Showing papers on "Tangent stiffness matrix published in 2005"

Enhanced stiffness modeling, identification and characterization for robot manipulators

Abstract: This paper presents the enhanced stiffness modeling and analysis of robot manipulators, and a methodology for their stiffness identification and characterization. Assuming that the manipulator links are infinitely stiff, the enhanced stiffness model contains: 1) the passive and active stiffness of the joints and 2) the active stiffness created by the change in the manipulator configuration, and by external force vector acting upon the manipulator end point. The stiffness formulation not accounting for the latter is known as conventional stiffness formulation, which is obviously not complete and is valid only when: 1) the manipulator is in an unloaded quasistatic configuration and 2) the manipulator Jacobian matrix is constant throughout the workspace. The experimental system considered in this study is a Motoman SK 120 robot manipulator with a closed-chain mechanism. While the deflection of the manipulator end point under a range of external forces is provided by a high precision laser measurement system, a wrist force/torque sensor measures the external forces. Based on the experimental data and the enhanced stiffness model, the joint stiffness values are first identified. These stiffness values are then used to prove that conventional stiffness modeling is incomplete. Finally, they are employed to characterize stiffness properties of the robot manipulator. It has been found that although the component of the stiffness matrix differentiating the enhanced stiffness model from the conventional one is not always positive definite, the resulting stiffness matrix can still be positive definite. This follows that stability of the stiffness matrix is not influenced by this stiffness component. This study contributes to the previously reported work from the point of view of using the enhanced stiffness model for stiffness identification, verification and characterization, and of new experimental results proving that the conventional stiffness matrix is not complete and is valid under certain assumptions.

A spatially curved‐beam element with warping and Wagner effects

Abstract: This paper presents a new spatially curved-beam element with warping and Wagner effects that can be used for the non-linear large displacement analysis of members that are curved in space. The non-linear behaviour of members curved in space shows that the Wagner effects are substantial in the large twist rotation analysis. Most existing finite beam element models, such as ABAQUS and ANSYS cannot predict the non-linear large displacement response of members curved in space correctly because the Wagner effects, viz. the Wagner moment and the corresponding finite strain terms, have not been considered in these finite beam elements. As a consequence, these finite beam elements do not provide correct predictions for the out-of-plane buckling and postbuckling behaviour of arches as well. In this paper, the symmetric tangent stiffness matrix has been derived based on the finite rotations parameterized by the conventional displacements. The warping and Wagner effects: both the Wagner moment and the corresponding finite strain terms and their constitutive relationship, are included in the spatially curved-beam element. Two components of the initial curvature, the initial twist and their interactions with the displacements are also considered in the spatially curved-beam element. This ensures that the large twist rotation analysis for the members curved in space is accurate. Comparisons with existing experimental, analytical and numerical results show that the spatially curved-beam element is accurate and efficient for the non-linear elastic analysis of curved members, buckling and postbuckling analysis of arches, and in its ability to predict large deflections and twist rotations in more arbitrarily curved members. Copyright © 2005 John Wiley & Sons, Ltd.

An enhanced co‐rotational approach for large displacement analysis of plates

Abstract: This paper presents a new co-rotational approach for the large displacement analysis of plates employing 4-noded quadrilateral flat shell elements. The proposed approach benefits from (i) a simple local co-rotational system invariant to the element nodal ordering, (ii) the choice of the two smallest components of the nodal normal vector as global rotational degrees of freedom, and (iii) the use of hierarchic freedoms, that are unaffected by the co-rotational transformations, for higher-order accuracy. Important additional benefits that arise from the aforementioned features include symmetry of the tangent stiffness matrix and complete insensitivity of the large displacement transformations to the size of the incremental step. The applicability of the new approach to moderately thick as well as thin plates is illustrated by considering two alternative local formulations based on the Reissner–Mindlin and discrete Kirchhoff hypotheses. Several examples are finally presented which demonstrate the accuracy, step-insensitivity and computational benefits of the proposed co-rotational approach for large displacement analysis of plate structures. Copyright © 2005 John Wiley & Sons, Ltd.

Mixed finite element method for coupled thermo-hydro-mechanical process in poro-elasto-plastic media at large strains

Abstract: SUMMARY A mixed finite element for coupled thermo-hydro-mechanical (THM) analysis in unsaturated porous media is proposed. Displacements, strains, the net stresses for the solid phase; pressures, pressure gradients, Darcy velocities for pore water and pore air phases; temperature, temperature gradients, the total heat flux are interpolated as independent variables. The weak form of the governing equations of coupled THM problems in porous media within the element is given on the basis of the Hu–Washizu three-filed variational principle. The proposed mixed finite element formulation is derived. The non-linear version of the element formulation is further derived with particular consideration of the THM constitutive model for unsaturated porous media based on the CAP model. The return mapping algorithm for the integration of the rate constitutive equation, the consistent elasto-plastic tangent modulus matrix and the element tangent stiffness matrix are developed. For geometrical non-linearity, the co-rotational formulation approach is utilized. Numerical results demonstrate the capability and the performance of the proposed element in modelling progressive failure characterized by strain localization and the softening behaviours caused by thermal and chemical effects. Copyright 2005 John Wiley & Sons, Ltd.

Nonlinear finite-element analysis of stranded conductors with variable bending stiffness using the tangent stiffness method

Abstract: Stranded conductors are widely used structural components. Owing to their construction in layers, their bending stiffness may vary according to their tension, curvature and deformation history. Recently, a sound and practical model of variable bending stiffness using the secant stiffness method became available. Based on the same physical assumptions, This work presents the development of a variable bending stiffness model using the tangent stiffness method and its implementation in a classical finite-element formulation adapted for nonlinear analysis under arbitrary loading. This extends its use to a general finite-element program. Comparisons with static and dynamic tests on short-span substation conductors show that the model computes a representative bending stiffness for such cases and yields adequate predictions of tractions generated at their ends, in both static and dynamic regimes.

A formulation of the non linear discrete Kirchhoff quadrilateral shell element with finite rotations and enhanced strains

Abstract: This paper presents a new formulation of the non-linear discrete Kirchhoff quadrilateral shell element applicable for the analysis of geometrically nonlinear structures undergoing finite rotations. The shell director is directly interpolated and the exact linearization of the discreet form of the equilibrium equations is derived in closed form. The consistent tangent stiffness matrix is symmetric and is given explicitly in this paper. Two or three rotational variables are used at each node. To improve the in-plane deformation enhanced incompatible modes are introduced. The formulation is then illustrated by a comprehensive set of numerical experiments selected from the literature.

Tangent Geometric Stiffness of Inclined Cables Under Self-Weight

Abstract: An exact formula is derived for the tangent geometric stiffness of inclined cables under self-weight. Comparisons are made with the well-known approximate formula by Ernst. It is found that the accuracy of Ernst's formula deteriorates with increasing cable slackness and inclination. The error can be significant for certain configurations of the cable that are commonly used in such structures as cable-stayed bridges, guyed towers, electric transmission lines, and conductor cables connecting electrical substation equipment. An accurate and simple approximate formula is developed, which is applicable to a wide range of parameter values.

Efficient Modification Scheme of Stress-Strain Tensor for Wrinkled Membranes

Abstract: A new modification scheme of the stress-strain tensor for wrinkled membranes is presented on the basis of the tension field theory. The scheme is applicable to finite element analysis of partly wrinkled membranes with arbitrary shapes. Derivation of the modification scheme rests on an introduction of the so-called wrinkle strain and a simplification of the virtual work equation of wrinkled membranes. Because all of the modifications required to account for wrinkling are totally confined within the stress-strain relations of membranes, the scheme can be easily implemented with existing finite element codes. Furthermore, the modified stress-strain tensor automatically leads to the consistent tangent stiffness matrix, where changes in both the wrinkling direction and the amount of wrinkliness are taken into account. Three numerical examples are treated to show the accuracy and effectiveness of the proposed modification scheme.

A Tangent Stiffness MLPG Method for Atom/Continuum Multiscale Simulation

Abstract: The main objective of this paper is to develop a multiscale method for the static analysis of a nano-system, based on a combination of molecular mechanics and MLPG methods. The tangent-stiffness formulations are given for this multiscale method, as well as a pure molecular mechanics method. This method is also shown to naturally link the continuum local balance equation with molecular mechanics, directly, based on the stress or force. Numerical results show that this multiscale method quite accurate. The tangent-stiffness MLPG method is very effective and stable in multiscale simulations. This multiscale method dramatically reduces the computational cost, but it still can provide reasonable accuracy in some regions of the model. keyword: Molecular mechanics, Multiscale method, continuum mechanics, MLPG.

Geometric Characterization of the Configuration Space of Rigid Body Mechanisms in Regular and Singular Points

Abstract: The kinematics of rigid body mechanisms is considered from a differential-geometric perspective Geometric properties of a mechanism are intrinsically determined by the topology of its configuration space — the solution set of closure functions The mechanism kinematics is usually characterized by the tangent space and tangent cone to the configuration space, ie by locally considering its topology There are, however, mechanisms for which this is not sufficient Generally, beside the topology, a complete picture of the kinematics needs both, the configuration space and the ideal generated by the closure functions Tangent spaces/cones are differential-geometric objects associated to a variety Two additional objects are introduced in this paper: the kinematic tangent space and the kinematic tangent cone Three locally equivalent models for the mechanism kinematics are introduced Due to their different mathematical nature the different models admit to apply specific mathematical tools The analysis of model I is based on Lie group and screw algebraic methods, while model II and III are analyzed using methods from algebraic geometry A computationally efficient algorithm for the construction of the kinematic tangent cone is presented Its application is shown for several examples A novel mechanism is presented of which the differential and local degree of freedom are different in regular points, so-called ‘paradox-in-the-small’Copyright © 2005 by ASME

Finite element simulation of the squaring circular tube

Abstract: The squaring tube process is examined by an incremental elasto-plastic finite-element method based on an updated Lagrangian formulation in which a sliding-sticking friction mode is specifically considered. The high nonlinearity of the process due to the geometric changes, the inelastic constitutive behavior, and the deformation-dependent boundary conditions are taken into account in an incremental manner. A static explicit approach to the solution is applied, the tangent stiffness matrix equation is solved without iteration and a weighting factor rmin is employed to limit the step size to linear relation. The simulated geometries of squaring clearly demonstrate the processes of square tube until unloading. The formation of squaring defects both collapse and asymmetry are reported in a theoretical manner. Accordingly, the effects of various parameters of the process, such as geometric ratio R/t, strain hardening exponent n, and the friction coefficient μ, on the occurrence of collapse (collapse ratio C/t) and on the extent of asymmetry (deviation ratio C1/C2) for the squaring process are discussed and interpreted in simulation. Mainly it is expected that formation of a square tube for industrial use that does not collapse will be found during the design stage, before trials begin. The present work may be expected to improve the understanding of the formation of the square tube .

Tangent stiffness of elastic continua on manifolds

Abstract: Non-linear models of beams, shells and polar continua are addressed from a general point of view with the aim of providing a clear motivation of the fact that the tangent stiffness of these structural models may be nonsymmetric. Classical and polar models of continua are investigated and a critical analysis of the commonly adopted strain measures is performed. It is emphasized that the kinematic space of a polar continuum is a non-linear differentiable manifold. Accordingly, by choosing a connection on the manifold, the Hessian operator of the elastic potential is defined as the second covariant derivative of the elastic potential. The Hessian operator can be expressed as the difference between the second directional derivative along the trial and test fields and the first directional derivative in the direction of the covariant derivative of the test field along the trial field. It follows that the evaluation of the Hessian operator requires the extension of the local virtual displacement to a vector field over the non-linear kinematic manifold. In any case the tensoriality of the Hessian operator ensures that the result is independent of the choice of the extension, and its symmetry depends on whether or not the assumed connection is torsionless. Conservative and nonconservative loads are considered and it is shown that, at equilibrium points, the tangent stiffness is independent of the chosen connection on the fiber manifold and symmetry holds for conservative loads.

Cable Element Based on Exact Analytical Expressions

Abstract: The two-node catenary cable element,derived using exact analytical expressions for the elastic catenary,is perfected,including cable element with external concentrating loads acting between cable ends and with rigid arms at two ends.Then the stiffness factor perpendicular to the element plane is derived directly from the formulae for suspended cables,and it is shown that tangent stiffness matrix of 3-D cable element can be formed from that of 2-D cable element.The cable element subjected to combined uniformly distributed loads is also included through transfer matrix.The element can be used in nonlinear finite element method analysis of suspension bridge,cable-stayed bridge etc.Its efficiency and accuracy are demonstrated by two numerical examples.

A finite element development for ball bearing nonlinear stiffness modelization

Abstract: A formulation and algorithmic treatment of a three-dimensional bearing stiffness is presented. This stiffness formulation is based on the non-linear equilibrium balance of forces and moments on the rolling elements, in a ball bearing, exerted by the inner and the outer races including contact forces. Newton-Raphson Method is used to solve the resulting non linear equation system. A non-linear two nodes finite element linking up a node of the inner race to another node of the outer race is developed. The tangent stiffness matrix of this element is deduced from the Jacobean of the convergent equilibrium. A numerical study is presented showing the influence of some parameters variation and the coupling between all stiffness terms. (Accepted by previous Editorial Team.)

Shape design sensitivity analysis of nonlinear structural systems

Abstract: Recent developments in design sensitivity analysis of nonlinear structural systems are presented. Various aspects, such as geometric, material, and boundary nonlinearities are considered. The idea of variation in continuum mechanics is utilized in differentiating the nonlinear equations with respect to design variables. Due to the similarity between variation in design sensitivity analysis and linearization in nonlinear analysis, the same tangent stiffness is used for both sensitivity and structural analyses. It has been shown that the computational cost of sensitivity calculation is a small fraction of the structural analysis cost. Such efficiency is due to the fact that sensitivity analysis does not require convergence iteration and it uses the same tangent stiffness matrix with structural analysis. Two examples are presented to demonstrate the accuracy and efficiency of the proposed sensitivity calculation method in nonlinear problems.

Nonlinear Truss Analysis by One Matrix Inversion

Abstract: A new method for nonlinear structural analysis has been developed. The novelty of the method is that only one stiffness matrix inversion is required without the need for updating and reinverting the matrix at every load increment. This stiffness matrix is not necessarily the real stiffness matrix of the structure. Instead any stiffness matrix compatible with the geometry and the constraints of the truss can be used. The advantage of this option is that if the design of some members is revised the already inverted and stored matrix is used for the analysis of the revised structure. Nonlinearities due to strain hardening, strain softening, buckling, breaking, and stiffness degradation are handled by iterations involving only multiplications of the banded matrix with a transformed force vector. The inversion of the half-banded original stiffness matrix is done using Gauss elimination performed on the half-banded matrix without destroying the bandedness, and the inverted matrix replaces the original without the need for additional storage. The coefficients for the transformation of the force vector are stored permanently in a new matrix of size equal to the size of the half-banded original. Thus, the total storage needed is equal to the storage for the banded original stiffness. Because, after the Gauss elimination, only multiplications of a matrix with a vector are involved, the method is computationally efficient. The method is not a step-by-step procedure. Any load increment can be applied, therefore, proportional, nonproportional, and cyclic loads are treated in a unified way. The energy dissipation and the residual stresses and strains after one or more cycles are readily available, and thus the method can be used in quasi-dynamic analysis (e.g. pushover) for an evaluation of the dynamic parameters of the structure.

Nonlinear Stability Analysis of Long-Span Continuous Rigid Frame Bridge with Thin-Wall High Piers

Abstract: By utilizing the current finite element program ANSYS, two types of finite element models (FEM), the beam model (BM) and shell model (SM), are established for the nonlinear stability analysis of a practical rigid frame bridge-Longtanhe Great Bridge. In these analyses, geometrical and material nonlinearities are simultaneously taken into account. For geometrical nonlinearity, updated Lagrangian formulations are adopted to derive the tangent stiffness matrix. In order to simulate the nonlinear behavior of the plastic hinge of the piers, the multi-lines spring element COMBIN39 is used in the SM while the bilinear rotational spring element COMBIN40 is employed in the BM. Numerical calculations show that satisfying results can be obtained in the stability analysis of the bridge when the double coupling nonlinearity effects are considered. In addition, the conclusion is significant for practical engineering.

Determination of Effective Buckling Length of Plane Frames using Elastic and Inelastic System Buckling Analysis

Abstract: An improved method for evaluating effective buckling lengths of beam-column members in plane frames is newly proposed based on system inelastic buckling analysis. To this end, the tangent stiffness matrix of be am-column elements is first calculated using stability functions and then the inelastic buckling analysis method is presented. The scheme for determining effective length of individual members is also addressed. Design examples and numerical results ?uc presented to show the validity of the proposed method.

Stiff arc length constraint in nonlinear FEA

Abstract: Arc length constraints enable iterative solution procedures in nonlinear finite element analysis to converge even at critical points. The arc length constraint replaces the conventional m×m stiffness matrix with an augmented (m+1)×(m+1) stiffness matrix. Its use is referred to as arc length control, in contrast to load control which furnishes the conventional stiffness matrix. In the current article, an apparently new arc length constraint is introduced. It identifies arc length parameters maximizing the stiffness (absolute value of the determinant) of the augmented matrix. The parameters, viewed as a vector, must be perpendicular to the rows of the stiffness matrix, likewise considered vectors. The augmented stiffness matrix is nonsymmetric and lacks the small bandwidth of the conventional stiffness matrix. However, using a block triangularization, it is demonstrated that a solution may be attained by standard finite element operations, namely triangularization of a banded nonsingular portion of the stiffness matrix followed by forward and backward substitutions involving banded lower and upper triangular matrices. The proposed constraint is expected to permit convergence under longer arc lengths than currently implemented methods. A simple example is given illustrating the application of the constraint.

A new hierarchical technique for the multiscale modeling of carbon nanostructures

Abstract: We present a new hierarchical modeling technique called the Consistent Atomic-scale Finite Element (CAFE) method [1]. Unlike traditional approaches for linking the atomic structure to its equivalent continuum [2-7], this method directly connects the atomic degrees of freedom to a reduced set of finite element degrees of freedom without passing through an intermediate homogenized continuum. As a result, there is no need to introduce stress and strain measures at the atomic level. This technique partitions atoms to masters and salves and reduces the total number of degrees of freedom by establishing kinematic constraints between them [5-6]. The Tersoff-Brenner interatomic potential [8] is used to calculate the consistent tangent stiffness matrix of the structure. In this finite element formulation, all local and non-local interactions between carbon atoms are taken into account using overlapping finite elements (Figure 1b). In addition, a consistent hierarchical finite element modeling technique is developed for adaptively coarsening and refining the mesh over different parts of the model (Figure 2a, 2b). The stiffness of higher-rank elements is approximated using the stiffness of lower-rank elements and kinematic constraints. This process is consistent with the underlying atomic structure and, by refining the mesh, molecular dynamic results will be recovered. This method is valid across the scales and can be used to concurrently model atomistic and continuum phenomena so, in contrast with most other multiscale methods [4-7], there is no need to introduce artificial boundaries for coupling atomistic and continuum regions. Effect of the length scale of the nanostructure is also included in the model by building the hierarchy of elements from bottom up using a finite size atom cluster as the building block (Figures 2a, 2b). In this method by introducing two independent field variables, the so-called inner displacement is taken into account (Fig. 3b). Applicability of the method is shown with several examples of deformation of carbon nanostructures such as graphene sheet, nanotube, and nanocone, subjected to different loads and boundary conditions.Copyright © 2005 by ASME

Application of vectorial rotational variables in large displacement analysis of structures

Abstract: Publisher Summary The chapter explains that the introduction of vectorial rotational variables in the large displacement analysis brings several advantages: these vectorial rotational variables are commutative and additive, so a consistent result can be easily achieved during updating the rotation description in the incremental/iterative solution procedure; all local variables can be obtained from global variables by applying a vector transformation matrix; symmetric-consistent tangent stiffness matrices can be achieved in the local and global systems, provided the equations of equilibrium are work-conjugate with the adopted displacement and rotation parameters; large incremental step can be adopted in the incremental loading process; and the rotational variables can describe the element large displacement response accurately. Verification examples provided in the chapter demonstrate that the proposed unified co-rotational approach provide accurate predictions of the large displacement response of two-dimensional/three-dimensional framed structures as well as curved shell problems, providing computational efficiency through the use of a symmetric tangent stiffness matrix and step-insensitivity.

Second-order analysis and design of cables and cable-frames

Abstract: Cable structures are lightweighted, simple to fabricate and reusable. They provide effective solutions for large-span structures. Analysis of cables is complex because of their highly geometrically nonlinear behavior. Based on the Lagrangian formulation and a fourth-order polynomial displacement function, the tangent stiffness matrix for a five-node curved cable element is derived and statically condensed to a simple form readily for incorporation into a frame analysis computer program. The program uses the pointwise–equilibrium–polynomial (PEP) element with initial imperfection and the "Nonlinear Integrated Design and Analysis (NIDA)" method for design and nonlinear analysis of cabled structures. Numerical examples demonstrate the robustness and practicality of the proposed method.