Showing papers on "Virtual work published in 2014"

Derivation of microstructured continua from lattice systems via principle of virtual works: the case of masonry-like materials as micropolar, second gradient and classical continua

Abstract: The description of the mechanical behaviour of brick/block masonry through equivalent continua is presented here as a paradigmatic example of the problem of gross modelling of discontinuous and heterogeneous materials as continua with microstructure. The approaches reported in the literature differ for the way identification of the continuum is carried out or the nature of the continuum itself. In this paper, continuous models equivalent to rigid particle systems with free or constrained rotations are derived within the general framework of the principle of virtual work. In particular, an integral equivalence procedure is used to derive micropolar, second gradient and classical models. The non-classical models have in the field equations non-standard kinematic and static descriptors accounting for the presence of the material internal structure. The differences in the material responses of the various continua are identified referring to their internal work formulas. For the reference material, it is shown that, unlike the Cauchy continuum, both micropolar and second gradient models are effective in the presence of load and geometrical singularities, which involve significant scale effects on the material response. On the other hand, the second gradient model, as well as the classical model, disregards the role of relative rotation between the local rigid rotation (macrorotation) and the microrotation, which is related to the presence of non-symmetric strains. This circumstance, significant in strongly anisotropic systems, allow us to point out the advantages of the micropolar modelling especially for orthotropic masonry assemblies made of elements of any size. These statements are discussed by means of selected numerical examples of masonry panels differing in size, shape and arrangement, under shear loading conditions.

Comparison of the absolute nodal coordinate and geometrically exact formulations for beams

Abstract: The modeling of flexibility in multibody systems has received increase scrutiny in recent years. The use of finite element techniques is becoming more prevalent, although the formulation of structural elements must be modified to accommodate the large displacements and rotations that characterize multibody systems. Two formulations have emerged that have the potential of handling all the complexities found in these systems: the absolute nodal coordinate formulation and the geometrically exact formulation. Both approaches have been used to formulate naturally curved and twisted beams, plate, and shells. After a brief review of the two formulations, this paper presents a detailed comparison between these two approaches; a simple planar beam problem is examined using both kinematic and static solution procedures. In the kinematic solution, the exact nodal displacements are prescribed and the predicted displacement and strain fields inside the element are compared for the two methods. The accuracies of the predicted strain fields are found to differ: The predictions of the geometrically exact formulation are more accurate than those of the absolute nodal coordinate formulation. For the static solution, the principle of virtual work is used to determine the solution of the problem. For the geometrically exact formulation, the predictions of the static solution are more accurate than those obtained from the kinematic solution; in contrast, the same order of accuracy is obtained for the two solution procedures when using the absolute nodal coordinate formulation. It appears that the kinematic description of structural problems offered by the absolute nodal coordinate formulation leads to inherently lower accuracy predictions than those provided by the geometrically exact formulation. These observations provide a rational for explaining why the absolute nodal coordinate formulation computationally intensive.

Coupled thermoelastic effect in free vibration analysis of anisotropic multilayered plates and FGM plates by using a variable-kinematics Ritz formulation

Abstract: A fully coupled thermoelastic formulation is developed to deal with free vibration analysis of anisotropic composite plates and isotropic/sandwich FGM plates. The proposed formulation is developed by combining refined hierarchical plate models and a trigonometric Ritz method. The temperature is considered as a primary variable and allows the evaluation of the temperature field effects in the free vibration analysis. The temperature profile across the plate thickness is always modeled with a layer-wise kinematics description, nevertheless both equivalent single layer and layer-wise approaches are properly and effectively used for the displacement variables. In the 2D and quasi-3D higher-order variable-kinematics plate theories, each displacement variable, in the displacement field, is treated independently from the others. Such artifice allows to select scrupulously each expansion order for each primary variable regarding to the required accuracy and the computational cost. So-called Ritz fundamental primary nuclei related to the coupled thermal and mechanical fields are generated by virtue of an unconventional principle of virtual displacement accounting for the internal thermal virtual work to reproduce the coupling effect. Each fundamental primary nucleus is mathematically invariant with respect to the used kinematics description, the employed expansion orders and the chosen Ritz functions. The thermoelastic coupling is investigated in terms of natural frequencies and the effect of stacking sequence and length-to-thickness ratio for lower and higher modes is discussed.

On a consistent finite-strain plate theory based on three-dimensional energy principle

Abstract: This paper derives a finite-strain plate theory consistent with the principle of stationary three-dimensional potential energy under general loadings with a fourth-order error. Starting from the three-dimensional nonlinear elasticity (with both geometrical and material nonlinearity) and by a series expansion, we deduce a vector plate equation with three unknowns, which exhibits the local force-balance structure. The success relies on using the three-dimensional field equations and bottom traction condition to derive exact recursion relations for the coefficients. Associated weak formulations are considered, leading to a two-dimensional virtual work principle. An alternative approach based on a two-dimensional truncated energy is also provided, which is less consistent than the first plate theory but has the advantage of the existence of a two-dimensional energy function. As an example, we consider the pure bending problem of a hyperelastic block. The comparison between the analytical plate solution and available exact one shows that the plate theory gives second-order correct results. Compared with existing plate theories, it appears that the present one has a number of advantages, including the consistency, order of correctness, generality of loadings, applicability to finite-strain problems and no involvement of non-physical quantities.

On a consistent finite-strain plate theory based on 3-D energy principle

Abstract: This paper derives a finite-strain plate theory consistent with the principle of stationary three-dimensional (3-D) potential energy under general loadings with a third-order error. Staring from the 3-D nonlinear elasticity (with both geometrical and material nonlinearity) and by a series expansion, we deduce a vector plate equation with three unknowns, which exhibits the local force-balance structure. The success relies on using the 3-D field equations and bottom traction condition to derive exact recursion relations for the coefficients. Associated weak formulations are considered, leading to a 2-D virtual work principle. An alternative approach based on a 2-D truncated energy is also provided, which is less consistent than the first plate theory but has the advantage of the existence of a 2-D energy function. As an example, we consider the pure bending problem of a hyperelastic block. The comparison between the analytical plate solution and available exact one shows that the plate theory gives second-order correct results. Comparing with existing plate theories, it appears that the present one has a number of advantages, including the consistency, order of correctness, generality of the loadings, applicability to finite-strain problems and no involvement of unphysical quantities.

Dynamic simulation of frictional contacts of thin beams during large overall motions via absolute nodal coordinate formulation

Abstract: The aim of this study is to develop an approach of simulating the frictional contact dynamics of thin beams with large deformations and continuous contact zones of large size during their large overall motions. For this purpose, the thin beams are meshed via initially straight and gradient deficient thin beam elements of the absolute nodal coordinate formulation (ANCF) degenerated from a curved beam element of ANCF. A detection strategy for contact zone is pro- posed based on the combination of the minimal distance criterion and master-slave approach. By making use of the minimal distance criterion, the closest points of two thin beams can be found efficiently. The master-slave approach is employed to determine the continuous con- tact zone. The generalized frictional contact forces and their Jacobians are derived based on the principle of virtual work. Gauss integration is used to integrate the contact forces over the continuous contact zone. The generalized-alpha method is used to solve the dynamic equations of contacting beams. Numerical simulations of four static and dynamic contact prob- lems, including those with continuous contact zones of large size, are completed to validate the high perfor- mance of the approach.

A novel analytical model for flexure-based proportion compliant mechanisms

Abstract: This paper proposes a novel analytical model for flexure-based proportion compliant mechanisms The displacement and stiffness calculations of such flexure-based compliant mechanisms are formulated based on the principle of virtual work and pseudo rigid body model (PRBM) According to the theory and method, a set of closed-form equations are deduced in this paper, which incorporate the stiffness characteristics of each flexure hinge, together with the other geometric and material properties of the compliant mechanism The rotation center point for a corner-filleted flexure hinge is investigated based on the finite element analysis (FEA) and PRBM An empirical equation for the rotational angle is fitted in this paper in order to calculate accurately the position of the end-point of the flexure hinge The displacement proportion equation for such mechanisms is derived according to the new approach Combining the new proposed design equation and the existed stiffness equation, a new proportion compliant mechanism with corner-filleted flexure hinges is designed by means of the least squares optimization The designed models are verified by finite element analysis

Mathematical Modeling of Large Elastic-Plastic Deformations

Abstract: The paper is devoted to development and numerical implementation for a method for investigation of stress-strain state of the solids with large elastic-plastic deformations. Calculation algorithm is based on the linearized equation of virtual work, defined to actual state. The arc-length method is used. A spatial discretization is based on the finite element method. The developed algorithm of investigation of large elastic-plastic deformations is tested on the solution of the necking of circular bar.

Hysteretic Finite Elements for the Nonlinear Static and Dynamic Analysis of Structures

Abstract: A new numerical analysis procedure is presented for the nonlinear analysis of structures. The proposed methodology is developed within the framework of the direct stiffness method and the hysteretic formulation of finite elements. The derived numerical scheme relies on the natural evolution of localized inelastic quantities within the element, that is, the plastic deformation evaluated at properly defined collocation points rather than the evaluation of global and varying state matrices. This is accomplished by considering the additive decomposition of the total strain rate into elastic and plastic parts. Using the principle of virtual work, an equilibrium expression is derived in which the total applied load is equilibrated by an elastic internal force vector and an additional term acting as a nonlinear correction to the elastic component. The evolution of the plastic components is based on a smooth multiaxial hysteretic law that is derived within the framework of classical plasticity. Examples are presented that demonstrate the validity of the proposed method and its computational advantages with respect to existing methods of inelastic analysis.

An isogeometric approach for the analysis of composite steel–concrete beams

Abstract: An isogeometric approach based on non-uniform rational B-spline (NURBS) basis functions is presented for the analysis of composite steel–concrete beams. A refined high-order theory is considered in deriving the governing equations using the principle of virtual work. The employed theory satisfies all the kinematic and stress continuity conditions at the layer interfaces and considers effects of the transverse normal stress and transverse flexibility. The global displacement components, described by polynomial or combinations of polynomial and exponential expressions, are superposed on local ones chosen based on the layerwise concepts. The present isogeometric formulation does not need incorporating any shear correction factor. Moreover, in the present isogeometric formulation, the number of unknowns is independent of the number of layers. The proposed isogeometric formulation is validated by comparing the present results with the available published and the three-dimensional (3D) finite element results. In addition to correctly predicting the distribution of all stress components of the composite steel–concrete beams, the proposed formulation is computationally very economic.

Finite element method for stability and free vibration analyses of non-prismatic thin-walled beams

Abstract: In this paper, a numerical method is presented for the free vibration and stability analyses of tapered thin-walled beams with arbitrary open cross sections. The proposed method takes the flexural–torsional coupling effect of tapered thin-walled beams with arbitrary open cross sections into account. The total potential energy is derived for an elastic behavior from the strain energy, the kinetic energy and work of the loads applied on the cross section contour. Free vibration is considered in the presence of harmonic excitations. The effects of the initial stresses and load eccentricities are also considered in stability analysis. The governing equilibrium equations, motion equations and the associated boundary conditions are derived from the stationary condition. As in the presence of tapering, stiffness quantities are not constant; therefore, the power series approximation is used to solve the fourth-order differential equations. Displacement components and cross-section properties are expanded in terms of power series of a known degree. Then, the shape functions are obtained by deriving the deformation shape of tapered thin-walled member as power series form. Finally, stiffness and mass matrices are carried out by means of the principle of virtual work along the member׳s axis. In order to measure the accuracy and check the validity of this method, the natural frequencies and buckling loads of non-prismatic thin-walled beams with web and flange tapering and various boundary conditions are obtained and compared to the results of finite element analysis using Ansys software and those of other available numerical and analytical ones. It can be seen that the results of present study are in a good agreement with other available theoretical and analytical methods.

Kinematics and Dynamics of Deployable Structures with Scissor-Like-Elements Based on Screw Theory

Abstract: Because the deployable structures are complex multi-loop structures and methods of derivation which lead to simpler kinematic and dynamic equations of motion are the subject of research effort, the kinematics and dynamics of deployable structures with scissor-like-elements are presented based on screw theory and the principle of virtual work respectively. According to the geometric characteristic of the deployable structure examined, the basic structural unit is the common scissor-like-element(SLE). First, a spatial deployable structure, comprised of three SLEs, is defined, and the constraint topology graph is obtained. The equations of motion are then derived based on screw theory and the geometric nature of scissor elements. Second, to develop the dynamics of the whole deployable structure, the local coordinates of the SLEs and the Jacobian matrices of the center of mass of the deployable structure are derived. Then, the equivalent forces are assembled and added in the equations of motion based on the principle of virtual work. Finally, dynamic behavior and unfolded process of the deployable structure are simulated. Its figures of velocity, acceleration and input torque are obtained based on the simulate results. Screw theory not only provides an efficient solution formulation and theory guidance for complex multi-closed loop deployable structures, but also extends the method to solve dynamics of deployable structures. As an efficient mathematical tool, the simper equations of motion are derived based on screw theory.

A method for modeling three-dimensional flexible mechanisms based on an equivalent rigid-link system

Abstract: Accurate modeling of flexible mechanisms is an open research topic, and different models have been presented since the 1970s. In this work, a novel approach for modeling of three-dimensional flexible mechanisms is presented, based on an equivalent rigid-link system, with respect to which elastic deformations are defined and computed. Concepts of three-dimensional kinematics are used in order to define an effective relationship between the rigid body and the elastic motion. The model is based on a compact kinematic formulation and, for a specific mechanism, there is no need for customizing the formulation. By using the principle of virtual work, a coupled dynamic formulation is found. A crucial advantage of this method is that it is not necessary to explicitly formulate the compatibility equations expressing the link connections, since they are included in the matrices of the system dynamics. The model was applied to a specific three-dimensional flexible mechanism. The results, compared with the Adams-Flex...

Free vibration analysis of inflatable beam made of orthotropic woven fabric

Abstract: The free vibration of inflatable beams was studied using the dynamic stiffness method. A 3D Timoshenko beam with a homogeneous orthotropic woven fabric (OWF) was considered. Using the usual total Lagrangian form of the virtual work principle, the model took the geometric nonlinearities and the inflation pressure follower force effect into account. The nonlinear equilibrium equations were then linearized around the prestressed reference configuration. The exact dynamic stiffness matrix was developed by directly solving the governing differential equations of a 3D loaded inflatable beam in a free vibration. The effects of the inflation pressure, fabric mechanical properties and the boundary conditions on the natural frequencies and mode shapes of the inflatable beams were demonstrated. The proposed model was validated favorably through its comparison with a 3D thin shell finite element model and an isotropic fabric model found in the literature.

Free Vibration Behaviour of Functionally Graded Plates Using Higher-Order Shear Deformation Theory

Abstract: The prime aim of the present study is to develop analytical formulations and solutions for the free vibration analysis of functionally graded plates (FGPs) using higher order shear deformation theory (HSDT) without enforcing zero transverse shear stress on the top and bottom surfaces of the plate. The theoretical model presented herein incorporates the transverse extensibility which accounts for the transverse effects. The equations of equilibrium and boundary conditions are derived using the principle of virtual work. Solutions are obtained for FGPs in closed-form using Navier's technique and solving the eigen value equation. The present results are compared with the solutions of the other HSDTs available in the literature. It can be concluded that the proposed theory is accurate and efficient in predicting the vibration behaivour of functionally graded plates.

Interfaces endowed with nonconstant surface energies revisited with the d’Alembert–Lagrange principle

Abstract: The equation of motions and the conditions on surfaces and edges between fluids and solids in presence of non-constant surface energies, as in the case of surfactants attached to the fluid particles at the interfaces, are revisited under the principle of virtual work. We point out that adequate behaviors of surface concentrations may drastically modify the surface tension which naturally appears in the Laplace and the Young-Dupre equations. Thus, the principle of virtual work points out a strong difference between the two revisited concepts of surface energy and surface tension.

Mathematics of the Masonry–Like model and Limit Analysis

Abstract: These notes present a brief introduction to the mathematics of equilibrium of no–tension (masonry–like) materials. We first review the constitutive equations using the idea that the stress of the no–tension material must be always negative semidefinite. The strain tensor is naturally split into the sum of the elastic strain and fracture strain. The stress depends linearly on the elastic strain via the fourth–order tensor of elasticities. Then we consider a body made of a no–tension material, introduce the loads and the total energy of the deformation with is the sum of the internal energy and the energy of the loads. Then we examine the question whether the total energy is bounded from below. That brings us to the important notion of the strong compatibility of loads. The loads are strongly compatible if they can be equilibrated (in the sense of the principle of virtual work) by a square integrable negative semidefinite stress field. The total energy is bounded from below if and only if the loads are strongly compatible. The notion of strong compatibility of loads is central in the limit analysis and in a strengthened form in the theory of existence of equilibrium states. Roughly speaking, if the loads are strongly compatible, then the body is safe, while otherwise strongly incompatible loads lead to the collapse of the body. To determine whether the loads are strongly compatible, it is not necessary to solve the full displacement problem, it suffices to find the negative semidefinite square integrable stress field, which is independent of the constitutive theory.

Diffuse Interface Methods for Multiple Phase Materials: An Energetic Variational Approach

Abstract: In this paper, we introduce a diffuse interface model for describing the dynamics of mixtures involving multiple (two or more) phases. The coupled hydrodynamical system is derived through an energetic variational approach. The total energy of the system includes the kinetic energy and the mixing (interfacial) energies. The least action principle (or the principle of virtual work) is applied to derive the conservative part of the dynamics, with a focus on the reversible part of the stress tensor arising from the mixing energies. The dissipative part of the dynamics is then introduced through a dissipation function in the energy law, in line with the Onsager principle of least energy dissipation. The final system, formed by a set of coupled time-dependent partial differential equations, reflects a balance among various conservative and dissipative forces and governs the evolution of velocity and phase fields. To demonstrate the applicability of the proposed model, a few two-dimensional simulations have been carried out, including (1) the force balance at the three-phase contact line in equilibrium, (2) a rising bubble penetrating a fluid-fluid interface, and (3) a solid particle falling in a binary fluid. The effects of slip at solid surface have been examined in connection with contact line motion and a pinch-off phenomenon.

Kinematic, Stiffness, and Dynamic Analyses of a Compliant Tensegrity Mechanism

Abstract: The objective of this study is to present an analytical procedure for analysis of a compliant tensegrity mechanism focusing on its stiffness and dynamic characteristics. The screw calculus is used to derive the static equations and stiffness matrix of a full degree-of-freedom tensegrity mechanism, and the equations of motion are derived based on the principle of virtual work. Finally, some numerical examples are solved for the inverse dynamics of the mechanism.

Equivalence and dynamic analysis for jointed trusses based on improved finite elements

Abstract: Deployable trusses that are widely applied in space missions utilize many hinged members to adapt to different geometrical shapes and work conditions. This paper presents two improved finite elements for a link and a bending beam to calculate dynamic characteristics of non-jointed and jointed trusses. First, the axial and transverse wave motion equations of a beam are used to get the shape functions of the link and the beading beam in their axial and transverse directions. Second, the dynamic matrices of stiffness and mass for a link and a bending beam are established on the basis of virtual work principle. The dynamic matrix and a theoretical equation are used to calculate the natural frequency of a cantilever bar. The comparison of the two results shows the effectiveness of the dynamic stiffness matrix. These improved elements are proved to be more accurate in calculating the responses of structures at high frequencies than common elements. Equivalent models of non-jointed and jointed structures are obt...

Nonlinear dynamical behavior analysis on rigid-flexible coupling mechanical arm of hydraulic excavator

Abstract: In order to accurately describe the dynamic model of hydraulic excavator's mechanical arm,modal functions were adopted to describe the elastic deformation of the niechanic'al arm.Lagrange theorem and the principle of virtual work were used to establish rigid-flexible coupling nonlinear dynamic equations of the arm frame system.The dynamic equations were numerically solved by MATLAB.The rigid-flexible coupling model of the arm was established and simulated by the simulation software ADAMS and NASTRAN.It is shown that the modeling method of dynamic equations adopted in this paper is correct by comparing the both results.The modal responses were numerically calculated the sensitivity of the first natural frequency to the geometric parameters related was investigated and the main modal parameters were analyzed which influence dynamic characteristics of the arm.The results provide the basis for the further optimization of excavator structure and the motion error control.