Showing papers on "Tangent stiffness matrix published in 2007"

The stiffness matrix in elastically articulated rigid-body systems

Abstract: Discussed in this paper is the Cartesian stiffness matrix, which recently has received special attention within the robotics research community. Stiffness is a fundamental concept in mechanics; its representation in mechanical systems whose potential energy is describable by a finite set of generalized coordinates takes the form of a square matrix that is known to be, moreover, symmetric and positive-definite or, at least, semi-definite. We attempt to elucidate in this paper the notion of “asymmetric stiffness matrices”. In doing so, we show that to qualify for a stiffness matrix, the matrix should be symmetric and either positive semi-definite or positive-definite. We derive the conditions under which a matrix mapping small-amplitude displacement screws into elastic wrenches fails to be symmetric. From the discussion, it should be apparent that the asymmetric matrix thus derived cannot be, properly speaking, a stiffness matrix. The concept is illustrated with an example.

Corotational non‐linear analysis of thin plates and shells using a new shell element

Abstract: A new three-node triangular shell element is developed by combining the optimal membrane element and discrete Kirchhoff triangle (DKT) plate bending element, and is modified for laminated composite plates and shells so as to include the membrane-bending coupling effect. Using appropriate shape functions for the bending and membrane modes of the element, the 'inconsistent' stress stiffness matrix is formulated and the tangent stiffness matrix is determined. Non-linear analysis of thin-walled structures with geometric non-linearity is conducted using the corotational method. The new element is thoroughly tested by solving few popular benchmark problems. The results of the analysis are compared with those obtained using existing membrane elements.

Hermite polynomial smoothing in beam-to-beam frictional contact

Abstract: In this paper a smoothing procedure is suggested for the 3D beam-to-beam contact. A smooth segment is defined basing on current position vectors of three nodes limiting two adjacent finite elements. The approximated fragment of a beam axis as a 3D curve spans between the centre points of these elements. The curve is described parametrically using three Hermite polynomials. The four boundary conditions necessary to determine the coefficients for each of these polynomials involve co-ordinates and slopes at the curve ends. The slopes are defined in terms of the element nodal co-ordinates, too. There is no dependence on nodal rotations so this formulation can be embedded in a beam analysis using any type of beam finite element. This geometric representation of the curve is incorporated into the 3D beam-to-beam frictional contact model with the penalty method used to enforce contact constraints. The residual vector and the corresponding tangent stiffness matrix are determined for the normal part of contact and for the stick or slip state of friction. A few numerical examples are presented to show the performance of the suggested smoothing procedure in the cases featuring large frictional sliding.

Accurate Stiffness Matrix for Nonprismatic Members

Abstract: In this note, the transfer matrix method is adopted to deduce general expressions for the components of the stiffness matrix and equivalent node load vector of nonprismatic members, and the effect of warping is not considered. State vectors are introduced to describe the nodal forces and displacements of a structural member. The relation between the state vectors of the left node and the right node of the member is given by a matrix referred to as the transfer matrix. It is found that the stiffness matrix of the member can be expressed in terms of the transfer matrix. Therefore, an accurate expression for the stiffness matrix can be obtained as long as the corresponding transfer matrix can be accurately determined. The method proposed is a general procedure for the stiffness matrix derivation of both continuous nonprismatic members and discontinuous nonprismatic members. The correctness of the obtained stiffness expressions is verified by two simple numerical examples.

An efficient co‐rotational formulation for curved triangular shell element

Abstract: A 6-node curved triangular shell element formulation based on a co-rotational framework is proposed to solve large-displacement and large-rotation problems, in which part of the rigid-body translations and all rigid-body rotations in the global co-ordinate system are excluded in calculating the element strain energy. Thus, an element-independent formulation is achieved. Besides three translational displacement variables, two components of the mid-surface normal vector at each node are defined as vectorial rotational variables; these two additional variables render all nodal variables additive in an incremental solution procedure. To alleviate the membrane and shear locking phenomena, the membrane strains and the out-of-plane shear strains are replaced with assumed strains in calculating the element strain energy. The strategy used in the mixed interpolation of tensorial components approach is employed in defining the assumed strains. The internal force vector and the element tangent stiffness matrix are obtained from calculating directly the first derivative and second derivative of the element strain energy with respect to the nodal variables, respectively. Different from most other existing co-rotational element formulations, all nodal variables in the present curved triangular shell formulation are commutative in calculating the second derivative of the strain energy; as a result, the element tangent stiffness matrix is symmetric and is updated by using the total values of the nodal variables in an incremental solution procedure. Such update procedure is advantageous in solving dynamic problems. Finally, several elastic plate and shell problems are solved to demonstrate the reliability, efficiency, and convergence of the present formulation. Copyright © 2007 John Wiley & Sons, Ltd.

Computational aspects of tangent moduli tensors in rate-independent crystal elastoplasticity

Abstract: The computational efficiencies of the continuum and consistent (algorithmic) tangent moduli tensors in rate-independent crystal elastoplasticity are examined in conjunction with the available implicit state update algorithms. It is, in this context, shown that the consistent tangent moduli associated with the state update algorithm with the exponential mapping coincide with the continuum tangent moduli. After verifying the reported performance of the exponential mapping algorithm in preserving the incompressibility of plastic deformation in a single crystal grain, we carry out numerical experiments to understand the convergence trends of the global Newton–Raphson iterative procedure with different kinds of tangent moduli tensors. Having done this, we are concerned with the performance of those tangent moduli tensors for the micro-scale analysis of a polycrystalline aggregate, which is regarded as a representative volume element, subjected to macro-scale uniform deformation in the context of the two-scale homogenization method.

Smooth Frictional Contact between Beams in 3D

Abstract: In this paper a smoothing procedure for the 3D beam-to-beam contact is presented. A smooth segment is based on current position vectors of three nodes for two adjacent finite elements. The approximated fragment of a 3D curve modeling a beam axis spans between the centre points of these elements. The curve is described parametrically using three Hermite polynomials. The four boundary conditions used to determine the coefficients for each of these polynomials involve co-ordinates and slopes at the curve ends. The slopes are defined in terms of the element nodal co-ordinates, too, so there is no dependence on nodal rotations and this formulation can be embedded in a beam analysis using any type of beam finite element. This geometric representation of the curve is incorporated into the 3D beam-to-beam frictional contact model with the penalty method used to enforce contact constraints. The residual vector and the corresponding tangent stiffness matrix are determined for the normal part of contact and for the stick or slip state of friction. A numerical example is presented to show the performance of the suggested smoothing procedure in a case of large frictional sliding.

Implementation of the modified compression field theory in a tangent stiffness-based finite element formulation

Abstract: A finite element implementation of the modified compression field theory (MCFT) using a tangential formulation is presented in this work. Previous work reported on implementations of MCFT has concentrated mainly on secant formulations. This work describes details of the implementation of a modular algorithmic structure of a reinforced concrete constitutive model in nonlinear finite element schemes that use a Jacobian matrix in the solution of the nonlinear system of algebraic equations. The implementation was verified and validated using experimental and analytical data reported in the literature. The developed algorithm, which converges accurately and quickly, can be easily implemented in any finite element code.

Influences of negative stiffness on a two-dimensional hexagonal lattice cell

Abstract: The overall viscoelastic responses and stability were investigated of a hexagonal lattice cell containing pre-chosen negative stiffness components. The study was motivated by the need for fundamental understanding of negative stiffness effects on structures consisting of many degrees of freedom, as well as the symmetry of the structures. Theoretically, use of negative stiffness inclusions to obtain high stiffness and high damping composites has been extensively studied in Hashin-Shtrikman models and one-dimensional discrete systems. Studies on two-dimensional structures containing negative stiffness components and their stability are limited. The finite-element method, combined with the technique of the state space representation, was used to analyze two-dimensional, nested hexagons composed of two-force components that obey the constitutive relation of the standard linear solid. Effective stiffness and damping anomalies in terms of peaks and anti-peaks are observed in the metastable regime when the amount of negative stiffness is selectively tuned. According to the Lyapunov indirect stability theorem, the system as a whole is stable when the tuning stiffness is greater than a critical negative value, but metastable otherwise. The degree of metastability depends on viscosity in the components. Furthermore, stiffness anomalies may be easier to observe experimentally from compression tests than hydrostatic or shear tests.

Explicit formulations of tangent stiffness tensors for isotropic materials

Abstract: A compact explicit expression for the tangent stiffness tensor is presented. Throughout the analysis, the formulation holds for general isotropic elastic materials and does not require solving eigenvector problems. On the theoretical side, a very simple solution of a tensor equation is obtained. Then the expressions for the derivatives of general symmetric isotropic tensor functions of a symmetric tensor are developed. On the computational side, particular attention is given to the consideration of the special case, Green elastic materials, in which the strain energy does not admit a closed-form expression in terms of principal invariants. Finally, a simple formulation of the tangent stiffness tensor for Ogden material model is supplied. Copyright © 2006 John Wiley & Sons, Ltd.

Tangent elastic modulus of Duncan-Chang model for different stress paths

Abstract: In practical applications,the soil might be subjected to different stress paths,such as axial unloading lateral loading and lateral unloading.But the well-known Duncan-Chang model was derived only under the axial loading condition.Thus the scope of its application was restricted.It was attempted to further develop the Duncan-Chang model and to derive a formula of tangent elastic modulus under different stress paths.The new formula was verified for various problems which could not be well handled by the traditional Duncan-Chang model.

Improved tangent modulus of nonlinear soil model

Abstract: Tangent modulus was essential to nonlinear soil model,and it was necessary to understand it properly during numerical simulation.First,a new concept namely half strength index was introduced to build the mathematic properties equations of stress-strain relationship,then the deficiency of hyperbolic expression of the relationship was indicated in terms of it.Second,according to laboratory test data,differential control equation of soil stress-strain relationship was presented and a new 3-parameter stress-strain expression was established.It was shown that the hyperbolic expression was the simplified type of the 3-parameter expression.Third,an improved tangent modulus of soil was obtained based on the new stress-strain expression,and methods to determine its parameters were discussed.Finally,simulation of triaxial test data using the improved tangent modulus was conducted.

A Tangent Plane Analysis of the Static Voltage Stability Region Boundary

Abstract: The algorithm of the static voltage stability region's tangent plane through the characteristic vector method or the implicit function derivative method is presented in the total parameter space,which can provide quantitatively the influence degree of all kinds of parameters on the voltage stability region.The implicit function derivative method can calculate the partial derivative of state variables to independent parameter variables.Hyper-plane approximation in the cut-set space can be derived analytically based on the partial derivative through the mapping from power injection space to cut-set space,which can bypass the data fitting algorithm and improve the calculating speed.The practical voltage stability region's tangent plane,which is formed by constraints of the power system,is also presented.Three kinds of voltage stability indices based on the tangent plane analysis method are presented to assess the voltage stability level comprehensively.Simulations of the IEEE 9-bus and IEEE 39-bus system show that the research on the static voltage stability region's tangent plane can assess the voltage stability level objectively and lay the foundation for correct decision-making for secure and stable operation of power systems.

Tangent plane approximation and some of its generalizations

Abstract: A review of the tangent plane approximation proposed by L.M. Brekhovskikh is presented. The advantage of the tangent plane approximation over methods based on the analysis of integral equations for surface sources is emphasized. A general formula is given for the scattering amplitude of scalar plane waves under an arbitrary boundary condition. The direct generalization of the tangent plane approximation is shown to yield approximations that include a correct description of the Bragg scattering and allow one to avoid the use of a two-scale model.

Influência das tensões residuais na resistência de pilares de aço considerando a análise avançada com plasticidade distribuída

Abstract: Residual stresses due to manufacturing process in steel sections reduce the column strength This reduction is more important for medium slenderness ratio(40 < l/r < 120) So, the column strength curve should be based on analysis that includes the effect of residual stresses through the cross section This paper presents a geometrically exact finite element formulation to consider material and geometric nonlinearities of steel plane frames, including distributed-plasticity analysis The Corrotacional technique is used to obtain the element's tangent stiffness matrix, considering self equilibrated residual stresses The formulation accuracy is showed in the examples The strengths predicted by the proposed formulation are compared with those predicted by the NBR 8800/06 review project, proving the efficiency of this approach as an Advanced Analysis Method

Generalized Stiffness Gear-Mesh Matrix Including EHD Stiffness

Abstract: This paper presents a method to compute gear mesh stiffness based on the EHD behavior by combined finite element solution of the Reynolds Equation with the elastic contact model. It is shown that this solution requires iterative procedure to balance the computed pressure profile with the external nominal transmission load. This mesh stiffness is load dependent and therefore is a nonlinear phenomenon. The nominal stiffness value is utilized to model a full (12×12) gear mesh matrix for a linear dynamic model of rotor bearing systems including gears to evaluate system dynamics and coupling between lateral/torsional vibrations.Copyright © 2007 by ASME

Study of generalized tangent bulk modulus

Abstract: Tangent bulk modulus of soil in the triaxial state of stress was researched.The different influences on tangent bulk modulus of conventional triaxial test and isotropic compression test were considered.On the basis of Hooke′s law,volume work is divided into plastic volume work and elastic volume work and then the influence coefficients on tangent bulk modulus of two kinds of triaxial tests were calculated.Using the values of tangent bulk modulus from two types of triaxial tests and the influence coefficients,the generalized tangent bulk modulus could be acquired.The results of verification show that generalized tangent bulk modulus is more typical and reliable.

Nonlinear finite element analysis and design optimization of thin-walled structures

Abstract: In this study, an efficient, accurate and robust methodology for nonlinear finite element analysis and design optimization of thin-walled structures is presented. Main parts of this research are: formulation and development of an accurate and efficient shell element, a robust nonlinear finite element analysis technique, and an efficient optimization methodology. In the first part, a new three-node triangular shell element is developed by combining the optimal membrane element and discrete Kirchhoff triangle (DKT) plate bending element, and is then modified for laminated composite plates and shells so as to include the membrane-bending coupling effect. Also, a moderately thick shell element is developed in a similar manner by combining the discrete Kirchhoff-Mindlin triangular (DKMT) plate bending element and the optimal membrane element. Using appropriate shape functions for the bending and membrane modes of the element, the "inconsistent" stress stiffness matrix is formulated and the tangent stiffness matrix is determined. In the second part, a robust nonlinear finite element analysis program based on the corotational technique is developed to analyze thin-walled structures with geometric nonlinearity. The new element is thoroughly tested by solving few popular benchmark problems. The results of the analyses are compared with those obtained based on other membrane elements. In the third part, optimization algorithms based on the optimality criteria are developed and then combined with the nonlinear finite element analysis to optimize different types of thin-walled structures with geometric nonlinearity. The optimization problem considers the thickness or geometry design variables, and aims to maximize the critical load of the structure subject to constant total mass, or minimize the total mass subject to constant applied loads. The optimization results based on the developed design optimization algorithm are compared with those based on the gradient-based sequential quadratic programming method to demonstrate the efficiency and accuracy of the developed procedure. An application of the thickness optimization for locating the potential places to add the stiffeners in stiffened panels is also presented. Also a method is presented to efficiently incorporate the effects of local buckling and mode switching during optimization process for stiffened panels.

Nonlinear analysis of two -node accurate catenary cable element with arbitrary rigid arms

Abstract: In order to solve the problem of rigid anchor connection when using catenary cable element, based on the tangent stiffness matrix of two-node catenary cable element deduced with catenary equations, a more exact expression of tangent stiffness matrix is derived for two-node catenary cable element with arbitrary rigid arms. The iteration technique for initial cable tension and cable erection is also analyzed. Analysis and numerical example demonstrate that the new catenary cable element can simulate the rigid connection successfully, and the FEM procedure of the new element is the same as the old one. The new tangent stiffness matrix may degenerate into the old one automatically if the catenary cable element has no rigid arms. The new element can be used in cable structures extensively.

Influência das tensões residuais na resistência de pilares de aço considerando a análise avançada com plasticidade distribuída (Influence of the residual stresses on the strength of steel columns considering advanced analysis with distributed plasticity)

Abstract: Residual stresses due to manufacturing process in steel sections reduce the column strength. This reduction is more important for medium slenderness ratio(40 ≤ l/r ≤ 120). So, the column strength curve should be based on analysis that includes the effect of residual stresses through the cross section. This paper presents a geometrically exact finite element formulation to consider material and geometric nonlinearities of steel plane frames, including distributed-plasticity analysis. The Corrotacional technique is used to obtain the element’s tangent stiffness matrix, considering self equilibrated residual stresses. The formulation accuracy is showed in the examples. The strengths predicted by the proposed formulation are compared with those predicted by the NBR 8800/06 review project, proving the efficiency of this approach as an Advanced Analysis Method.

Research on consistent tangent modulus for unified viscoplastic constitutive equations

Abstract: Unified viscoplasticity constitutive equations are a system of integral or incremental equations with implied internal variables.So,it is difficult to provide a well-defined expression of the consistent tangent modulus.Based on the backward Euler's integral for unified viscoplasticity constitutive equations,a new expression of consistent tangent modulus was derived for rate-dependent plasticity.The constitutive equations and consistent tangent modulus expression were implemented in a commercial finite element code-MARC.The numerical examples of finite elements verify the feasibility of consistent tangent modulus.

A fully variational strong discontinuity approach at finite strains

Abstract: A novel, fully variational three-dimensional finite element formulation for the modeling of locally embedded strong discontinuities at finite strains is presented. The proposed numerical model is based on the Enhanced Assumed Strain concept with an additive decomposition of the displacement gradient into a conforming and an enhanced part. The discontinuous component of the displacement field which is associated with the failure in the modeled structure is isolated in the enhanced part of the deformation gradient. In contrast to previous works, a variational constitutive update is used. The internal variables are determined by minimizing a pseudo-elastic potential. The advantages of such a formulation are well known, e.g. the tangent stiffness matrix is symmetric, standard optimization algorithms can be applied and it represents a natural basis for error estimation and mesh adaption. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Electro-mechanical Coupling in MEMS: Modeling and Experimental Validation

Abstract: This paper presents the advantages of a strong coupled formulation to model the electro-mechanical coupling appearing in MEMS. The classical modeling approach is to use a staggered methodology iterating between two different programs to obtain the solution of the coupled problem. In this research a strong coupled formulation is proposed and a tangent stiffness matrix of the whole problem is computed. Using this matrix, nonlinear algorithms such as the Riks-Crisfield algorithm may be applied to solve the static nonlinear problem and accurately determine the static pull-in voltage. Moreover, the natural frequencies may be computed around each equilibrium positions. The dynamic behavior of the structure may also be studied and two new parameters are defined: the dynamic pull-in voltage and the dynamic pull-in time. This strong coupled methodology deriving from variational principle may also be used for topology optimization and extended finite elements.