Showing papers on "Tangent stiffness matrix published in 2006"

Stability conditions of prestressed pin-jointed structures

Abstract: Pin-jointed structures are first classified to trusses, tensile structures, and tensegrity structures in view of their respective stability properties. A sufficient condition for stability of an equilibrium state is derived for tensegrity structures. The condition is based on the bilinear forms of the linear and geometrical stiffness matrices considering the flexibility of members. The stability is defined by the positive definiteness of the tangent stiffness matrix, whereas the definition of prestress-stability is based on the geometrical stiffness matrix and the infinitesimal mechanisms. Numerical examples verify that the so-called super-stability condition might not be satisfied by a stable tensegrity structure, and that a prestress-stable structure can be unstable if the prestresses are moderately large.

Monolithic modelling of electro‐mechanical coupling in micro‐structures

Abstract: The purpose of the present work is to model and to simulate the coupling between the electric and mechanical fields. A new finite element approach is proposed to model strong electro-mechanical coupling in micro-structures with capacitive effect. The proposed approach is based on a monolithic formulation: the electric and the mechanical fields are solved simultaneously in the same formulation. This method provides a tangent stiffness matrix for the total coupled problem which allows to determine accurately the pull-in voltage and the natural frequency of electro-mechanical systems such as MEMs. To illustrate the methodology results are shown for the analysis of a micro-bridge.

Implicit algorithm for finite deformation hypoelastic–viscoplasticity in fcc metals

Abstract: An implicit objective stress update algorithm is proposed for a hypoelastic–viscoplastic model. A thermal/dynamic yield function, which is derived based on the thermal activation analysis and dislocation interaction mechanisms, is used, along with the Consistency approach and the framework of additive viscoplasticity, in deriving the proposed model for fcc metals. The corotational formulation approach is utilized in developing the proposed model in the finite deformation field. For the case of the Newton–Raphson iteration method, a new expression for the consistent (algorithmic) tangent stiffness matrix of rate-dependent metals is derived by direct linearization of the stress update algorithm. Finite element simulations are performed by implementing the proposed viscoplasticity constitutive models in the commercial finite element program ABAQUS. Numerical implementation for a simple tensile problem is used for validating the material parameters of the OFHC Copper under low and high strain rates and temperatures. The numerical results of the adiabatic true stress–true strain curves compare very well with the experimental data. The effectiveness of the present approach is tested by studying strain localization in a simple plane strain problem. Results indicate excellent performance of the present framework in describing the strain localization problem and in obtaining mesh-independent results. Copyright © 2006 John Wiley & Sons, Ltd.

A co-rotational formulation for 3D beam element using vectorial rotational variables

Abstract: Based on a co-rotational framework, a 3-noded iso-parametric element formulation of 3D beam was presented, which was used for accurate modelling of frame structures with large displacements and large rotations. Firstly, a co-rotational framework was fixed at the internal node of the element, it translates and rotates with the node rigidly; then, vectorial rotational variables were defined, they are three smaller components of the cross-sectional principal vectors at each node, sometimes they represent different components of the cross-sectional principal vectors in incremental solution procedure so as to avoid the occurrence of ill-conditioned tangent stiffness matrix; thereafter, the internal force vector and tangent stiffness matrix in local system was derived from the strain energy of the element as its first partial derivative and second partial derivative with respect to local variables, respectively, and a symmetric tangent stiffness matrix was achieved; finally, several examples were analysed to illustrate the reliability and accuracy of this procedure.

A mixed co‐rotational formulation of 2D beam element using vectorial rotational variables

Abstract: Based on a co-rotational framework, a three-noded iso-parametric mixed element formulation for large displacement analysis of 2D beam was presented, this formulation is free of membrane/shear locking problems. Firstly, the co-rotational framework was fixed at the internal node of the element, which translates and rotates rigidly with this node, but does not deform with the element. Then, a set of vectorial rotational variables were defined to describe accurately the element deformation, these rotational variables may have different connotations in an incremental solution procedure. Thirdly, Green strain and independently assumed strain fields were introduced in calculating the mixed Hellinger–Reissner functional, and the internal force vector was derived by enforcing the stationarity to the mixed functional, thereafter, the element tangent stiffness matrix was calculated from the internal force vector by differentiating it with respect to local variables, and a symmetric element tangent stiffness matrix was achieved. Finally, several examples were analysed to illustrate the reliability and efficiency of the proposed procedure in modelling of 2D frame structures with large displacements and large rotations. Copyright © 2006 John Wiley & Sons, Ltd.

A sub-stepping approach for elasto-plasticity with rotational hardening

Abstract: Within the framework of the finite element method, this paper presents new algorithms implementing implicit stress integration and consistent tangent matrix calculations for an elasto-plastic model with rotational hardening. The sub-stepping technique is used for both the numerical integration of the constitutive relations and determination of the consistent tangent matrix in order to overcome the convergence difficulty arising from the complexity of the elasto-plastic model with rotational hardening. The integration of the constitutive relations and the computation of the consistent tangent matrix are incorporated into a unique procedure. Numerical tests are carried out and discussed to demonstrate the global accuracy and stability of the presented algorithms.

Elasto‐plastic finite element analysis of shells with damage due to microvoids

Abstract: This paper presents a non-linear finite element analysis for the elasto-plastic behaviour of thick/thin shells and plates with large rotations and damage effects. The refined shell theory given by Voyiadjis and Woelke (Int. J. Solids Struct. 2004; 41:3747–3769) provides a set of shell constitutive equations. Numerical implementation of the shell theory leading to the development of the C0 quadrilateral shell element (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted) is used here as an effective tool for a linear elastic analysis of shells. The large rotation elasto-plastic model for shells presented by Voyiadjis and Woelke (General non-linear finite element analysis of thick plates and shells. 2006, submitted) is enhanced here to account for the damage effects due to microvoids, formulated within the framework of a micromechanical damage model. The evolution equation of the scalar porosity parameter as given by Duszek-Perzyna and Perzyna (Material Instabilities: Theory and Applications, ASME Congress, Chicago, AMD-Vol. 183/MD-50, 9–11 November 1994; 59–85) is reduced here to describe the most relevant damage effects for isotropic plates and shells, i.e. the growth of voids as a function of the plastic flow. The anisotropic damage effects, the influence of the microcracks and elastic damage are not considered in this paper. The damage modelled through the evolution of porosity is incorporated directly into the yield function, giving a generalized and convenient loading surface expressed in terms of stress resultants and stress couples. A plastic node method (Comput. Methods Appl. Mech. Eng. 1982; 34:1089–1104) is used to derive the large rotation, elasto-plastic-damage tangent stiffness matrix. Some of the important features of this paper are that the elastic stiffness matrix is derived explicitly, with all the integrals calculated analytically (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted). In addition, a non-layered model is adopted in which integration through the thickness is not necessary. Consequently, the elasto-plastic-damage stiffness matrix is also given explicitly and numerical integration is not performed. This makes this model consistent mathematically, accurate for a variety of applications and very inexpensive from the point of view of computer power and time. Copyright © 2006 John Wiley & Sons, Ltd.

A Beam Finite Element for Shear-Critical RC Beams

Abstract: The proposed beam finite element is capable of describing the response of reinforced concrete members under the interaction of axial force, shear, and bending moment. The element is based on a mixed formulation and does not exhibit the shear locking problems of displacement-based elements. The material model of the Modified Compression Field Theory by Vecchio and Collins (1986) was used to describe the biaxial response of the concrete at monitoring points across several monitoring sections along the element axis. With this model the proposed beam element is able to capture well the overall monotonic response of shear critical beams, while equally satisfactory agreement is obtained with local response measures, such as crack orientations. In spite of this promising agreement with experimental results the numerical performance of the material model was rather slow for the lack of a consistent tangent stiffness matrix. This fact coupled with the material model's inability to represent cyclic loading point out the need for a numerically consistent, robust and efficient constitutive model.

Apparatus for Testing Concrete under Multiaxial Compression at Elevated Temperature (mac2T)

Abstract: This paper describes a new test facility for determining material mechanical properties of structural concrete. The novel facility subjects 100 mm cubic concrete specimens to true multiaxial compression (σ1 ≠ σ2 ≠ σ3) up to 400 MPa at temperatures of up to 300°C. Forces are delivered through three independent loading frames equipped with servo-controlled hydraulic actuators creating uniform displacement boundary conditions via rigid platens. Specimen deformation is calculated from displacements measured to an accuracy of 10−6 m using a system of six laser interferometers. The combination of stiff loading frames, rigid platens, an accurate and reliable strain measurement system and a fast control system enables investigation of the material response in the post-peak range. The in-house developed control software allows complex multi-stage experiments involving (i) load and temperature cycling, (ii) small stress probes and (iii) arbitrary (pre-defined) loading paths. The program also enables experiments in which the values of the control parameters and the execution of the test sequences depend on the response of the specimen during the test. The capabilities of the facility are illustrated in this paper by experiments determining the effects of different heat-load regimes on the strength and stiffness of the material and tests identifying the tangent stiffness matrix of the material and the associated changes in the acoustic tensor under multiaxial compression.

Stability and nonlinear dynamic behaviour of cross-ply laminated heated cylindrical shells

Abstract: THE STABILITY OF POSTBUCKLED EQUILIBRIUM CONFIGURATIONS AND THE NONLINEAR DYNAMIC CHARACTERISTICS OF CROSS-PLY LAMINATED HEATED CYLINDRICAL SHELLS ARE INVESTIGATED EMPLOYING SEMIANALYTICAL SHELL ¯NITE ELEMENT. THE PRESENCE OF ASYMMETRIC PERTURBATION IN THE FORM OF SMALL MAGNITUDE LOAD SPATIALLY PROPORTIONAL TO THE LINEAR BUCKLING MODE SHAPE IS CONSIDERED TO INITIATE THE BIFURCATION OF THE SHELL DEFORMATION FROM AXISYMMETRIC MODE TO ASYMMETRIC ONE. THE FREQUENCIES OF SMALL OSCILLATIONS ABOUT EQUILIBRIUM CONFIGURATION ARE OBTAINED BY SOLVING THE EIGENVALUE PROBLEM FORMULATED USING TANGENT STIFFNESS MATRIX OF THE CONVERGED EQUILIBRIUM CON¯GURATION AND MASS MATRIX. THE STUDY REVEALS THAT THE PREDICTION OF THE POSTBUCKLING EQUILIBRIUM CON¯GURATION FROM NONLINEAR STATIC ANALYSIS DEPENDS ON THE NATURE (LONGITUDI-NALLY SYMMETRIC/ANTISYMMETRIC) OF INITIAL DISTURBANCE. THE LONGITUDINALLY ANTISYMMETRIC POSTBUCKLED EQUILIBRIUM CONFIGURATION IS STABLE WHEREAS THE LONGITUDINALLY SYMMETRIC ONE IS UNSTABLE. THE NONLINEAR DYNAMIC RESPONSE SHOWS THAT THE SHELL WITH LONGITUDINALLY SYMMETRIC DISTURBANCE JUMPS FROM SYMMETRIC MODE TO ANTISYMMETRIC MODE AND THE PREDICTED EQUILIBRIUM CON¯GURATION IS OF ANTISYMMETRIC NATURE IRESPECTIVE OF THE TYPE OF INITIAL DISTURBANCE. THE NONLINEAR FORCED DYNAMIC RESPONSE OF THE HEATED SHELL IN THE PREBUCKLING REGION DIFFERS SIGNIFICANTLY FROM THAT IN THE POSTBUCKLING REGION.

The use of tangent stiffness to characterize the resilient response of unbound crushed aggregates

Abstract: Repeated-load triaxial tests were conducted on crushed granitic base-course material to study the resilient response under different stress paths and compaction states. It has been established that the resilient response of this prestrained unbound granular material is best defined in terms of tangent stiffness (Et) and vertical stress (σv). The data also revealed the existence of a threshold value of tangent stiffness that is essentially dependent on initial confining stress for a given compaction state. When the tangent modulus exceeds this threshold value, a unique relationship between tangent stiffness and vertical stress exists for mobilized shear resistance ratios less than 0.4. This Et–σv relationship is independent of stress path. A simple power law model can be used to predict the resilient response of unbound base-course material and an approximate value of resilient modulus for any desired stress path and initial stress condition. The use of the tangent stiffness – vertical stress model for pav...

Numerical structural analysis system based on the load-transfer-path method

Abstract: The purpose of this invention is to reduce the calculation time in the numerical structure analysis system based on load-transfer-path method. The parameters are set in the condition that the supporting point B in the objective structure is fixed and the load is applied to the specific loading point A. The FEM calculation means 2 calculates the deformation of the objective structure according to the structural stiffness matrix in the stiffness matrix holding means 1 to find the basic data such as the displacement of each point and so on. The FEM calculation means calculates each deformation to find the displacement under the condition that the specific loading point A and the supporting point B are fixed and three inspection loadings are applied to the variable loading point C. The partial stiffness matrix calculation means 3 solves the multidimensional simultaneous linear equation based upon the internal stiffness matrix of the objective structure, the load value and the displacement to find the partial stiffness matrix K AC . The stiffness parameter calculation means 8 calculates the value of the stiffness parameter U* according to the partial stiffness matrix K AC and the displacement in the basic data and so on. The value of U* of each point is calculated with changing the variable loading point C as to follow sequentially all the necessary points in the objective structure.

Computation of multiple bifurcation point

Abstract: Purpose – The aim of this paper is to develop a new method for finding multiple bifurcation points in structures.Design/methodology/approach – A brief review of nonlinear analysis is presented. A powerful method (called arc‐length) for tracing nonlinear equilibrium path is described. Techniques for monitoring critical points are discussed to find the rank deficiency of the stiffness matrix. Finally, by using eigenvalue perturbation of tangent stiffness matrix, load parameter associated with multiple bifurcation points is obtained.Findings – Since other methods of finding simple bifurcation points diverge in the neighborhood of critical points, this paper introduces a new method to find multiple bifurcation points. It should be remembered that a simple bifurcation point is a multiple bifurcation point with rank deficiency equal to one. Therefore, the method is applicable to simple critical points as well.Practical implications – Global buckling of the structures should be considered in design. Many structu...

Dynamic stiffness for thin-walled structures by power series

Abstract: The dynamic stiffness method is introduced to analyze thin-walled structures including thin-walled straight beams and spatial twisted helix beam. A dynamic stiffness matrix is formed by using frequency dependent shape functions which are exact solutions of the governing differential equations. With the obtained thin-walled beam dynamic stiffness matrices, the thin-walled frame dynamic stiffness matrix can also be formulated by satisfying the required displacements compatibility and forces equilibrium, a method which is similar to the finite element method (FEM). Then the thin-walled structure natural frequencies can be found by equating the determinant of the system dynamic stiffness matrix to zero. By this way, just one element and several elements can exactly predict many modes of a thin-walled beam and a spatial thin-walled frame, respectively. Several cases are studied and the results are compared with the existing solutions of other methods. The natural frequencies and buckling loads of these thin-walled structures are computed.

Fourier series based finite element analysis of tube hydroforming: An axisymmetric model

Abstract: Purpose – The purpose of this paper is to provide an axisymmetric model of tube hydroforming using a Fourier Series based finite element method.Design/methodology/approach – Fourier series interpolation function, which considerably reduces the size of the global stiffness matrix and the number of variables, is employed to approximate displacements. The material of the tube is assumed to be elastic‐plastic and to satisfy the plasticity model that takes into account the rate independent work hardening and normal anisotropy. Numerical solution obtained from an updated Lagrangian formulation of the general shell theory is employed. The axial displacement stroke (a.k.a. axial feed) during tube hydroforming is incorporated using Lagrange multipliers. Contact constraints and boundary friction condition are introduced into the formulation based on the penalty function, which imposes the constraints directly into the tangent stiffness matrix. A forming limit curve based on shear instability and experimental measur...

A Constitutive Model for Magnetostrictive Materials - Theory and Finite Element Implementation

Abstract: A 3D macroscopic constitutive law for hysteresis effects in magnetostrictive materials is presented and a finite element implementation is provided The novel aspect of the thermodynamically consistent model is an additive decomposition of the magnetic and the strain field in a reversible and an irreversible part Employing the irreversible magnetic field is advantageous for a finite element implementation, where the displacements and magnetic scalar potential are the nodal degrees of freedom To consider the correlation between the irreversible magnetic field and the irreversible strains a one-to-one relation is assumed The irreversible magnetic field determines as internal variable the movement of the center of a switching surface This controls the motion of the domain walls during the magnetization process The evolution of the internal variables is derived from the magnetic enthalpy function by the postulate of maximum dissipation, where the switching surface serves as constraint The evolution equations are integrated using the backward Euler implicit integration scheme The constitutive model is implemented in a 3D hexahedral element which provides an algorithmic consistent tangent stiffness matrix A numerical example demonstrates the capability of the proposed model to reproduce the ferromagnetic hysteresis loops of a Terfenol-D sample (© 2006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim)

Closed-Form Stiffness Matrix for the Four-Node Quadrilateral Element with a Fully Populated Material Stiffness

Abstract: This technical note presents closed-form finite-element stiffness formulations for the four-node quadrilateral element with a fully populated material stiffness, which is required for the nonlinear analysis of reinforced concrete membrane structures. With the material stiffness matrix accounting for anisotropy of the materials and prestrain effects, the developed closed-form element stiffness can be incorporated into a nonlinear finite-element algorithm. Through use of the developed explicit expressions, the examples provided show that the computational effort required to form the stiffness matrix is greatly reduced, compared to either the conventional numerical integration scheme or the elastic-material-stiffness-oriented Griffths’ FORTRAN subroutine.

Theory and Design of Statically Balanced Tensegrity Mechanisms

Abstract: The fields of static balancing and tensegrity structures are combined into statically balanced tensegrity mechanisms. This combination results in a new class of prestressed structures that behave like mechanisms: although member lengths and orientations change, they can be deformed into a wide range of positions, while continuously remaining in equilibrium; in other words, the structures have zero stiffness. The key to these structures is the use of zero-free-length springs as tension members. The tools of structural engineering were used to search for, and understand, zero-stiffness modes in the tangent stiffness matrix of prestressed pin-jointed bar frameworks. To this end the recently uncovered parallels between structural engineering and mathematical rigidity theory were exploited. Mathematical literature described that affine transformations preserve the equilibrium of a tensegrity structure; these findings gained value when translated from a mathematical concept into the engineering terms rigid-body motions, shear and dilation. Not only did these transformations prove to be instrumental for describing zero stiffness, but it also provided new insight in the form-finding methods for tensegrity structures: the minimum nullity requirement for the stress matrix is formed by the affine transformations. In this research it was shown that affine transformations of the structure that preserve the length of conventional members are zero-stiffness modes valid over finite displacements: these are statically balanced zero-stiffness modes. What is more, for prestress stable structures with a positive semi-definite stress matrix of maximal rank -- meaning there are only affine transformations in its nullspace -- those are the only possible zero-stiffness modes. The length-preserving affine transformations exist if and only if the directions of the conventional members lie on a conic at infinity. If all conventional member directions lie on a conic, the number of independent length-preserving affine transformations can then be found with a simple counting rule. A systematic analysis of the zero-stiffness modes in the tangent stiffness matrix of a prestressed pin-jointed bar framework yielded several interesting scenarios that warrant further attention, as they cannot be fully described within the currently developed framework. Finally, a demonstration prototype was designed and constructed to illustrate the properties of statically balanced tensegrity mechanisms; the prototype serves as a proof of concept, not as a practically applicable design. Prior to construction, the range of motion of the tensegrity used for the prototype was extensively analysed using the analytic equilibrium conditions. The results were instrumental in dimensioning the prototype.

Implicit and Explicit Implementation of the Dynamic Relaxation Method for the Definition of Initial Equilibrium Configurations of Flexible Lines

Abstract: Recent activities in the offshore oil exploitation industry require new structural concepts employing flexible lines (both mooring lines and risers). Such systems present increasingly complex configurations, with dynamic nonlinear behaviour; therefore, the use of efficient numerical solution procedures, based on the Finite Element Method, becomes mandatory for their analysis. The usual analysis procedure for flexible lines by the FEM is based in the calculation of an initial, stable static equilibrium configuration in order to define the finite element mesh. Usually this configuration is obtained by the classic catenary equations. However, in more complex problems these equations cannot be applied. Therefore, the objective of this work is to present the use of a more general finite element approximation, associated to dynamic relaxation algorithms. Such algorithms can be started from arbitrary configurations, not necessarily in equilibrium. The resulting procedure is accurate, robust, and avoids numerical problems such as the ill-conditioning of the tangent stiffness matrix, allowing the static equilibrium configuration to be obtained in an efficient way.Copyright © 2006 by ASME

Parallel Implementation of Structural Dynamic Analysis for Parachute Simulation

Abstract: An iterative parallel finite element procedure for the simulation of nonlinear structural behavior of parachute systems is presented. The Hilber-Hughes-Taylor method is employed in this procedure for nonlinear implicit time integration. Parallel techniques derived from the geometrical parallelism algorithm are developed in a finite element code. This involves the parallel implementation of generating the tangent stiffness matrix and the effective force vector, solving the resulting linearized simultaneous system equations and other required operations. Two sample problems are presented to demonstrate the inherent features of parachute simulations and to analyze the efficiency of the parallel algorithms.

Continuum‐based design sensitivity analysis and optimization of nonlinear shell structures using meshfree method

Abstract: SUMMARY A continuum-based shape and configuration design sensitivity analysis (DSA) method for a finite deformation elastoplastic shell structure has been developed. Shell elastoplasticity is treated using the projection method that performs the return mapping on the subspace defined by the zero-normal stress condition. An incrementally objective integration scheme is used in the context of finite deformation shell analysis, wherein the stress objectivity is preserved for finite rotation increments. The material derivative concept is used to develop a continuum-based shape and configuration DSA method. Significant computational efficiency is obtained by solving the design sensitivity equation without iteration at each converged load step using the same consistent tangent stiffness matrix. Numerical implementation of the proposed shape and configuration DSA is carried out using the meshfree method. The accuracy and efficiency of the proposed method is illustrated using numerical examples. Copyright 2006 John Wiley & Sons, Ltd.