Virtual Work Approach Research Articles

Structural stability of a lightsail for laser-driven interstellar flight

The structural stability of a lightsail under the intense laser flux necessary for interstellar flight is studied analytically and numerically. A sinusoidal perturbation is introduced into a two-dimensional thin-film sail to determine if the sail remains stable or if the perturbations grow in amplitude. A perfectly reflective sail material that gives specular reflection of the laser illumination is assumed in determining the resulting loading on the sail, although other reflection models can be incorporated as well. The quasi-static solution of the critical point between shape stability and instability is found by equating the bending moments induced on the sail due to radiation pressure with the restoring moments caused by the strength of the sail material and the tension applied at the edges of the sail. From this quasi-static solution, analytical expressions for the critical value of elastic modulus and boundary tension magnitude are found as a function of sail properties (e.g., thickness) and the amplitude and wave number of the initial sinusoidal perturbation. These same expressions are also derived from a more formal variational energy (virtual work) approach. A numerical model of the complete lightsail dynamics is developed by discretizing the lightsail into rectangular finite elements. By introducing torsional and rectilinear springs between the elements into the numerical model, a hierarchy of models is produced that can incorporate the effects of bending and applied tension. The numerical models permit the transient dynamics of a perturbed lightsail to be compared to the analytic results of the quasi-static analysis, visualized as stability maps that show the rate of perturbation growth as a function of sail thickness, elastic modulus, and applied tension. The analytic theory is able to correctly predict the stability boundary found in the numerical simulations. The stiffness required to make a thin lightsail stable against uncontrolled perturbation growth appears to be unfeasible for known materials, however, a relatively modest tensioning of the sail (e.g., via an inflatable structure or spinning of the sail) is able to maintain the sail shape under all wavelengths and amplitudes of perturbations.

Thermal Postbuckling of Temperature-Dependent Functionally Graded Nanocomposite Annular Sector Plates Reinforced by Carbon Nanotubes

In this study, the thermal buckling and postbuckling of functionally graded (FG) nanocomposite annular sector plates reinforced by carbon nanotubes (CNTs) are numerically analyzed. The effective material properties of FG nanocomposite are temperature-dependent (TD) and evaluated via the modified micromechanical method and rule of mixture. Based on the higher-order shear deformation theory (HSDT) and using the principle of virtual work and variational differential quadrature (VDQ) approach, the unified weak form of discretized nonlinear governing equilibrium equations is derived. Then, by using the linear part of equations and solving the derived eigenvalue problem, the critical temperature rise and associated mode shapes are obtained, which are used as the initial guess in solving the nonlinear thermal postbuckling problem. The pseudo-arc-length method and an iterative solver are employed to obtain the nonlinear thermal postbuckling equilibrium path of the FG nanocomposite annular sector plates. The influences of geometrical parameters, boundary conditions (BCs), CNT volume fraction, and CNT distribution pattern on the critical temperature rise and thermal postbuckling behavior of the FG nanocomposite annular sector plates are evaluated and discussed. Also, comparisons are made between the results considering the TD and temperature-independent (TID) properties. It is demonstrated that for higher values of sector angle, the effect of sector angle on the critical temperature rise and thermal postbuckling path is negligible. Moreover, by increasing the sector angle, the effect of BCs of straight edges vanishes, and the critical temperature rise and thermal postbuckling curves of for BCs of CSCS and SCSC approach those for CCCC and SSSS ones.

Bending analysis of thick functionally graded piezoelectric rectangular plates using higher-order shear and normal deformable plate theory

In this paper, bending-stretching analysis of thick functionally graded piezoelectric rectangular plates is studied using the higher-order shear and normal deformable plate theory. On the basis of this theory, Legendre polynomials are used for approximating the components of displacement field. Also, the effects of both normal and shear deformations are encountered in the theory. The governing equations are derived using the principle of virtual work and variational approach. It is assumed that plate is made of piezoelectric materials with functionally graded distribution of material properties. Hence, exponential function is used to modify mechanical and electrical properties through the thickness of the plate. Finally, the effect of material properties, electrical boundary conditions and dimensions are investigated on the static response of plate. Also, it is shown that results of the presented model are close to the three dimensional elasticity solutions.

A Virtual Work Approach to Modeling the Nonlinear Behavior of Steel Frames

A Virtual Work Approach to Modeling the Nonlinear Behavior of Steel Frames

Rejection and Correction of the Fracture Mechanics Singularity Approach with its Associated Tree Modes of Crack Separation

Limit Analysis is an prescribed exact approach of Wood Science, what is shown to also apply for wood Fracture Mechanics. Knowledge of the gradual elastic to plastic behavior and of the imitation by non-linear elasticity (and J-integral) is shown to be not needed. The linear – full plastic limit approach delivers an elastic lower bound, up to this full plastic boundary, the fracture- or yield criterion, where ultimate load behavior is described, by virtual work approach and “flow” by the normality rule. This delivers the possibility to look at any equilibrium system, which satisfies compatibility and boundary conditions and nowhere exceeds this “flow” criterion and is verified by test data. Because the accepted singularity approach does not deliver a right mixed mode fracture criterion, it is necessary to make comparisons with other possible Airy stress functions. Therefore, the derivation of the accepted, general applied, elementary singularity solution with its 3 failure modes, is discussed and compared with new theory. This new limit analysis theory is based on an older, forgotten, Airy stress function, and shows e.g., by the new approach and application to wood, that there is no real difference between strength theory and fracture mechanics and between linear and non-linear theory. It delivers the, empirical verified, exact mixed mode failure criterion for wood; shows that stresses in the isotropic wood matrix also have to be regarded separately, to explain the, only by isotropy, possible, extremely high triaxial hydrostatic stress, and stress increase by the stress spreading effect. Therefore the stresses and strengths of the isotropic wood matrix are derived. The transformation to total stresses, including the reinforcement is shortly given for the empirical verification and for literature reference. Therefore, only the derivation of the necessary corrections of the singularity approach, for isotropic material, is regarded. This leads to a necessary rejection of the, tree failure modes, singularity approach of Irwin and of associate equations. By the splitting in 3 modes, there is no compatibility and no mixed mode fracture criterion. Instead there are tree, each excluding, Airy stress functions, thus 3, each excluding, compatibility equations. Necessary is one mixed mode solution for the total load. Then the solution also is known for separate acting (thus non zero) loading components. This is done in § 4, necessarily in elliptic coordinates, to know failure, by the highest tangential, uniaxial tensile stress in the crack boundary. The tangential direction in polar coordinates is not tangential to the elliptic first expanded of the crack boundary. Therefore then not the right KIIc values are obtained. The expression in elliptic coordinates delivers (by the highest empirical correlation) the failure criterion of wood for every load combination. Transformation of this mixed mode solution to polar coordinates gives the corrected singularity method based on a mixed mode failure criterion and delivers also the definition of the stress intensity factor. This last also gives another interpretation of the Bazant curve, which is shown to be the initial mode I yield criterion.

Dynamic modeling of the Stewart platform for the NanShan Radio Telescope

The NanShan Radio Telescope is a 26-m fully steerable radio telescope, and it adopts a 6-UPU Stewart platform with electric motors to adjust and align the position of the subreflector. In order to analyze the actual dynamic performance and control the Stewart platform of the NanShan Radio Telescope, this article models the inverse dynamic of the Stewart platform using the virtual work approach. The model improves the accuracy of the dynamic equations and considered the pitching motion of the base platform in the practical application of the radio telescope. Dynamic simulations of the Stewart platform are implemented, the conditions of the passive rotation of the piston of actuators are considered, and the results show that the effect of the passive rotation of the pistons of the actuators is important to obtain more accurate result. The conditions of the system under the different elevation angles of the radio telescope are also considered, and the results show that the change of the elevation angles of the radio telescope has a great impact on the driving forces of the Stewart platform. It is known from the analysis that the passive rotation of pistons of actuators and the elevation movement of the primary reflector of the radio telescope are not ignorable for the precise analysis and control of the Stewart platform of the NanShan Radio Telescope.

The Mechanics of Belt Friction Revisited

A new insight into the mechanics of belt friction is given. A conceptual and methodological drawback in the presentation of the classical derivation of the force required to pull the belt over a fixed drum against the hold-force and the friction between the belt and a drum is pointed out, corrected and discussed. The total forces due to pressure and friction ( P and F) are evaluated in magnitude and direction. It is shown that not only the local friction force is proportional to the local pressure ( f = μ p), but also their resultants ( F = μ P), where μ is the coefficient of static friction. The magnitude of the pressure force is P = FR/(1 + μ2)1/2, where FR is the magnitude of the resultant of the pull- and hold-forces applied at two ends of the belt. Different methodological approaches to the analysis are presented, which are particularly appealing from the educational point of view. These include the local and integral equilibrium considerations, the virtual work approach, and the dimensional analysis.

Dry Stone Masonry Walls in Bending—Part II: Analysis

With the objective of developing analytical models for evaluation of the lateral strength of dry-stack masonry (DSM) walls subject to horizontal inertial loading, simple collapse mechanisms are proposed, which include a combination of out-of-plane flexural deformation along the main panel and in-plane shear deformation in the connecting return walls. The lateral strength is calculated using a virtual work approach incorporating moment capacities along different types of crack lines and shear resistance of in-plane panels participating in the mechanisms. The proposed treatment of the active in-plane panels assumes that they undergo frictional shear deformation (versus rigid body rocking), in consistency with the zero tensile strength nature of DSM. Allowance is also made for the presence of either restrained or unrestrained overburden loads at the top of the wall, which can generate either a strengthening or weakening effect, respectively. The accuracy of the analysis is assessed by comparing its predictions to the results of experimental tests on DSM reported in the companion study. These comparisons are favorable with respect to both the predicted values of strength and the critical failure mechanisms.

FEEDFORWARD SEMI-ACTIVE MODEL-BASED CONTROL OF A PLATE CARRYING CONCENTRATED MASSES

The multiobjective task of optimal control of vibration response of an elastic plate is considered. An application of a genetic algorithm for determination of the optimum compensating force frequency dependence and parameters of concentrated masses for different boundary conditions is described. The principle of virtual work and Ritz approach are employed for investigation of dynamic behaviour of mass-loaded plates, which are subjected to any number of forces. The optimisation problem is formulated as a constrained task. Optimization provided the reduction of both total acceleration level and compensating force. Numerical results show the appropriateness of the model for optimization of concentrated masses values and their location on a plate. Interpolation of optimal compensating force parameters frequency dependence is used for the design of feedforward control system.

Geometric nonlinear formulation for thermal-rigid-flexible coupling system

This paper develops geometric nonlinear hybrid formulation for flexible multibody system with large defor- mation considering thermal effect. Different from the con- ventional formulation, the heat flux is the function of the ro- tational angle and the elastic deformation, therefore, the cou- pling among the temperature, the large overall motion and the elastic deformation should be taken into account. Firstly, based on nonlinear strain-displacement relationship, varia- tional dynamic equations and heat conduction equations for a flexible beam are derived by using virtual work approach, and then, Lagrange dynamics equations and heat conduction equations of the first kind of the flexible multibody system are obtained by leading into the vectors of Lagrange mul- tiplier associated with kinematic and temperature constraint equations. This formulation is used to simulate the thermal included hub-beam system. Comparison of the response be- tween the coupled system and the uncoupled system has re- vealed the thermal chattering phenomenon. Then, the key parameters for stability, including the moment of inertia of the central body, the incident angle, the damping ratio and the response time ratio, are analyzed. This formulation is also used to simulate a three-link system applied with heat flux. Comparison of the results obtained by the proposed for- mulation with those obtained by the approximate nonlinear model and the linear model shows the significance of con- sidering all the nonlinear terms in the strain in case of large deformation. At last, applicability of the approximate non- linear model and the linear model are clarified in detail.

Robot based on task-space dynamical model

This study deals with the dynamic modelling and task space control of the Stewart–Gough platform. The dynamic equations were obtained using the virtual work approach, and include all the 13 presented in the platform, that is, two bodies per each leg and the upper ring. The results show some similarities in the dynamic model of serial and parallel robots, making it possible to extend the control theory of serial robot to the Stewart–Gough platform. To show this, an adaptive control law is proposed to control the parallel platform. The law is obtained from the dynamical equation and its performance is tested using a sinusoidal path for position and orientation of the upper ring of parallel robot. Some additional simulations are carried out to demonstrate a well-behaved controller.

Numerical modeling of a MEMS actuator considering several magnetic force calculation methods

PurposeThe purpose of this paper is to investigate the accuracy of different force calculation methods and their impact on mechanical deformations. For this purpose, a micrometer scaled actuator is considered, which consists of a micro‐coil and of a permanent magnet (PM) embedded in a deformable elastomeric layer.Design/methodology/approachFor the magnetic field evaluation a hybrid numerical approach (finite element method/boundary element method (FEM/BEM) coupling and a FEM/BEM/Biot‐Savart approach) is used, whereas FEM is implemented for the mechanical deformation analysis. Furthermore, for the magneto‐mechanical coupling several force calculation methods, namely the Maxwell stress tensor, the virtual work approach and the equivalent magnetic sources methods, are considered and compared to each other and to laboratory measurements.FindingsThe numerically evaluated magnetic forces and the measured ones are in good accordance with each other with respect to the normal force acting on the PM. Nevertheless, depending on the used method the tangential force components differ from each other, which leads to slightly different mechanical deformations.Research limitations/implicationsSince the force calculations are compared to measurement data, it is possible to give a suggestion about their applicability. The mechanical behavior of the actuator due to the acting forces is solely calculated and therefore only an assumption concerning the deformation can be given.Originality/valueA new kind of micrometer scaled actuator is numerically investigated by using two different hybrid approaches for the magnetic field evaluation. Based on those, the results of several force calculation methods are compared to measurement data. Furthermore, a subsequent structural analysis is performed, which shows slightly different mechanical deformations depending on the used force calculation method.

Consistent virtual work approach for the nonlinear and postbuckling analysis of steel frames under thermal and mechanical loadings

The aim of this paper is to provide a consistent virtual work formulation for the nonlinear and postbuckling analysis of steel frames at high temperatures. Central to this study is the derivation of the virtual work terms for the thermal stage, in addition to those for the loading stage, based on the updated Lagrangian formulation. The incremental stiffness equation derived for the beam element, considering both the geometrical and thermal effects, is qualified by the rigid body test. The generalized displacement control (GDC) method is adopted as the path-tracing scheme for postbuckling response. Eurocode-3 reduction factors and transformed section method are both adopted for steel I-sections. Two loading cases are studied. For structures loaded gradually under constant temperature, the critical or ultimate loading strength is obtained from the load-deflection curve. For structures heated gradually under constant loading, the critical or maximum temperature that can be sustained by the structure is computed. Conclusions are drawn for the examples studied in this paper.

Analytical Model of Unconstrained Nonlocal Higher-Order Nano-Plates for Bending Analysis

This paper presents a higher-order nonlocal plate model and its formulation for bending analysis of nanoplates via variational principle and virtual work approach based on Leung’s unconstrained higher-order plate theory and Eringen’s nonlocal continuum theory. Bending of the simply supported rectangular higher-order nano-plate is investigated in comparison with the lower-order plate models. The numerical examples show that nonlocal nanoscale parameters increase the deflections of the plate as the rotary inertia and the transverse shear deformation do.

Non-linear in-plane postbuckling of arches with rotational end restraints under uniform radial loading

This paper presents a thorough and comprehensive investigation of non-linear buckling and postbuckling analyses of pin-ended shallow circular arches subjected to a uniform radial load and which have equal elastic rotational end-restraints. The differential equations of equilibrium for non-linear buckling and postbuckling are established based on a virtual work approach. Exact solutions for the non-linear bifurcation, limit point and lowest buckling loads are obtained; in particular, exact solutions for the non-linear postbuckling equilibrium paths are derived. The criteria for switching between fundamental buckling and postbuckling modes are developed in terms of critical values of a geometric parameter for an arch, with exact solutions for these critical values of geometric parameter being obtained. Analytical solutions of non-linear buckling and postbuckling problems for arches with rotational end-restraints are of great interest, since they constitute one of the very few closed-form analyses of buckling and postbuckling behaviour of continuous structural systems. These exact solutions are a contribution to the non-linear structural mechanics of arches, as well as providing useful benchmark solutions for verifying non-linear numerical analyses.

Material and Spatial Motion Problems in Nonlinear Electro- and Magneto-elastostatics

In this paper, material and spatial motion problems of the coupled nonlinear problem of electro-and magneto-elastostatics are discussed in the context of non-potential loading where mechanical loads are not assumed to be derived explicitly from some potential. A virtual work approach is used to derive the corresponding balance equations and boundary conditions of the material motion problems.

Sensitivity of unsteady collapsible channel flows to modelling assumptions

AbstractWe investigate the influences of modelling assumptions on the dynamic behaviour of collapsible channel flows. The elastic wall is modelled in various different ways: as a large strain Bernoulli–Euler beam, as a small strain Timoshenko beam, and as a 2D‐solid model derived from a general virtual work approach, using small strain or large strain assumptions. Different inlet boundary conditions are also considered. The in‐house finite element codes and the commercial finite element package ADINA 8.4 are used. The steady results agree very well when using the different models/approaches. The unsteady results, on the other hand, can be quite different. The dynamic behaviour of the system is analysed for a set of chosen parameters, using the full numerical solvers, the linear stability analysis, and the Fourier transform. It is found that the system stability is highly sensitive to the solid modelling assumptions used, numerical solvers adopted, or the boundary conditions imposed. Accuracy of the numerical schemes also has an impact on the system's unsteady behaviour. However, despite the high sensitivity of the unsteady solutions to the modelling assumptions, a cascade stability structure previously revealed by the authors seems to exist when different numerical approaches are used. Copyright © 2009 John Wiley & Sons, Ltd.

Material force method: theoretical and numerical aspects in nonlinear electro‐elastostatics

AbstractFormulations of the spatial and material motion problem in nonlinear electroelastostatics are considered in this work by a virtual work approach. Based on these formulations, a finite element discretization is realized and a numerical example is presented to demonstrate possible uses of the material force method in studying the closing process of cracks. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A FURTHER STUDY OF FLEXURAL-TORSIONAL BUCKLING OF ELASTIC ARCHES

This paper uses both a virtual work approach and a static equilibrium approach to study the elastic flexural-torsional buckling of circular arches under uniform bending, or under uniform compression. In most studies of the elastic flexural-torsional buckling of arches under uniform compression produced by uniformly-distributed radial loads, the directions of the radial loads are conventionally assumed not to change but to remain parallel to their initial directions during buckling. In practice, the uniform compression may be produced by hydrostatic loads or by uniformly-distributed radial loads that are directed to a specific point during buckling. In addition, there are discrepancies between existing solutions for the elastic flexural-torsional buckling moment and load of arches under uniform bending or under uniform compression which need to be clarified. Closed form solutions for the buckling moment and load are developed. The discrepancies among the existing solutions for the elastic flexural-torsional buckling moment and load of arches are clarified and the sources for the discrepancies are identified. It is found that the lateral components of hydrostatic loads and of uniformly-distributed radial loads that are always directed toward the center of the arch increase the flexural-torsional buckling resistance of an arch under uniform compression. It is also found that first-order buckling deformations are sufficient for static equilibrium approaches for the flexural-torsional buckling analysis of arches. The rational static equilibrium approach for the flexural-torsional buckling in the present study is effective.

Static Analysis of Tensegrity Structures

Two problems are addressed in this paper. First, the mathematical model to perform the static analysis of an antiprism tensegrity structure subjected to a wide variety of external loads is presented. The virtual work approach is used to deduce the equilibrium equations and a method based on Newton’s Third Law is used to verify the numerical results. Two numerical examples are provided to demonstrate the use of the mathematical model, as well as the verification method. The second problem deals with the development of a mathematical model to perform the static analysis of a prestressed antiprism tensegrity structure subjected to an arbitrary length reduction of its connecting ties. Again, a virtual work approach is used to deduce the equilibrium equations and the numerical results are verified using a Newtonian approach. One example is provided to illustrate the mathematical model.