Finite element response sensitivity analysis using three‐field mixed formulation: general theory and application to frame structures

This paper presents a method to compute response sensitivities of finite element models of structures based on a three-field mixed formulation. The methodology is based on the direct differentiation method (DDM), and produces the response sensitivities consistent with the numerical finite element response. The general formulation is specialized to frame finite elements and details related to a newly developed steel-concrete composite frame element are provided. DDM sensitivity results are validated through the forward finite difference method (FDM) using a finite element model of a realistic steel-concrete composite frame subjected to quasi-static and dynamic loading. The finite element model of the structure considered is constructed using both monolithic frame elements and composite frame elements with deformable shear connection based on the three-field mixed formulation. The addition of the analytical sensitivity computation algorithm presented in this paper extends the use of finite elements based on a three-field mixed formulation to applications that require finite element response sensitivities. Such applications include structural reliability analysis, structural optimization, structural identification, and finite element model updating.

[1]  T. Pian.  Derivation of element stiffness matrices by assumed stress distributions , 1964 .

[2]  W. J. Hall,et al.  Dynamics of Structures—Theory and Applications to Earthquake Engineering , 1996 .

[3]  Joel P. Conte,et al.  Finite element response sensitivity analysis using force-based frame models , 2004 .

[4]  R. Haftka,et al.  Elements of Structural Optimization , 1984 .

[5]  Ted Belytschko,et al.  Limitation principles for mixed finite elements based on the Hu-Washizu variational formulation , 1985 .

[6]  Minho Kwon,et al.  A 3D hypoplastic model for cyclic analysis of concrete structures , 2001 .

[7]  Walter D. Pilkey,et al.  State-of-the-art surveys on finite element technology , 1983 .

[8]  P. Franchin.  Reliability of uncertain inelastic structures under earthquake excitation , 2004 .

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

[10]  John F. Abel,et al.  On mixed finite element formulations and stress recovery techniques , 2000 .

[11]  Vitelmo V. Bertero,et al.  Local bond stress-slip relationships of deformed bars under generalized excitations , 1982 .

[12]  Andrea Dall'Asta,et al.  Composite beams with weak shear connection , 2001 .

[13]  Nathan M. Newmark,et al.  A Method of Computation for Structural Dynamics , 1959 .

[14]  Joel P. Conte,et al.  Finite Element Response Sensitivity Analysis of Steel-Concrete Composite Beams with Deformable Shear Connection , 2005 .

[15]  Sherif El-Tawil,et al.  Nonlinear Analysis of Steel-Concrete Composite Structures: State of the Art , 2004 .

[16]  鷲津 久一郎.  Variational methods in elasticity and plasticity , 1982 .

[17]  Andrea Dall'Asta,et al.  Three-field mixed formulation for the non-linear analysis of composite beams with deformable shear connection , 2004 .

[18]  D. Malkus,et al.  Mixed finite element methods—reduced and selective integration techniques: a unification of concepts , 1990 .

[19]  Ashraf Ayoub,et al.  MIXED FORMULATION OF NONLINEAR STEEL-CONCRETE COMPOSITE BEAM ELEMENT , 2000 .

[20]  Chien Wei-zang.  Method of high-order lagrange multiplier and generalized variational principles of elasticity with more general forms of functionals , 1983 .

[21]  E. P. Popov,et al.  CONSTITUTIVE MODEL FOR 3D CYCLIC ANALYSIS OF CONCRETE STRUCTURES , 1997 .

[22]  Mark A. Bradford,et al.  Elementary Behaviour of Composite Steel and Concrete Structural Members , 2000 .

[23]  Andrea Dall'Asta,et al.  Non-linear analysis of composite beams of a displacement approach , 2000 .

[24]  Hwang Chien,et al.  Large deflection of circular plate under compound load , 1983 .

[25]  Andrea Dall'Asta,et al.  Slip locking in finite elements for composite beams with deformable shear connection , 2004 .

[26]  Duong Hien Tran,et al.  Parameter Sensitivity in Nonlinear Mechanics: Theory and Finite Element Computations , 1997 .

[27]  Henrik O. Madsen,et al.  Structural Reliability Methods , 1996 .

[28]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .

[29]  A. Zona,et al.  Comparison and validation of displacement and mixed elements for the non-linear analysis of continuous composite beams , 2004 .

[30]  Michael H. Scott,et al.  Response sensitivity for nonlinear beam-column elements , 2004 .

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

[32]  Joel P. Conte,et al.  Consistent Finite-Element Response Sensitivity Analysis , 2003 .

[33]  R. G. Slutter,et al.  Shear Strength of Stud Connectors in Lightweight and Normal-Weight Concrete , 1971, Engineering Journal.

[34]  B. D. Veubeke.  Displacement and equilibrium models in the finite element method , 1965 .

[35]  Pc Pandey,et al.  Analytical shape sensitivity computation using hybrid finite elements , 1999 .

[36]  A. Kiureghian,et al.  Dynamic response sensitivity of inelastic structures , 1993 .

[37]  Robert L. Taylor,et al.  A mixed finite element method for beam and frame problems , 2003 .

[38]  N. M. Newmark,et al.  Test and analysis of composite beams with incomplete interaction , 1951 .

[39]  Joel P. Conte,et al.  Finite Element Response Sensitivity Analysis: a comparison between force-based and Displacement-Based Frame Element Models , 2005 .