Showing papers on "Tangent stiffness matrix published in 2012"

Generalized Eigenvalue Analysis of Symmetric Prestressed Structures Using Group Theory

Abstract: As conventional approaches for calculating natural frequencies do not make full use of the inherent symmetry of a structure, the rising degree of freedoms often leads to significant increase in computational demand. In this study, a simplified technique for analyzing dynamic characteristics of symmetric prestressed structures is described using group theory. First, the generalized eigenvalue equation of a prestressed structure based on tangent stiffness matrix and lumped mass matrix is built to get natural frequencies and the corresponding vibration shapes in which the contribution of initial prestresses is considered. A symmetry-adapted coordinate system for the structure is adopted to block-diagonalize the stiffness and mass matrices. The complexity of generalized eigenvalue equation is reduced by solving the mutually independent subspaces, and thus natural frequencies and the corresponding vibration modes could be obtained. Illustrative examples point out the general procedure, and show the sup...

Towards a Nonlinear Elastic Representation of Finite Compression and Instability of Boron Carbide Ceramic

Abstract: A nonlinear constitutive model invoking third-order anisotropic elasticity is developed for boron carbide single crystals subjected to potentially large compressive stresses. The model makes use of limited available published data from various experimental and theoretical (i.e., quantum or ab initio) studies. The model captures variations in second-order tangent elastic moduli and loss of elastic mechanical stability with increasing compression. In particular, reduced stability of boron carbide single crystals compressed normal to the c-axis (i.e., [0001]-direction) relative to higher stability in spherical compression is represented. Different stability criteria proposed in the literature are examined for boron carbide under spherical and uniaxial compression; model predictions show that the most critical criterion corresponds to a vanishing eigenvalue of a particular tangent stiffness matrix (i.e., incremental modulus) derived exactly in the present work. Model constants are proposed for CCC (less elast...

Analytical buckling analysis of an inflatable beam made of orthotropic technical textiles

Abstract: This paper is devoted to the linear eigen and nonlinear buckling analysis of an inflatable beam made of orthotropic technical textiles. The method of analysis is based on a 3D Timoshenko beam model with a homogeneous orthotropic woven fabric. The finite element model established here involves a three-noded Timoshenko beam element with C0-type continuity for the transverse displacement and quadratic shape functions for the bending rotation and the axial displacement. In the linear buckling analysis, a mesh convergence test on the beam critical load was carried out by solving the linearized eigenvalue problem. The stiffness matrix in this case is generally assumed not to be a function of displacements, while in the nonlinear buckling problem, the tangent stiffness matrix includes the effect of changing the geometry as well as the effect of the stress stiffening. The nonlinear finite element solutions were investigated by using the straightforward Newton iteration with the adaptive load stepping for tracing the load–deflection response of the beam. To assess the effect of geometric nonlinearities and the inflation pressure on the stability behavior of inflatable beam: a simply supported beam was studied. The influence of the beam aspect ratios on the buckling load coefficient was also pointed out. To check the validity and the soundness of the results, a 3D thin-shell finite element model was used for comparison. For a further validation, the results were also compared with those from experiments at low inflation pressures.

A consistent total Lagrangian finite element for composite closed section thin walled beams

Abstract: This work presents a consistent geometrically exact finite element formulation of the thin-walled anisotropic beam theory. The present formulation is thought to address problems of composite beams with nonlinear behavior. The constitutive formulation is based on the relations of composite laminates and thus the cross sectional stiffness matrix is obtained analytically. The variational formulation is written in terms of generalized strains, which are parametrized with the director field and its derivatives. The generalized strains and generalized beam forces are obtained by introducing a transformation that maps generalized components into physical components. A consistent tangent stiffness matrix is obtained by parametrizing the finite rotations with the total rotation vector; its derivation is greatly simplified by obtention of the derivatives of the director field via interpolation of nodal triads. Several numerical examples are presented to show the accuracy of the formulation and also its frame invariance and path independence.

Multiobjective Hybrid Optimization–Antioptimization for Force Design of Tensegrity Structures

Abstract: Properties of Pareto optimal solutions considering bounded uncertainty are first investigated using an illustrative example of a simple truss. It is shown that the nominal values of the Pareto optimal solutions considering uncertainty are slightly different from those without considering uncertainty. A hybrid approach of multiobjective optimization and antioptimization is next presented for force design of tensegrity structures. We maximize the lowest eigenvalue of the tangent stiffness matrix and minimize the deviation of forces from the specified target distribution. These objective functions are defined as the worst values due to the possible errors in the fabrication and construction processes. The Pareto optimal solutions are found by solving the two-level optimization-antioptimization problems using a nonlinear programming approach for the upper optimization problem and enumeration of the vertices of the uncertain region for the lower antioptimization problem.

Finite element corotational formulation for geometric nonlinear analysis of thin shells with large rotation and small strain

Abstract: Based on the consistent symmetrizable equilibrated (CSE) corotational formulation, a linear triangular flat thin shell element with 3 nodes and 18° of freedom, constructed by combination of the optimal membrane element and discrete Kirchhoff triangle (DKT) bending plate element, was extended to the geometric nonlinear analysis of thin shells with large rotation and small strain. Through derivation of the consistent tangent stiffness matrix and internal force vector, the corotational nonlinear finite element equations were established. The nonlinear equations were solved by using the Newton-Raphson iteration algorithm combined with an automatic load controlled technology. Three typical case studies, i.e., the slit annular thin plate, top opened hemispherical shell and cylindrical shell, validated the accuracy of the formulation established in this paper.

Dynamic Analysis of Suspension Bridges and Full Scale Testing

Abstract: This paper is concerned with the earthquake analysis of suspension bridges, in which the effects of large deflections are taken into account. The first part of the study deals with an iteration scheme for the nonlinear static analysis of suspension bridges by means of tangent stiffness matrices. The concept of tangent stiffness matrix is then introduced in the frequency equation governing the free vibration of the system. At any equilibrium stage, the vibrations are assumed to take place tangent to the curve representing the force-deflection characteristics of the structure. The bridge is idealized as a three dimensional lumped mass system and subjected to three orthogonal components of earthquake ground motion producing horizontal, vertical and torsional oscillations. By this means a realistic appraisal is achieved for torsional response as well as for the other types of vibration. The modal response spectrum technique is applied to evaluate the seismic loading for the combination of these vibrations. Various numerical examples are introduced in order to demonstrate the method of analysis. The procedure described enables the designer to evaluate the nonlinear dynamic response of suspension bridges in a systematic manner.

Initial prestress distribution and natural vibration analysis of tensegrity structures based on group theory

Abstract: As conventional approaches for morphology and natural vibration analysis do not make full use of the symmetry of structures, the computational cost is significantly raised with increasing number of nodes. In this paper, we propose a simplified technique used to analyze initial prestress distribution and natural vibration of tensegrity structures based on group theory. First, the conditions of symmetry and equilibrium equations for tensegrity structures were established on the basis of the symmetry-adapted coordinate systems found by group theory. Then the initial prestress modes could be found from the null space of the independent sub-matrix of symmetry-adapted equilibrium matrix. Subsequently, the tangent stiffness matrix and the lumped mass matrix were block-diagonalized using symmetry. The generalized eigenvalue problems were simplified by solving the mutually independent subspaces, with the corresponding natural frequencies and vibration modes obtained. Two illustrative examples demonstrate the general procedure, and show the superiority in reducing the difficulty of initial prestress distribution and natural vibration analysis. When compared with numerical results obtained by Abaqus and those of Murakami, the proposed method is shown to be more accurate and efficient.

Stiffness analysis of a wire-driven parallel manipulator

Abstract: The positioning accuracy of a wire-driven parallel manipulator mainly depends on its stiffness when forces act on a mobile platform. In this paper the stiffness of the wire-driven parallel manipulator is theoretically studied, and a whole analytic expression is deduced based on the differential transformation principle. The stiffness matrix of the wire-driven parallel manipulator is composed of two parts which are structural parameter stiffness matrix and wire tension stiffness matrix. The simulation results show that the stiffness of the manipulator relies heavily not only on its structural parameters, but also on the tension of wires. The stiffness of the wire-driven parallel manipulator can be adjusted by changing the wire tension when its structural parameters are unchanged.

Analysis of Elastic-Plastic Contact Performance of Rigid Sphere Against a Deformable Flat-Effect of Strain Hardness

Abstract: Problem statement: The present study considers an elastic-plastic contact analysis of a rigid sphere with a deformable flat (Rigid Sphere-model) using finite element analysis. The effect of tangent modulus on the contact behavior of a no adhesive frictionless elastic-plastic contact was analyzed using commercial finite element software ANSYS. Approach: Different materials, in terms of the ratio of Young's modulus to yield strength, had been considered to study the effect of tangent modulus. The Finite Element (FE) contact analysis was carried out by incorporating the various tangent modulus values of different materials. Results: The result clearly shows that for different tangent modulus the material hold different stress values. When this modulus increases the strain hardness value of material was also increased. Conclusion: With increase in tangent modulus, strain hardening resistance to deformation of a material is increased and the material becomes capable of carrying higher amount of load in a smaller contact area.

A General Formulation for the Stiffness Matrix of Parallel Mechanisms

Abstract: Starting from the definition of a stiffness matrix, the authors present a new formulation of the Cartesian stiffness matrix of parallel mechanisms. The proposed formulation is more general than any other stiffness matrix found in the literature since it can take into account the stiffness of the passive joints, it can consider additional compliances in the joints or in the links and it remains valid for large displacements. Then, the validity, the conservative property, the positive definiteness and the relation with other formulations of stiffness matrices are discussed theoretically. Finally, a numerical example is given in order to illustrate the correctness of this matrix.

Nonlinear Structural Identification of a Three-Story Infilled Frame Using Instantaneous Modal Parameters

Abstract: Even though modal analysis theory is not applicable to nonlinear dynamic structural systems, such systems can be characterized by their time-varying amplitude-dependent instantaneous modal parameters In this study, the nonlinear behavior of a large-scale test structure is characterized based on its time-varying instantaneous modal parameters identified during an earthquake The test structure is a 2/3-scale, three-story, two bay, masonry-infilled reinforced concrete frame, tested on the University of California, San Diego (UCSD) outdoor shake table Deterministic stochastic subspace identification (DSI) method is used for estimation of instantaneous modal parameters of the structure based on sliding time windows of input-output data during a seismic base excitation These identified time-variant modal parameters are used to estimate the effective stiffness of different components of the test structure corresponding to its tangent stiffness matrix through a linear finite element (FE) model updating strategy Variation of the identified stiffness as a function of maximum displacement can be used to characterize the hysteretic behavior at element/substructure level

Contact analysis of spherical ball and a deformable flat model with the effect of tangent modulus

Abstract: The paper is on contact analysis of a spherical ball with a deformable flat, considering the effect of tangent modulus on the contact parameters of a non-adhesive frictionless elastic-plastic contact. The contact analysis of this model has been carried out using analysis software Ansys and Abaqus. The contact parameters such as area of contact between two consecutive steps, volume of bulged material are evaluated from the formulated equations. The effect of the tangent modulus is considered for determining these parameters. The tangent modulus are accounted between 0.1E and 0.5E of materials E/Y value greater than 500 and less than 1750. Result shows that upto an optimal tangent modulus values the elastic core push up to the free surface in the flat. The simulation is also carried out in Abaqus and result provide evidence for the volume of bulged material in the contact region move up and flow into the free surface of the flat from the contact edge between the ball and flat. The strain energy of the whole model is varied between 20 to 40 percentage of the stipulated time for analysis.

Characteristics of the Solution of the Consistently Linearized Eigenproblem for Lateral Torsional Buckling

Abstract: The consistently linearized eigenproblem has proved to be a powerful mathematical tool for classification of buckling, based on the percentage bending energy of the total strain energy. Of particular interest are prebuckling states with a constant percentage strain energy. The two limiting cases of such states are membrane stress states and states of pure bending. Buckling at pure bending, referred to as lateral torsional buckling, is the topic of this work. The transfer matrix method is used to derive a secant stiffness matrix in analytical form. Formulation of the consistently linearized eigenproblem by means of this matrix yields the same solution as would be obtained by a formulation based on the tangent stiffness matrix which is an essential ingredient of nonlinear Finite Element Analysis. This remarkable finding permits analytical verification of hypothesized subsidiary conditions for lateral torsional buckling. © 2012 Bull. Georg. Natl. Acad. Sci.

Wrinkle generation mechanism without buckling in sheared rectangular membrane

Abstract: The objective of this study is to clarify wrinkle generation in membranes. It is generally considered that wrinkles are generated by local buckling. The bifurcation buckling is a singular point in numerical computation and the post-buckling solution is difficult to calculate. Types of buckling can be predicted in advance by asymptotic theory. Thus, we introduce a path analysis method into the finite element software package ABAQUS. This analysis uses eigenvectors corresponding to the zero eigenvalues of the tangent stiffness matrix in order to calculate the post-buckling equilibrium path. Using this method, we calculate wrinkle generation in a rectangular membrane. As the result, this study shows a novel wrinkle generation mechanism without buckling and clarifies the bifurcation structure of the system.

Closed-loop Stiffness Modeling and Stiffness Index Analysis for Multi-axis Machining System

Abstract: During multi-axis machining process,complex surface can be machined flexibly while cutter's posture is changeable in available workspace,which directly affects general stiffness characteristics of multi-axis machining system and also machining performance.By establishing general stiffness model related to cutter posture,the distribution of stiffness performance of whole machining system is analyzed.Based on multi-body small deflection theory,a semi-analytic method about closed-loop stiffness modeling for multi-axis machining system is proposed,in which Jacobi matrix method,point transformation matrix method and finite element method are applied.The force ellipsoid corresponding to the decoupling stiffness model is established in 3D space.Stiffness index is derived from force ellipsoid,which is used to plot isolines of general stiffness performance of multi-axis machining system.The result of analysis about the distribution of stiffness performance can help direct tool motion planning.

Beam element for large displacement analysis of elasto-plastic frames

Abstract: ABSTRA CT. The present paper develops a non-linear beam element for analysis of elastoplastic frames under large displacements. The finite element formulations are derived by using the co-rotational approach and expression of the virtual work. The Gauss quadrature is employed for numerically computing the element tangent stiffness matrix and internal force vector. A bilinear stress-strain relationship with isotropic hardening is adopted to update the stress. The arc-length technique based on the Newton-Raphson iterative method is employed to compute the equilibrium paths. A number of numerical examples is employed to assess the performance of the developed element. The effects of plastic action on the large displacement behavior of the structures as well as the expansion of plastic zones in the loading process are discussed.

Non-linear vibrations of tensegrity structures

Abstract: A study has been done on different methods to solve the linear and non-linear problems with single or multi degrees of freedom structures. To do that direct time integration methods are used to solve the dynamic equilibrium equations. It has been tried to perform general methods to apply in most structures. In this thesis structures are made of cables and bars but the solution methods which is presented can be applied for any structure, and to utilize those implicit and explicit methods for all structures it is needed to know the tangent stiffness matrix and mass matrix for the structure. Then, it would be possible to analyze the dynamic response of structures under general loads and by general it can be understood that by application of arbitrary forces on different nodes of structure, the method generates result based on applied forces. It is crucial to get the right tangent stiffness matrix and mass matrix to know the dynamic equation in each node. Hence, the method will then work correctly to solve the dynamic problems. More, a parametric study has been done to see the effects of times step, stiffness of elements, length of elements, and other mechanical properties of elements, and this parametric study enables one to produce new results by changing every parameter. Also, continuation of the study on x-frame tensegrity has been done by solving them to check out dynamic response of structure with proposed methods of this thesis. Moreover, a method is presented to use the codes of solver methods of current thesis to apply them for other structures. Hence, as a future work, one can combine the codes of structures and solver codes of this thesis for dynamic response of structure. In fact the main effort of this thesis is on presenting different methods to solve various structures.

Comprehensive Stiffness of Prestressed Lattice Materials

Abstract: Several approaches to obtain the comprehensive stiffness of finite frameworks are present in literature; yet, the formulation has not been addressed for lattice materials and infinitely periodic structures. The objective of this paper is to introduce a systematic method to calculate the comprehensive stiffness of prestressed, infinitely periodic, structures and lattice materials with pin- and rigid-jointed connectivity. We first derive the comprehensive stiffness of a finite framework through the superposition of its material and nonlinear geometrical stiffness. By using the Bloch's theorem, we derive the irreducible form of the stiffness system of the finite framework, which represents the stiffness behaviour of the corresponding infinite, periodic assembly. Finally, the comprehensive stiffness of the infinite lattice is homogenized to generate the stiffness characteristics of the lattice material. A detailed example is provided to show the application of the methodology. Closed-form expressions of the elastic properties are presented for 12 planar lattices.

Tangent Stiffness Matrix of Single-Layer Reticulated Shell’s Members with Rigid Ends

Abstract: In the paper, the axial stiffness and bending stiffness of single-layer reticulated shell’s joint are considering together, non-linear beam-column element with rigid springs and rigid ends is taken as the analysis model of members of single-layer reticulated shell, a tangent stiffness matrix of members of single-layer reticulated shell considering joint’s stiffness is derived on the basis of the beam-column theory. In this matrix, not only coupling effects of bending in two axes but also joint’s stiffness and joint’s size are considered, not only the effect of axial force on bending but also the effect of axial force on torsion are considered. All higher order terms in the displacement function are considered. So this matrix is perfect and more precise than the tangent stiffness matrix from C.Oran, and this model can be suited to the non-linear stablity analysis of single-layer reticulated shell.

Stiffness Analysis of 6-SPS Parallel Manipulator

Abstract: This paper describes a stiffness analysis of a 6-RSS parallel manipulator by using matrix structural analysis. The stiffness analysis is based on standard concepts of static elastic deformations. The formulation has been implemented in order to obtain the stiffness matrix that can be numerically computed by defining a suitable model of the manipulator, which takes into account the stiffness properties of each element such as links and joints. The obtained stiffness matrix was used to map the end-effector compliant displacements when external forces and torques are applied on it. Finally experimental test were performed to verify the proposed methodology. Keywords: Parallel Manipulators, Stiffness Analysis, Matrix

The shock response of hyperbolic tangent and tangent comprehensive model on cushion system considering rotary motion

Abstract: In this paper,the three regions: linear elastic deformation region,collapse plateau region and the densification region of the common cushion materials are expressed by hyperbolic tangent and tangent comprehensive nonlinear model.Considering the rotary degree of freedom caused by eccentricity of package product structure,the dynamics equations of comprehensive package system model under the action of half-sine acceleration pulse are deduced,and the couple laws of model of the packaging system with translational and rotary motions are researched.The numerical results show that it results in excessive packaging to some degree regardless of the rotary motion.The establishment of the model and the dynamic equation considering coupling effect between translational motion and rotary motion,which are suitable for engineering application,can provide reference for optimization design of the cushion packaging structure,and this method can be directly used in the cushion design for multiple degrees of freedom systems.

Compressed storage algorithm of 3D-FEM stiffness matrix

Abstract: To improve the efficiency and reduce the storage space of finite element analysis,compression and storage algorithm of 3D-FEM stiffness matrix is studied.The relationship between "generalized adjacent double nodes" and the non-zero sub-matrix in stiffness matrix is researched for getting distribution of non-zero sub-matrix in stiffness matrix.A new algorithm of stiffness matrix of compressed storage-"improved CSR storage method" is proposed.Based on the algorithm,the generation process of stiffness matrix is given and iterative solution of linear equations method is proposed to improve the efficiency of solving linear equations.The algorithm is applied to the three-dimensional bulk forming finite element analysis software and the numerical results show that the algorithm can effectively decrease the storage space and improve the computation efficiency.