Showing papers on "Virtual work published in 1991"

Finite Element Analysis

Abstract: Mathematical Models and Engineering Decisions. Generalized Solutions Based on the Principle of Virtual Work. Finite Element Discretizations in One Dimension. Extensions and Their Convergence Rates in One Dimension. Two-Dimensional Linear Elastostatic Problems. Element-Level Basis Functions in Two Dimensions. Computation of Stiffness Matrices and Load Vectors for Two Dimensional Elastostatic Problems. Potential Flow Problems. Assembly, Constraint Enforcement, and Solution. Extensions and Their Convergence Rates in Two Dimensions. Computation of Displacements, Stresses and Stress Resultants. Computation of the Coefficients of Asymptotic Expansions. Three-Dimensional Linear Elastostatic Problems. Models for Plates and Shells. Miscellaneous Topics. Estimation and Control of Errors of Discretization. Mathematical Models. Appendices. Index.

A gradient flow theory of plasticity for granular materials

Abstract: A flow theory of plasticity for pressure-sensitive, dilatant materials incorporating second order gradients into the flow-rule and yield condition is suggested. The appropriate extra boundary conditions are obtained with the aid of the principle of virtual work. The implications of the theory into shear-band analysis are examined. The determination of the shear-band thickness and the persistence of ellipticity in the governing equations are discussed.

Mechanics of solids and fluids

Abstract: 1. Kinematics.- 2. Statics, Systems of Forces, Hydrostatics.- 3. Mechanical Work, Power, Potential Energy.- 4. Constitutive Equations.- 5. Principle of Virtual Work.- 6. Selected Topics of Elastostatics.- 7. Dynamics of Solids and Fluids, Conservation of Momentum of Material and Control Volumes.- 8 First Integrals of the Equations of Motion, Kinetic Energy.- 9. Stability Problems.- 10. D'Alember's Principle and Lagrange Equations of Motion.- 11. Some Approximation Methods of Dynamics and Statics.- 12. Impact.- 13. Elementary Supplements of Fluid Dynamics.- 14. Selected Problems.- Table A. Some Average Values of Mechanical Material Parameters.- Table B. U.S. (Basic) Customary Units and Their SI Equivalents.

Dynamics of Structures

Abstract: 1. Introduction to elasto-dynamics: linear oscillators. 2. The equations of motion and virtual work methods in dynamics. 3. The nature of the inertia forces and the mass matrix. 4. The natural vibrations of undamped systems. 5. Free vibrations of undamped systems. 6. Forced vibrations of undamped systems. 7. The nature of damping forces modal damping. 8. Random vibrations of modally damped systems. 9. Dynamic analysis of structures with arbitrary viscous damping. 10. Direct integration methods for the equation of dynamic equilibrium. 11. Aspects of non-linear structural dynamics.

Weighted Integral Method. II: Response Variability and Reliability

Abstract: After obtaining in a companion paper an exact expression of the stochastic stiffness matrix in terms of random variables called weighted integrals, the response variability and the safety index of stochastic frame structures are calculated in this paper. The response variability is calculated using a first-order Taylor’s expansion and the safety index using the advanced first-order second-moment approach. It is concluded that the potential energy and virtual work approaches produce identical results for the mean value and the variance of nodal displacements and internal forces. On the contrary, the two approaches produce different values for the safety index of both nodal displacements and internal forces. These values for the safety index obtained using the two approaches compare very well. It is noted that the stochastic stiffness matrix obtained using the potential energy approach is an approximation of the corresponding one obtained using the virtual work approach. Finally, the effect of statistical dependence or independence among the stochastic fields of different elements is examined.

Transient Waves in Anisotropic Laminated Plates, Part 1: Theory

Abstract: A hybrid numerical method in which the finite element method and the method of Fourier transforms are combined is proposed. The plate is divided into N plate elements, and the principle of virtual work is used to develop approximate dynamic equilibrium equations for three- and two-dimensional problems

傾斜機能材料平板におけるラム波伝播とその衝撃応答 : 第一報,解析手法

Abstract: The numerical methods which have been proposed by the present authors for Lamb waves in an anisotropic laminated plate and its transient responses are expanded for the wave propagation analysis of functionally gradient material (FGM) plates. The material properties of the plate change gradually in the thickness direction, and are anisotropic in the plane of the plate. The plate is divided into N plate elements. In the element, we assume that the material properties change linearly in the thickness direction, and that the displacement field is expressed by second-order polynomials. The principle of virtual work is used to develop approximate dynamic equilibrium equations. The dispersion relation and the mode shape of the Lamb waves are obtained by using the free boundary conditions. The energy velocities of the Lamb waves are formulated with the aid of the Rayleigh quotient. The method of Fourier transforms in conjunction with the modal analysis is used to determine the response of displacements. The formulation of the theory is described in this paper.

Non‐linear analysis of stiffened plates using super elements

Abstract: A new numerical technique for large deflection elasto-plastic analysis of stiffened plates is presented. The method uses super finite elements which are macro elements having analytical as well as the usual finite element shape functions, specially designed so that only one plate element per bay and one beam element per span are needed. The large deflection theory by von Karman and the von Mises yield criterion and associated flow rule are employed. The governing equations are derived using the principle of virtual work, integrated numerically using Gauss quadrature and solved by Newton–Raphson iteration. Numerical solutions are presented for simple beams and plates, and plates stiffened in one or two mutually perpendicular directions. Good approximations are obtained with only one-element representations of each plate bay or beam span with significant savings in computing time, costs and storage requirements as compared with using regular finite elements.

Adaptive ALE finite elements with particular reference to external work rate on frictional interface

Abstract: Simulation methods for material forming processes based on arbitrary Lagrangian-Eulerian (ALE) finite elements and adaptive mesh deployment techniques are developed. Special emphasis is on the ALE formulation with respect to the external virtual work rate on the friction interface. An extension of the solution algorithm from the explicit scheme to Newton iteration is also given. Successful strategies for the use of ALE finite element models in nonlinear problems are explored. The implementation of ALE finite element modeling to rolling material forming processes with friction and the fusion of ALE methods with adaptive finite elements is explored. Applications to metal rolling problems are then given.

Engineering mechanics : statics and dynamics

Abstract: 1. Fundamentals 2. Forces 3. Equilibrium of Particles 4. Moments of Forces 5. Equilibrium of Rigid Bodies 6. First Moments: Centroids and Centers of Gravity 7. Second Moments: Moments of Inertia 8. Structures 9. Friction 10. Virtual Work 11. Kinematics of Particles 12. Kinetics of Particles: Force and Acceleration 13. Kinetics of Particles: Work and Energy 14. Kinetics of Particles: Impulse and Monetum 15. Plane Kinematics of Rigid Bodies 16. Plane Kinetics of Rigid Bodies: Force and Acceleration 17. Plane Kinetics of Rigid Bodies: Work and Energy 18. Plane Kinetics of Rigid Bodies: Impulse and Momentum 19. Motion of Rigid Bodies in Three Dimensions 20. Vibrations Appendices A. Prior Basic Mathematics B. Review of Basic Mathematics C. Infinitesimal Angular Displacement D. Simultaneous Equations Solver E. Digital Root Finder F. Supplementary Software Answers to Selected Developmental Exercises and Problems Index 1. Fundamentals 2. Forces 3. Equilibrium of Particles 4. Moments of Forces 5. Equilibrium of Rigid Bodies 6. First Moments: Centroids and Centers of Gravity 7. Second Moments: Moments of Inertia 8. Structures 9. Friction 10. Virtual Work 11. Kinematics of Particles 12. Kinetics of Particles: Force and Acclleration 13. Kinetics of Particles: Work and Energy 14. Kinetics of Particles: Impulse and Momentum 15. Plane Kinematics of Rigid Bodies 16. Plane Kinetics: Force and Acceleration 17. Plane Kinetics: Work and Energy 18. Plane Kinetics: Impulse and Momentum 19. Motion of Rigid Bodies in Three Dimensions 20. Vibrations Appendices Prior Basic Mathematics Review of Basic Mathematics Infinitesimal Angular Displacement Simultaneous Equations Solver Digital Root Finder Supplementary Software Answers to Selected Developmental Exercises and Problems Index

A method for the geometrically nonlinear analysis of compressively loaded prismatic composite structures

Abstract: A method was developed for the geometrically nonlinear analysis of the static response of thin-walled stiffened composite structures loaded in uniaxial or biaxial compression. The method is applicable to arbitrary prismatic configurations composed of linked plate strips, such as stiffened panels and thin-walled columns. The longitudinal ends of the structure are assumed to be simply supported, and geometric shape imperfections can be modeled. The method can predict the nonlinear phenomena of postbuckling strength and imperfection sensitivity which are exhibited by some buckling-dominated structures. The method is computer-based and is semi-analytic in nature, making it computationally economical in comparison to finite element methods. The method uses a perturbation approach based on the use of a series of buckling mode shapes to represent displacement contributions associated with nonlinear response. Displacement contributions which are of second order in the model amplitudes are incorported in addition to the buckling mode shapes. The principle of virtual work is applied using a finite basis of buckling modes, and terms through the third order in the model amplitudes are retained. A set of cubic nonlinear algebraic equations are obtained, from which approximate equilibrium solutions are determined. Buckling mode shapes for the general class of structure are obtained using the VIPASA analysis code within the PASCO stiffened-panel design code. Thus, subject to some additional restrictions in loading and plate anisotropy, structures which can be modeled with respect to buckling behavior by VIPASA can be analyzed with respect to nonlinear response using the new method. Results obtained using the method are compared with both experimental and analytical results in the literature. The configurations investigated include several different unstiffened and blade-stiffening panel configurations, featuring both homogeneous, isotropic materials, and laminated composite material.

Some considerations on the elastic theory for nematic liquid crystals

Abstract: The elastic theory for nematic liquid crystals is critically analysed. After a review on the variational calculus formalism, the range of applicability of the Lagrangian method for the solution of practical problems is discussed. It is underlined that only a limited number of problems can be solved by means of a variational approach. The role at the Jacobi equation is also discussed. The importance of the non linear character of the K13-problem is analyzed in the framework of a simple molecular model. Finally, the principle of virtual work is applied to the elastic theory of nematic liquid crystals. Our analysis shows that the K13 elastic problem is an ill-posed one, since this problem can only be solved by means of a variational, or virtual work, approach by modifying the bulk elastic free energy and taking into account new terms quadratic in the second order deviatives. However it is necessary to remember that, in the proximity of a surface, a spatial variation of the density and of the scalar ...

A general force/torque relationship and kinematic representation for flexible link manipulators

Abstract: A technique to determine the relationship between the generalized joint torque and the generalized force exerted by the end-effector of a flexible link manipulator on the environment is presented. First, a systematic technique which accounts for the links' bending and torsion is used to determine the kinematic equations of the manipulator. A procedure to determine a “pseudo” Jacobian matrix which accounts for the links' flexibility is then developed. In the determination of the pseudo Jacobian matrix, partial derivatives are not used, thus reducing inaccuracies resulting from approximating the bending and torsion of the links. Using the principle of virtual work, a force and torque relationship between the joints and end-effector is obtained. To demonstrate the validity of the technique presented, computer simulations of the resulting force/torque relationship are compared to results obtained experimentally.

An efficient approximate analysis method based on an exact univariate model for the element loads

Abstract: Approximate analysis modules for structural design are usually based on a linear Taylor expansion of the nodal displacements in terms of the reciprocals of the design variables Direct approximations of the member forces have received lesser attention This paper describes an approach for the direct calculation of the member forces in a truss as a function of the design variables It is based on the exact expression of the member forces if only one design variable is allowed to vary at a time In the case of an arbitrary move in the design space the method gives approximate results of a very good quality This is obtained by enforcing zero order homogeneity of the element loads and by refining the results via a virtual work equation The method is illustrated with numerical results on previously published test cases for elastic trusses Preliminary results for an elastic frame are also presented This new approximate force model is shown to yield excellent results

Dynamics of magnetic domains and walls

Abstract: General equations are proposed to describe the simultaneous rotations of the magnetization vectors and the displacements of curved domain walls in one pair of magnetostatically coupled magnetic films separated by a variable distance. Leakage‐field energy is written in the ‘‘transmission‐line’’ approximation. The effects of dissipation and the constraint of flux continuity across a domain wall are handled by d’Alembert’s virtual work principle. The result is a set of coupled equations of the following kinds: (1) dynamic torque balance at each point inside a domain, (2) wall‐domain constraint due to flux continuity, (3) boundary condition on domain magnetization which depends on instantaneous wall positions, and (4) wall velocity. Within certain limitations these equations apply to the core of an inductive magnetic recording head.

Dynamics of domains and walls in soft magnetic films

Abstract: General equations are derived to describe the simultaneous nonuniform planar rotations of the magnetization vector and displacements of curved domain walls and their junctions in soft magnetic films. These equations take into account effects of exchange stiffness, magnetic anisotropy, external and either long- or short-range demagnetizing fields, wall energy, and dissipation. The case of a matched film pair using the capacitor or transmission-surface approximation for its short-range demagnetizing energy is considered. The theory is founded on energy and dissipation functionals including domain and wall terms. The constraint of wall-normal magnetization continuity across a domain wall is handled by a method of implementing d'Alembert's virtual work principle without introducing Lagrange multipliers. The result is a set of coupled equations expressing the dynamic torque balance at points inside domains, the wall-domain constraint due to wall-normal magnetization continuity, an additional boundary condition coupling domain magnetization and wall curves, and the wall velocity. >

Modélisation mécanique d'un milieu polyphasique par la méthode des puissances virtuelles

Abstract: Making use of the virtual work method, a mechanical model for a multiphase continuum is proposed. This method makes it possible to account in a consistent way for the interactions between the phases and leads to a mathematical description of the internal forces within each physe. The results thus obtained are applied to the particular case of a saturated porous medium.

Computer-aided modeling and control of multibody systems for robotics application

Abstract: A closed-form formalism for modeling an analysis of constrained multibody systems subject to control and its application to robot dynamics are presented. The algorithm is derived from the virtual work principle and is cast in a recursive and matrix-vector form. The effects of a moving reference frame are included allowing simulation of a multibody system such as a robot on a moving base. The formalism can be applied to holonomic and simple nonholonomic systems. A symbolic program that can formulate, simplify, and linearize the equations of motion has been developed. The procedure symbolically eliminates the constraint forces and redundant coordinates, and generates the equations in a closed form. The formalism is then used to simulate a biped robot walking on a vibrating platform. Optimal control is used to compute the joint torques. The study shows that the closed-form formalism holds promise in the simulation of robotic systems. >

A planar nonlinear model of the human spine

Abstract: In this study the derivation of a two-dimensional mathematical model of the human lumbar spine and its approximate solution using the method of finite elements is described. The computer model LUSP (Lumbar Spine) serves as a basis for studying the kinematic and load-bearing behaviour of the lumbar spine. The underlying working hypothesis is that the smallest spinal unit, the so called functional spinal unit (Junghanns, reflects the basic characteristic behaviour of the musclefree spine. On the basis of Lagrange's virtual work principle the nonlinear static and dynamic equations of motion for a sagitally symmetrical spine model of comprising rigid bodies, springs, beams and dampers are derived. The finite element method is used as an appropriate approximation scheme. Intensive research was conducted to provide the necessary geometrical and material input data. Special attention was paid to achieving a realistic description of the nonlinear stress-strain relationships for the soft tissue involved. A database-type preprocessor and a graphics-oriented postprocessor are made for convenient handling of the input and output data. The efficiency of the present computer model is demonstrated by means of an orthopaedic-biomechanical study on degenerative phenomena in so-called juxta-fused lumbosacral motion segments.

On a Correspondence between Mechanical and Thermal Fields in Composites with Slipping Interfaces

Abstract: The present paper is concerned with composites in which the constituent interfaces are weak in shear and therefore exhibit shear deformation associated with sliding. Thermomechanical loadings of such systems are considered which consist of homogeneous traction or displacement boundary conditions and a uniform temperature change on the outside surface of the composite. For binary systems with isotropic constituents, it is shown that the actual fields in the purely thermal problem can be uniquely determined from the solution of the purely mechanical problem. This correspondence relation is used to determine the effective thermal strain and stress tensors on the basis of the effective mechanical properties. For multi—phase systems with anisotropic constituents undergoing interface slip and separation, the theorem of virtual work is used to establish a similar relation between the effective thermal tensors and the mechanical concentration factors and constituent properties of the composite.

Statical-Kinematic Analysis

Abstract: The objective of this chapter is to modernize and further develop the basic concepts of statical-kinematic analysis, and to refine the classification of generic structural systems. This is done with a view to both a rigorous treatment of the topic and the development of analytical criteria amenable to computerized analysis.

Hierarchical three dimensional curved beam element based on p-version

Abstract: This paper presents a three node curved three dimensional beam element for linear static analysis where the element displacement approximation in the axial (ξ) and transverse directions (η and ζ) can be of arbitrary polynomial orders pξ, pη and pζ. This is accomplished by, first constructing one dimensional hierarchical approximation functions and the corresponding nodal variable operators in ξ, η and ζ directions using Lagrange interpolating polynomials and then taking the products (also called tensor product) of these hierarchical one dimensional approximation functions and the corresponding nodal variable operators. The resulting approximation functions and the corresponding nodal variables for the three dimensional beam element were hierarchical. The formulation guarantees C0 continuity. The element properties are established using the principle of virtual work. In formulating the properties of the element all six components of the stress and strain tensor are ratained. The geometry of the beam element is defined by the coordinates of the nodes located at the axis of the beam and node point vectors representing the nodal cross-sections. The results obtained from the present formulation are compared with analytical solutions (when available) and the h-models using isoparametric three dimensional solid elements. The formulation is equally effective for very slender as well as deep beams since no assumptions are made regarding such conditions during the formulation.

High-Order Mixture Homogenization of Fiber-Reinforced Composites

Abstract: : An asymptotic mixture theory of fiber-reinforced composites with periodic microstructure is presented for rate-independent inelastic responses, such as elastoplastic deformation. Key elements are the modeling capability of simulation critical interaction across material interfaces and the inclusion of the kinetic energy of microdisplacement. The construction of the proposed mixture model, which is deterministic, instead of phenomenological, is accomplished by resorting to a variational approach. The principle of virtual work is used for total quantities to derive mixture equations of motion and boundary conditions, while Reissner's mixed variational principle (1984, 1986), applied to the incremental boundary value problem yields consistent mixture constitutive relations. In order to assess the model accuracy, numerical experiments were conducted for static and dynamic loads. The prediction of the model in the time domain was obtained by an explicit finite element code. DYNA2D is used to furnish numerically exact data for the problems by discretizing the details of the microstructure. On the other hand, the model capability of predicting effective tangent moduli was tested by comparing results with NIKE2D. In all cases, good agreement was observed between the predicted and exact data for plastic, as well as, elastic responses. Keywords: Fiber reinforced composites; Mixture theory; Structural properties. (kt)

Assessment of Finite Element Approximations for Nonlinear Flexible Multibody Dynamics

Abstract: : Systematic investigation is made of effects of kinematic assumptions and finite element approximations in the context of nonlinear flexible multibody dynamics. Two nonlinear beam finite elements are consistently derived from virtual work principle using Bernoulli Euler and Timoshenko beam kinematics. Initial assessment is made by studying convergence properties of element formulations with eigenvalue problems: free vibration, static buckling, and dynamic buckling. Equations of motion are derived for rigid central body with flexible appendage using virtual work principle. Virtual work principle allows natural and consistent discretization of flexible appendage using finite element method. Nonlinearities in flexibility are explored through dynamics examples using beam finite elements. Application of dynamics formulation is made to a realistic scenario involving space shuttle remote manipulator arm with attached payload. Contribution of nonlinear theory, in both formulation and solution, is assessed.