A fibre Smart Displacement Based (FSDB) beam element for the nonlinear analysis of R/C members

Abstract Beam finite elements for non linear plastic analysis of beam-like structures are formulated according to Displacement Based (DB) or Force Based (FB) approaches. DB formulations rely on modelling the displacement field by means of displacement shape functions. Despite the greater simplicity of DB over FB approaches, the latter provide more accurate responses for those requiring a coarser mesh. In order to fill the existing gap between the two approaches, improvement of the DB formulation without the introduction of mesh refinement, is necessary. To this aim, the authors recently provided a contribution to the improvement of the DB approach by proposing new enriched, adaptive displacement shape functions, leading to the Smart Displacement Based (SDB) beam element. In this paper the SDB element is extended to include the axial force-bending moment interaction, crucial for the analysis of reinforced concrete (r/c) cross sections. The proposed extension requires the formulation of discontinuous axial displacement shape functions which are dependent on the diffusion of plastic deformations. The stiffness matrix of the extended smart element is provided explicitly and is dependent on the displacement shape functions updating. The axial force-bending moment interaction is approached by means of a fibre discretisation of the r/c cross section. The extended element, addressed as Fibre Smart Displacement Based (FSDB) beam element, is shown to be accurate furthermore, it is accompanied by an optional procedure proposal in order to verify the strong equilibrium of the axial force along the beam element, which is not usually accomplished by DB beam elements. Given a fixed mesh discretisation, the performance of the FSDB beam element is compared with the DB approach to show the higher degree of accuracy in the proposed element.

[1]  Amr S. Elnashai,et al.  Advanced Nonlinear Formulation for Reinforced Concrete Beam-Columns , 1994 .

[2]  Rui Pinho,et al.  Numerical Issues in Distributed Inelasticity Modeling of RC Frame Elements for Seismic Analysis , 2010 .

[3]  Filip C. Filippou,et al.  Evaluation of Nonlinear Frame Finite-Element Models , 1997 .

[4]  Bartolomeo Pantò,et al.  A smart displacement based (SDB) beam element with distributed plasticity , 2017 .

[5]  F. Filippou,et al.  Mixed formulation of nonlinear beam finite element , 1996 .

[6]  S. Caddemi,et al.  Closed-form solutions for stepped Timoshenko beams with internal singularities and along-axis external supports , 2013 .

[7]  Michael H. Scott,et al.  Plastic Hinge Integration Methods for Force-Based Beam¿Column Elements , 2006 .

[8]  Amr S. Elnashai,et al.  ADAPTIVE SPACE FRAME ANALYSIS. PART I: A PLASTIC HINGE APPROACH. , 1993 .

[9]  Katrin Beyer,et al.  Modelling Approaches for Inelastic Behaviour of RC Walls: Multi-level Assessment and Dependability of Results , 2016 .

[10]  J. Colombeau,et al.  New Generalized Functions and Multiplication of Distributions , 2012 .

[11]  S. Caddemi,et al.  Closed form solutions of Euler–Bernoulli beams with singularities , 2005 .

[12]  H. Bremermann,et al.  On Analytic Continuation, Multiplication, and Fourier Transformations of Schwartz Distributions , 1961 .

[13]  Frank McKenna,et al.  OpenSees: A Framework for Earthquake Engineering Simulation , 2011, Computing in Science & Engineering.

[14]  S. Lang,et al.  An Introduction to Fourier Analysis and Generalised Functions , 1959 .

[15]  Ivo Caliò,et al.  A novel beam finite element with singularities for the dynamic analysis of discontinuous frames , 2013 .

[16]  김지은,et al.  모멘트-곡률 관계에 기초한 철근콘크리트 보의 비선형 해석 = Nonlinear analysis of RC beams based on moment-curvature relations , 2002 .

[17]  M. Crisfield,et al.  Finite Elements and Solution Procedures for Structural Analysis , 1986 .

[18]  Bassam A. Izzuddin,et al.  An efficient beam–column formulation for 3D reinforced concrete frames , 2002 .

[19]  D. Addessi,et al.  A regularized force‐based beam element with a damage–plastic section constitutive law , 2007 .

[20]  Michael H. Scott,et al.  Numerically consistent regularization of force‐based frame elements , 2008 .

[21]  Domenico Liberatore,et al.  Force-Based Beam Finite Element (FE) for the Pushover Analysis of Masonry Buildings , 2015 .

[22]  Amr S. Elnashai,et al.  ADAPTIVE SPACE FRAME ANALYSIS. PART II: A DISTRIBUTED PLASTICITY APPROACH. , 1993 .

[23]  Afsin Saritas,et al.  Variational base and solution strategies for non-linear force-based beam finite elements , 2012 .

[24]  R. Park,et al.  Flexural Members with Confined Concrete , 1971 .

[25]  R. Taylor.  The Finite Element Method, the Basis , 2000 .

[26]  A. Zemanian,et al.  Distribution theory and transform analysis , 1966 .

[27]  F. Filippou,et al.  Geometrically Nonlinear Flexibility-Based Frame Finite Element , 1998 .

[28]  K. Bathe.  Finite Element Procedures , 1995 .

[29]  John Sharp,et al.  Microsoft Visual C♯ 2005 step by step , 2005 .

[30]  Giuseppe Giambanco,et al.  The multiple slope discontinuity beam element for nonlinear analysis of RC framed structures , 2018 .

[31]  M. Menegotto.  Method of Analysis for Cyclically Loaded R. C. Plane Frames Including Changes in Geometry and Non-Elastic Behavior of Elements under Combined Normal Force and Bending , 1973 .

[32]  Hongnan Li,et al.  Seismic response analysis of reinforced concrete frames using inelasticity‐separated fiber beam‐column model , 2018 .

[33]  F. Filippou,et al.  Nonlinear Analysis of Prestressed Concrete Structures , 1994 .

[34]  Rui Pinho,et al.  Adaptive force-based frame element for regularized softening response , 2012 .

[35]  Enrico Spacone,et al.  FIBRE BEAM–COLUMN MODEL FOR NON‐LINEAR ANALYSIS OF R/C FRAMES: PART I. FORMULATION , 1996 .

[36]  Katrin Beyer,et al.  Axially equilibrated displacement‐based beam element for simulating the cyclic inelastic behaviour of RC members , 2017 .

[37]  Ignacio Carol,et al.  Nonlinear time-dependent analysis of planar frames using an ‘exact’ formulation—I. Theory , 1989 .

[38]  Paola Ceresa,et al.  A fibre flexure–shear model for seismic analysis of RC‐framed structures , 2009 .

[39]  M. Menegotto,et al.  Method of analysis of cyclically loaded RC plane frames including changes in geometry and non-elastic behavior of elements under normal force and bending , 1973 .

[40]  S. Caddemi,et al.  Euler–Bernoulli beams with multiple singularities in the flexural stiffness , 2007 .

[41]  Amr S. Elnashai,et al.  Application of Adaptive Analysis to Reinforced Concrete Frames , 1994 .

[42]  Stephen A. Mahin,et al.  Analysis of Reinforced Concrete Beam-Columns under Uniaxial Excitation , 1988 .

[43]  Lawrence H. N. Lee.  Inelastic Behavior of Reinforced Concrete Members , 1955 .

[44]  Enrico Spacone,et al.  Localization Issues in Force-Based Frame Elements , 2001 .

[45]  Cv Clemens Verhoosel,et al.  Non-Linear Finite Element Analysis of Solids and Structures , 1991 .