied to the strain field at strain localization locations during strain softening. Strain localization locations are not a priori known and are identified through a localization criterion. This paper advances/refines the GI theory and GI FB element formulation by (1) eliminating (unintended and unphysical) discontinuities in the temporal response of section strains introduced during application of the localization condition; and (2) alleviating false identifications of localization locations in the case of arbitrary and combined (axial/flexural/shear) loading through a new robust localization criterion. Furthermore, this paper assesses (1) various mathematically admissible end boundary conditions to the section strain fields—proper selection of which has been a challenge in higher order gradient theories—in terms of their physical rationale and mesh convergence properties; and (2) higher-order nonlocality relations in terms of the spatial characteristics of the resulting section strain fields and mesh convergence properties. The performance of the proposed formulation is evaluated through examples on frame members subjected to monotonic, cyclic, and seismic loading, and comparisons with experimental data from quasi-static cyclic testing of a reinforced concrete column.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
Partial support for this research has been provided by the National Science Foundation (NSF) under Award No. CMMI 1538585. This support is gratefully acknowledged. The opinions, findings, and conclusions expressed in this publication are those of the authors and do not necessarily reflect the views of the NSF.
References
Addessi, D., and Ciampi, V. (2007). “A regularized force-based beam element with a damage-plastic section constitutive law.” Int. J. Numer. Methods Eng., 70(5), 610–629.
Aifantis, E. C. (1999). “Gradient deformation models at nano, micro, and macro scales.” J. Eng. Mater. Technol., 121(2), 189–202.
Aifantis, E. C. (2003). “Update on a class of gradient theories.” Mech. Mater., 35(3), 259–280.
Alemdar, Z. F. (2010). “Plastic hinging behavior of reinforced concrete bridge columns.” Ph.D. dissertation, Univ. of Kansas, Lawrence, KS.
Almeida, J. P., Das, S., and Pinho, R. (2012). “Adaptive force-based frame element for regularized softening response.” Comput. Struct., 102, 1–13.
Altan, S., and Aifantis, E. (1992). “On the structure of the Mode III crack-tip in gradient elasticity.” Scr. Metall. Mater., 26(2), 319–324.
Bažant, Z. P., and Belytschko, T. B. (1985). “Wave propagation in a strain-softening bar: Exact solution.” J. Eng. Mech., 381–389.
Bažant, Z. P., and Chang, T. P. (1987). “Nonlocal finite element analysis of strain-softening solids.” J. Eng. Mech., 89–105.
Bažant, Z. P., and Jirásek, M. (2002). “Nonlocal integral formulations of plasticity and damage: Survey of progress.” J. Eng. Mech., 1119–1149.
Bažant, Z. P., and Oh, B. H. (1983). “Crack band theory for fracture of concrete.” Mater. Struct., 16(3), 155–177.
Bažant, Z. P., and Planas, J. (1998). Fracture and size effect in concrete and other quasibrittle materials, CRC, Boca Raton, FL.
Benvenuti, E., Borino, G., and Tralli, A. (2002). “A thermodynamically consistent nonlocal formulation for damaging materials.” Eur. J. Mech. A/Solids, 21(4), 535–553.
Challamel, N. (2010). “A variationally based nonlocal damage model to predict diffuse microcracking evolution.” Int. J. Mech. Sci., 52(12), 1783–1800.
Challamel, N., Casandjian, C., and Lanos, C. (2009). “Some closed-form solutions to simple beam problems using nonlocal (gradient) damage theory.” Int. J. Damage Mech., 18(6), 569–598.
Challamel, N., and Hjiaj, M. (2005). “Non-local behavior of plastic softening beams.” Acta Mech., 178(3), 125–146.
Challamel, N., Lanos, C., and Casandjian, C. (2008). “Plastic failure of nonlocal beams.” Phys. Rev. E, 78(2), 026604.
Challamel, N., Lanos, C., and Casandjian, C. (2010). “On the propagation of localization in the plasticity collapse of hardening-softening beams.” Int. J. Eng. Sci., 48(5), 487–506.
Challamel, N., and Wang, C. (2008). “The small length scale effect for a non-local cantilever beam: A paradox solved.” Nanotechnology, 19(34), 345703.
Coleman, J., and Spacone, E. (2001). “Localization issues in force-based frame elements.” J. Struct. Eng., 1257–1265.
de Borst, R. (1993). “A generalisation of J2-flow theory for polar continua.” Comput. Methods Appl. Mech. Eng., 103(3), 347–362.
de Borst, R., Pamin, J., Peerlings, R., and Sluys, L. (1995). “On gradient-enhanced damage and plasticity models for failure in quasi-brittle and frictional materials.” Comput. Mech., 17(1), 130–141.
Dell’Isola, F., Andreaus, U., and Placidi, L. (2015a). “At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola.” Math. Mech. Solids, 20(8), 887–928.
Dell’Isola, F., Della Corte, A., Esposito, R., and Russo, L. (2016). “Some cases of unrecognized transmission of scientific knowledge: From antiquity to Gabrio Piola’s Peridynamics and generalized continuum theories.” Generalized continua as models for classical and advanced materials, Springer, Switzerland, 77–128.
Dell’Isola, F., Della Corte, A., and Giorgio, I. (2017). “Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives.” Math. Mech. Solids, 22(4), 852–872.
Dell’Isola, F., Seppecher, P., and Della Corte, A. (2015b). “The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: A review of existing results.” Proc. R. Soc. A, 471(2183), 20150415.
Dides, M. A., and De la Llera, J. C. (2005). “A comparative study of concentrated plasticity models in dynamic analysis of building structures.” Earthquake Eng. Struct. Dyn., 34(8), 1005–1026.
Engelen, R. A. B., Geers, M. G. D., and Baaijens, F. (2003). “Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour.” Int. J. Plasticity, 19(4), 403–433.
Eringen, A. C. (1983). “On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves.” J. Appl. Phys., 54(9), 4703–4710.
Eringen, A. C., and Edelen, D. G. B. (1972). “On nonlocal elasticity.” Int. J. Eng. Sci., 10(3), 233–248.
Filippou, F. C., Popov, E. P., and Bertero, V. V. (1983). “Effects of bond deterioration on hysteretic behavior of reinforced concrete joints.”, Earthquake Engineering Research Center, Berkeley, CA.
Fleck, N., and Hutchinson, J. (1997). Strain gradient plasticity (advances in applied mechanics), Vol. 33, Elsevier, New York.
Fleck, N., Muller, G., Ashby, M., and Hutchinson, J. (1994). “Strain gradient plasticity: Theory and experiment.” Acta Metall. Mater., 42(2), 475–487.
Gauthier, R. D., and Jahsman, W. E. (1975). “A quest for micropolar elastic constants.” J. Appl. Mech., 42(2), 369–374.
Germain, P. (1973). “The method of virtual power in continuum mechanics. II: Microstructure.” SIAM J. Appl. Math., 25(3), 556–575.
Gutkin, M. Y., and Aifantis, E. (1996). “Screw dislocation in gradient elasticity.” Scr. Mater., 35(11), 1353–1358.
Hillerborg, A., Modéer, M., and Petersson, P. E. (1976). “Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements.” Cem. Concr. Res., 6(6), 773–781.
Hwang, K., Jiang, H., Huang, Y., Gao, H., and Hu, N. (2002). “A finite deformation theory of strain gradient plasticity.” J. Mech. Phys. Solids, 50(1), 81–99.
Jirásek, M. (1998). “Nonlocal models for damage and fracture: Comparison of approaches.” Int. J. Solids Struct., 35(31), 4133–4145.
Jirásek, M., Rokoš, O., and Zeman, J. (2013). “Localization analysis of variationally based gradient plasticity model.” Int. J. Solids Struct., 50(1), 256–269.
Jirásek, M., and Rolshoven, S. (2003). “Comparison of integral-type nonlocal plasticity models for strain-softening materials.” Int. J. Eng. Sci., 41(13), 1553–1602.
Jirásek, M., and Zeman, J. (2015). “Localization study of a regularized variational damage model.” Int. J. Solids Struct., 69, 131–151.
Kaewkulchai, G., and Williamson, E. B. (2004). “Beam element formulation and solution procedure for dynamic progressive collapse analysis.” Comput. Struct., 82(7), 639–651.
Kafadar, C., and Eringen, A. C. (1971). “Micropolar media—I. The classical theory.” Int. J. Eng. Sci., 9(3), 271–305.
Lasry, D., and Belytschko, T. (1988). “Localization limiters in transient problems.” Int. J. Solids Struct., 24(6), 581–597.
Lee, C. L., and Filippou, F. C. (2009). “Efficient beam-column element with variable inelastic end zones.” J. Struct. Eng., 1310–1319.
Maugin, G. (1998). “On the structure of the theory of polar elasticity.” Philos. Trans. R. Soc. London A Math., Phys. Eng. Sci., 356(1741), 1367–1395.
Mazza, F., and Mazza, M. (2010). “Nonlinear analysis of spatial framed structures by a lumped plasticity model based on the Haar-Kàrmàn principle.” Comput. Mech., 45(6), 647–664.
McKenna, F., and Fenves, G. L. (2001). The OpenSees command language manual, Univ. of California, Berkeley, CA.
Menegotto, M., and Pinto, P. (1973). “Method of analysis for cyclically loaded reinforced concrete plane frames including changes in geometry and nonelastic behavior of elements under combined normal force and bending.” IABSE Symp. on Resistance and Ultimate Deformability of Structures Acted on by Well-Defined Repeated Loads, Final Rep., Lisbon, Portugal.
Mullapudi, T., and Ayoub, A. (2013). “Analysis of reinforced concrete columns subjected to combined axial, flexure, shear, and torsional loads.” J. Struct. Eng., 561–573.
Peerlings, R. H. J., De Borst, R., Brekelmans, W. A. M., and De Vree, J. H. P. (1996a). “Gradient enhanced damage for quasi-brittle materials.” Int. J. Numer. Methods Eng., 39(19), 3391–3403.
Peerlings, R. H. J., De Borst, R., Brekelmans, W. A. M., De Vree, J. H. P., and Spee, I. (1996b). “Some observations on localisation in non-local and gradient damage models.” Eur. J. Mech. A Solids, 15(6), 937–953.
Peerlings, R. H. J., Geers, M. G. D., de Borst, R., and Brekelmans, W. A. M. (2001). “A critical comparison of nonlocal and gradient-enhanced softening continua.” Int. J. Solids Struct., 38(44–45), 7723–7746.
Pijaudier-Cabot, G., and Bazant, Z. P. (1987). “Nonlocal damage theory.” J. Eng. Mech., 1512–1533.
Placidi, L. (2016). “A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model.” Continuum Mech. Thermodyn., 28(1–2), 119–137.
Polizzotto, C. (2001). “Nonlocal elasticity and related variational principles.” Int. J. Solids Struct., 38(42), 7359–7380.
Polizzotto, C. (2003). “Gradient elasticity and nonstandard boundary conditions.” Int. J. Solids Struct., 40(26), 7399–7423.
Powell, G. H., and Chen, P. F.-S. (1986). “3D beam-column element with generalized plastic hinges.” J. Eng. Mech., 627–641.
Ribeiro, F. L., Barbosa, A. R., Scott, M. H., and Neves, L. C. (2015). “Deterioration modeling of steel moment resisting frames using finite-length plastic hinge force-based beam-column elements.” J. Struct. Eng., 04014112.
Ristinmaa, M., and Vecchi, M. (1996). “Use of couple-stress theory in elasto-plasticity.” Comput. Methods Appl. Mech. Eng., 136(3), 205–224.
Rokoš, O., Zeman, J., and Jirásek, M. (2016). “Localization analysis of an energy-based fourth-order gradient plasticity model.” Eur. J. Mech. A/Solids, 55, 256–277.
Ru, C., and Aifantis, E. (1993). “A simple approach to solve boundary-value problems in gradient elasticity.” Acta Mech., 101(1–4), 59–68.
Salehi, M., Sideris, P., and Liel, A. (2017). “Seismic collapse analysis of RC framed structures using the gradient inelastic force-based element formulation.” 16th World Conf. on Earthquake Engineering (16WCEE), Earthquake Engineering Research Institute, Oakland, CA.
Santoro, M. G., and Kunnath, S. K. (2013). “Damage-based RC beam element for nonlinear structural analysis.” Eng. Struct., 49, 733–742.
Scott, B. D., Park, R., and Priestley, M. J. N. (1982). “Stress-strain behavior of concrete confined by overlapping hoops at low and high strain rates.” ACI J., 79(1), 13–27.
Scott, M. H., and Fenves, G. L. (2006). “Plastic hinge integration methods for force-based beam-column elements.” J. Struct. Eng., 244–252.
Scott, M. H., and Hamutçuoğlu, O. M. (2008). “Numerically consistent regularization of force-based frame elements.” Int. J. Numer. Methods Eng., 76(10), 1612–1631.
Sideris, P., and Salehi, M. (2016). “A gradient-inelastic flexibility-based frame element formulation.” J. Eng. Mech., 04016039.
Tanaka, H. (1990). “Effect of lateral confining reinforcement on the ductile behaviour of reinforced concrete columns.” Dept. of Civil Engineering, Univ. of Canterbury, Christchurch, New Zealand.
Triantafyllidis, N., and Aifantis, E. C. (1986). “A gradient approach to localization of deformation. I: Hyperelastic materials.” J. Elast., 16(3), 225–237.
Triantafyllou, A., Perdikaris, P. C., and Giannakopoulos, A. E. (2015). “Gradient elastodamage model for quasi-brittle materials with an evolving internal length.” J. Eng. Mech., 04014139.
Valanis, K. C. (1985). “On the uniqueness of solution of the initial value problem in softening materials.” J. Appl. Mech., 52(3), 649–653.
Valipour, H. R., and Foster, S. J. (2009). “Nonlocal damage formulation for a flexibility-based frame element.” J. Struct. Eng., 1213–1221.
Valipour, H. R., and Foster, S. J. (2010). “A total secant flexibility-based formulation for frame elements with physical and geometrical nonlinearities.” Finite Elem. Anal. Des., 46(3), 288–297.
Vardoulakis, I., and Aifantis, E. (1991). “A gradient flow theory of plasticity for granular materials.” Acta Mech., 87(3–4), 197–217.
Yang, Y., Ching, W., and Misra, A. (2011). “Higher-order continuum theory applied to fracture simulation of nanoscale intergranular glassy film.” J. Nanomechanics Micromech., 60–71.
Zervos, A., Papanastasiou, P., and Vardoulakis, I. (2001). “A finite element displacement formulation for gradient elastoplasticity.” Int. J. Numer. Methods Eng., 50(6), 1369–1388.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 143 • Issue 9 • September 2017
Copyright
©2017 American Society of Civil Engineers.
History
Received: Apr 9, 2016
Accepted: Feb 17, 2017
Published online: Jun 15, 2017
Published in print: Sep 1, 2017
Discussion open until: Nov 15, 2017
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.