Virtual Work Formulation Research Articles

Lateral-distortional buckling of frames composed of non-aligned I-members by a simple distortional beam element considering angling effect

Lateral-distortional buckling may occur in frames composed of non-aligned I-members, for which the buckling resistance may be overestimated if the distortional deformation of the cross-section was ignored. In this regard, the geometric stiffness matrix for a novel straight beam element with nine DOFs per node that incorporates the effect of angling (or concentrated transverse stresses induced by in-plane bending moments) between two non-aligned members via the symmetric and anti-symmetric distortion modes is derived. Specifically, the virtual work done by the induced moments by in-plane bending moments near the angled joints when, in buckling, undergoing the angle of twist and the anti-symmetric distortion, is consistently included in the virtual work formulation. Although the local geometric stiffness matrices appear to be asymmetric, the symmetry of the structural stiffness matrix is ensured in the global assembly process, provided that the compatibility and equilibrium conditions at joints in the deformed configuration are satisfied. The reliability of the present distortional beam element is verified by comparing the present solutions with those by the Abaqus shell element in several well-designed examples, by which the superiority of the present element to the non-distortional beam elements is also demonstrated. Comparison study shows that the effect of distortional deformation is extremely significant when the angled frame is composed of short members, or the arch is of high curvature, and when rigid flanges or flexible webs are adopted. Moreover, the distortional beam element excluding the angling effect will produce significant errors in some particular cases.

Nonlinear coupled moving-load excited dynamics of beam-mass structures

Investigated in this paper is the first on the moving-load-caused nonlinear coupled dynamics of beam-mass systems. A constant value load excites the beam-mass system where its position on the beam-mass system changes periodically. The energy contribution of the moving load is included via a virtual work formulation. The kinetic energy of the mass together with the beam as well as energy stored in the beam after deflection is formulated. Hamilton’s principle gives nonlinear equations of the beam-mass system under a moving load in a coupled transverse/longitudinal form. A weighted-residual-based discretisation gives a 20 degree of freedom which is numerically integrated via continuation/time integration along with Floquet theory techniques. The resonance dynamics in time, frequency, and spatial domains is investigated. As we shall see, torus bifurcations are present for some beam-mass structure parameters as well as travelling waves. A finite element analysis is performed for a simpler linear version of the problem for to-some-extend verifications.

Comparison of thrust line analysis, limit state analysis and distinct element modeling to predict the collapse load and collapse mechanism of a rammed earth arch

This paper assesses the suitability of two analytical and one numerical analysis techniques to determine the collapse load and the collapse mechanism of a rammed earth arch. The first method, based on thrust line analysis, is a graphic statics based approach that can predict collapse relying only on material properties of density and compressive strength, assuming that rammed earth has no tensile strength. The second method, limit state analysis, is based on a virtual work formulation to predict the collapse assuming the same set of material properties. Both analytical methods have been adapted from masonry analysis to take into account the limited compressive strength of rammed earth to better predict the rammed earth arch’s behavior and can be generalized to any material without tensile strength. The third method is based on a distinct element modeling technique. It is shown through a comparison with a load testing experiment of a 2m span rammed earth arch that thrust line analysis is an excellent tool to predict the collapse load, but that it cannot provide decisive information regarding collapse mechanisms. Limit state analysis, in contrast, is very suitable to determine the collapse mechanism but may underestimate the ultimate load capacity if the location where cracks can form is not known in advance. Distinct element modeling can provide accurate information on both collapse mechanism and collapse load, but is more computationally demanding and requires a comprehensive characterization of material properties. The application of these techniques to rammed earth is motivated by the rise of the design of new arched and curved rammed earth structures, while appropriate analysis tools are lacking.

Complete Tangent Stiffness for eXtended Finite Element Method by including crack growth parameters

SUMMARY The eXtended Finite Element Method (XFEM) is a useful tool for modeling the growth of discrete cracks in structures made of concrete and other quasi-brittle and brittle materials. However, in a standard application of XFEM, the tangent stiffness is not complete. This is a result of not including the crack geometry parameters, such as the crack length and the crack direction directly in the virtual work formulation. For efficiency, it is essential to obtain a complete tangent stiffness. A new method in this work is presented to include an incremental form the crack growth parameters on equal terms with the degrees of freedom in the FEM-equations. The complete tangential stiffness matrix is based on the virtual work together with the constitutive conditions at the crack tip. Introducing the crack growth parameters as direct unknowns, both equilibrium equations and the crack tip criterion can be handled within the same standard nonlinear iterations. This new solution strategy is believed to provide the modeling capabilities to deal with simultaneous growth of several cracks. A cohesive crack modeling is used. The method is applied to a partly cracked XFEM element of linear strain triangle type with the crack length as the unknown crack growth parameter. In this paper, two examples are given. The first example verifies the theory and the implementation. The second example is the benchmark test three point bending test, where the efficiency of the complete tangential behavior is shown. Copyright © 2013 John Wiley & Sons, Ltd.

Inelastic local buckling of curved plates with or without thickness-tapered sections using finite strip method

This paper addresses the inelastic local buckling of the curved plates using finite strip method in which buckling modes and displacements of the curved plate are calculated using sinusoidal shape functions in the longitudinal direction and polynomial functions in the transverse direction. A virtual work formulation is employed to establish the stiffness and stability matrices of the curved plate whilst the governing equations are then solved using a matrix eigenvalue problem. The accuracy and efficiency of the proposed finite strip model is verified with finite element model using ABAQUS as well as the results reported elsewhere while a good agreement is achieved. In order to illustrate the proposed model, a comprehensive parametric study is performed on the steel and aluminium curved plates in which the effects of curvature, the length of the curved plate as well as circumferential boundary conditions on the critical buckling stress are investigated. The developed finite strip method is also used to determine the buckling loads of the curved plates with thickness-tapered sections as well as critical stresses of the aluminium cylindrical sectors that are subjected to uniform longitudinal stresses.

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.

EFFECT OF SUPPORT STIFFNESS ON STABILITY OF SHALLOW ARCHES

The nonlinear behavior and in-plane stability of parabolic shallow arches with elastic rotational supports are investigated. A central concentrated load is applied to create the compression in the supports. Nonlinear buckling analysis based on the virtual work formulation is carried out to obtain the critical load for both symmetric snap-through buckling and anti-symmetric bifurcation buckling. It is found that the effect of rotational stiffness of the elastic supports on the critical loads is significant. The critical load increases as either the initial stiffness coefficient α or the stiffening rate β increases. The effect of the stiffening rate β on the critical loads decreases, as the initial stiffness coefficient α increases. In addition, the influence of rotational stiffness on the bifurcation buckling is larger than that on the snap-through buckling. The limit of the geometric parameter λ between the bifurcation and snap-through buckling modes also increases with the increase in the stiffening rate β. In addition, the snap-through buckling mode governs the buckling of the arch with larger stiffening rates.

Buckling of parabolic shallow arches when support stiffens under compression

The in-plane stability of parabolic shallow arches whose rotational stiffness of supports increases when loaded is investigated. The nonlinear buckling analysis based on the virtual work formulation is carried out to obtain the critical load for both snap-through buckling and bifurcation buckling. It is found that the critical load increases with either the initial stiffness coefficient or the stiffening rate. The influence of the stiffening rate on critical loads decreases with an increase of the initial stiffness. The values of the modified arch slenderness that delineates different buckling modes increase with the stiffening rate.

Generic nonlinear modelling of a steel beam in a frame sub-assembly at elevated temperatures

This paper presents a virtual work formulation that enables the elevated temperature response of a steel beam with translational and rotational restraint, which forms a component of a compartment sub-assembly in a steel frame structure, to be determined in the pre-yielded range of structural response. The differential equations for the temperature-induced transverse and longitudinal deformations are derived, and in particular the static boundary conditions which can sometimes be counter-intuitive in analyses under thermal actions eventuate from the virtual work formulation. In lieu of most advanced finite element packages whose application generally does not permit an overall ‘feeling’ for the underlying structural mechanics, the technique proposed here is generic and leads to solutions in closed form for arbitrary regimes of temperature loading prior to attaining first yield. The technique can further be extended to quantify the post-yield response in a generic fashion. The application of the method is then illustrated with a simple numerical example.

In-Plane Stability of Parabolic Arches with Horizontal Spring Supports. I: Theory

This paper investigates the in-plane elastic stability of a shallow parabolic arch with horizontal spring supports subjected to a uniformly distributed vertical load. A virtual work formulation is used to establish both the nonlinear equilibrium equations and the buckling equilibrium equation for shallow arches. Analytical solutions for the in-plane buckling loads of shallow arches subjected to this loading regime are obtained, and a closed-form solution for the in-plane antisymmetric bifurcation buckling load and an approximation to the symmetric snap-through buckling load of shallow arches are proposed. Comparisons with the finite-element predictions of these buckling loads, together with the measured buckling loads of arches tested in the laboratory and reported in the companion paper, demonstrate that the analytical solutions are accurate. It is found that the effects of the stiffness of the horizontal springs on the buckling load and buckling behavior of arches are significant, and that the buckling load of an arch increases with an increase in the flexibility of the horizontal springs at the supports. Values of the slenderness that delineate the buckling modes increase with an increase of the flexibility of the horizontal springs. Values of the slenderness that distinguish between an arch (for which buckling is possible) and a beam curved in elevation (that cannot buckle in plane) also increase with an increase of the flexibility of the horizontal springs.

Creep Buckling of Shallow Parabolic Concrete Arches

This paper addresses, both theoretically and experimentally, the buckling of shallow concrete arches that may occur due to the effects of creep and shrinkage deformations of the concrete. Under sustained loading, these viscoelastic effects cause time-dependent changes of the deflections of a shallow arch and can lead to the establishment of a geometric configuration at which the arch is no longer stable, and which heralds the onset of creep buckling. By using a virtual work formulation, the equilibrium and buckling equations of an arch with horizontal spring supports are derived in closed form, and using the algebraically tractable age-adjusted effective modulus method to model the time-dependent behavior of the concrete allows the buckling load and the prebuckled life of the arch to be determined using prescriptive equations. Long-term tests on three 4.25 m span arches under sustained loading are described. Each arch failed because of creep buckling several months after first loading. These tests are used to validate the theory developed in the paper and to provide benchmark data for calibration. The study shows that creep and shrinkage of the concrete have a profound influence on the structural behavior of shallow concrete arches.

A new finite element for modeling pneumatic muscle actuators

Pneumatic Muscle Actuators (PMAs) have recently been used as control devices in parachute soft landing systems and parachute steering control systems. In this paper, a special cable element is developed to model the mechanical behavior of PMAs. The new element provides relationships between the PMA internal pressure, fiber bias angle, PMA radius and length, and resultant axial force, based on the kinematic assumption of inextensible PMA fibers. The principle of virtual work and total Lagrange formulation are used to derive the element internal force vector and tangent stiffness matrix. The PMA element is implemented in a geometrically nonlinear, transient finite element program for simulating the structural dynamics of airdrop systems. Several numerical examples are given to validate the mechanical behavior of the new element. Two large-scale application problems are also presented to demonstrate the capabilities of the new element for simulating PMAs used in airdrop systems.

The coupled radial–axial deformations analysis of flexible pipes conveying fluid

AbstractThis paper presents large deformation analysis of pipes conveying fluid in which two complicated behaviours are taken into consideration. The first is the coupling between radial and axial deformations of pipe wall, and the other is the interaction between a deformed pipe and transported fluid having the variable internal flow velocity. The coupled radial–axial deformation theory of the pipes and the continuity theory of flow inside the moving deformed pipes are developed to undertake these coupling behaviours. All strong and weak forms of governing equations are obtained by carrying out the virtual work formulation. The hybrid‐finite element method is used to solve the highly non‐linear static problems, which configure the initial large deflection and large strain conditions of the pipes. The state‐space finite element model for use in analyses of non‐linear vibration and system stability is established as well as the suggested numerical solution procedures. The numerical studies of the pipes under circumstances of intense radial loads such as deep‐water risers demonstrate that even a slight change of the radial deformation has a significant effect in increasing non‐linear responses, and reducing stabilities of the pipes. Copyright © 2004 John Wiley & Sons, Ltd.

Thermodynamic framework for coupling of non-local viscoplasticity and non-local anisotropic viscodamage for dynamic localization problems using gradient theory

This study develops a general consistent and systematic framework for the analysis of heterogeneous media that assesses a strong coupling between rate-dependent plasticity and anisotropic rate-dependent damage for dynamic problems within the framework of thermodynamic laws and gradient theories. The proposed formulation includes thermo-elasto-viscoplasticity (rate-dependent plasticity) with anisotropic thermo-viscodamage (rate-dependent damage); a dynamic yield criterion of a von Mises type and a dynamic damage growth criterion; the associated flow rules; thermal softening; non-linear strain hardening; strain-rate hardening; strain hardening gradients; and strain-rate hardening gradients. Since the material macroscopic thermomechanical response under dynamic loading is governed by different physical mechanisms on the meso- and macroscale levels, the proposed three-dimensional kinematical model is introduced with manifold structure accounting for discontinuous fields of dislocation interactions (plastic flow) and crack and void interactions (damage growth). The gradient theory of rate-independent plasticity and rate-independent damage that incorporates macroscale interstate variables and their higher-order gradients is generalized here for rate-dependent plasticity and rate-dependent damage to properly describe the change in the internal structure and in order to investigate the size effect of statistical inhomogeneity of the evolution-related rate- and temperature dependent materials. The idea of bridging length-scales is made more general and complete by introducing spatial higher-order gradients in the temporal evolution equations of the internal state variables that describe hardening in coupled viscoplasticity and viscodamage models, which are considered here physically and mathematically related to their local counterparts. Furthermore, the constitutive equations for the damaged material are written according to the principle of strain energy equivalence between the virgin material and the damaged material; that is, the damaged material is modeled using the constitutive laws of the effective undamaged material in which the nominal stresses and strains are replaced by their effective ones. In addition, computational issues concerned with the current gradient-dependent formulation of initial-boundary value problems are introduced in a finite element context. A weak (virtual work) formulation of the non-local dynamic viscoplastic and viscodamage conditions is derived, which can serve as a basis for the numerical solution of initial boundary value problems in the sense of the finite element method. Explicit expressions for the generalized tangent stiffness matrix and the generalized nodal forces are given. The model presented in this paper can be considered as a feasible thermodynamic approach that enables one to derive various coupled gradient viscoplasticity and viscodamage theories by introducing simplifying assumptions.

Finite element formulation for modeling sliding cable elements

“Sliding cable elements” are developed to solve the general problem of constraining a string of cable elements to continuously pass through a prescribed moving node. These elements can be used for a wide variety of applications and in the current work are used to model various features in parachute systems. The principle of virtual work and total Lagrange formulation are used to derive the element internal force vector, tangent stiffness matrix, and time-dependent mass matrix and body forces. The element equations are implemented in a geometrically nonlinear, transient implicit finite element program. A searching algorithm is developed to track the relative movement of the elements, which is performed each iteration prior to formulation of the element equations. Two problems with analytical solutions are performed to validate the mechanical behavior of the new element and the searching algorithm. Several examples are presented where the sliding cable elements are used in parachute systems.

In-Plane Elastic Stability of Arches under a Central Concentrated Load

This paper is concerned with the in-plane elastic stability of arches with a symmetric cross section and subjected to a central concentrated load. The classical methods of predicting elastic buckling loads consider bifurcation from a prebuckling equilibrium path to an orthogonal buckling path. The prebuckling equilibrium path of an arch involves both axial and transverse deformations and so the arch is subjected to both axial compression and bending in the prebuckling stage. In addition, the prebuckling behavior of an arch may become nonlinear. The bending and nonlinearity are not considered in prebuckling analysis of classical methods. A virtual work formulation is used to establish both the nonlinear equilibrium conditions and the buckling equilibrium equations for shallow arches. Analytical solutions for antisymmetric bifurcation buckling and symmetric snap-through buckling loads of shallow arches subjected to this loading regime are obtained. Approximations for the symmetric buckling load of shallow arches and nonshallow fixed arches and for the antisymmetric buckling load of nonshallow pin-ended arches, and criteria that delineate shallow and nonshallow arches are proposed. Comparisons with finite element results demonstrate that the solutions and approximations are accurate. It is found that the existence of antisymmetric bifurcation buckling loads is not a sufficient condition for antisymmetric bifurcation buckling to take place.

A virtual work derivation of the scaled boundary finite-element method for elastostatics

The scaled-boundary finite element method is a novel semi-analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. This paper develops a new virtual work formulation and modal interpretation of the method for elastostatics. This formulation follows a similar procedure to the traditional virtual work derivation of the standard finite element method. As well as making the method more accessible, this approach leads to new techniques for the treatment of body loads, side-face loads and axisymmetry that simplify implementation. The paper fully develops the new formulation, and provides four examples illustrating the versatility, accuracy and efficiency of the scaled boundary finite-element method. Both bounded and unbounded domains are treated, together with problems involving stress singularities.

Non–self–similar decohesion along a finite interface of unilaterally constrained delaminations

We have presented a novel and unified approach for the analysis of delaminated structures in a compressive load environment. Previous studies have addressed delamination buckling, postbuckling and growth as three separate events with respect to remote load, with an appropriate criterion to demarcate the separate regimes. We show that a unified treatment of this problem is possible so that the evolution of the delaminated areas is obtained as a part of the calculation process. The unified approach is made possible by the introduction of an interface decohesion law that is appealing both as a computational device to regularize an otherwise singular problem, and as a physical model of the decohesion process. This type of modelling is known in fracture mechanics as Barenblatt–Dugdale (BD) models. Employing BD models, we were able to overcome some of the limitations of linear elastic fracture mechanics approaches in predicting general delamination growth. Using a virtual work formulation, the delamination–interphase–substrate system was modelled as one, resulting in a system of integral equations that was solved using an approximate method. The present treatment is new and demonstrates a successful departure from traditional fracture mechanics based concepts that require empirical relations for non–self similar delamination growth studies. The problem formulation places no distinction between the phenomena of buckling (and postbuckling) and non–self–similar growth. That is, the same equations were found to govern the entire behaviour from beginning (starting to load/displace the structure) to end (complete decohesion and/or loss of stiffness) without specification to certain regimes of validity. We have demonstrated that the use of nonlinear elastic foundation models to characterize unilateral constraints and the use of interphase models to analyse delamination decohesion and growth are indeed viable. Non–self–similar delamination growth patterns were simulated without resorting to fracture mechanics concepts and it was found that unilateral contact can occur at buckling or in the postbuckling regime, as well as prior to delamination growth or after delamination growth. Several examples are presented to illustrate this new treatment.

Semi-analytical finite elements in the higher-order theory of beams

The paper presents semi-analytical finite elements leading to the development of a higher-order theory for beams. An analysis of existing beam theories reveals their semi-analytical character. The semi-analytical finite element (SFE) method proposed assumes the cross-section of a solid beam to be an assemblage of certain semi-analytical elements. The SFE method is based on independent approximations of three-dimensional fields of displacements and stresses. The method reflects the multi-level nature of beams where conventional finite elements are transferred to the level of beam equations for heuristic considerations. A mathematical formulation of the linear theory of a solid beam involves the equations of compatibility, equilibrium and constitutive relationships, i.e. the elastic law as well as static and kinematic boundary conditions. Compatibility conditions are derived directly by using finite element approximations, while equilibrium equations, constitutive equations and boundary conditions may be derived by applying stationarity of a modified Hellinger-Reissner functional. This derivation may be also defined by straightforward application of a generalised virtual work formulation. The equations of a beam are formulated in terms of generalised variables — integral resultants of stresses and strains as well as of nodal displacements. The definition of generalised variables is assigned to the nodes of the finite element mesh. Bending, warping and other higher-order deformation modes are described automatically. Finally, the set of governing equations includes mixed algebraic-differential operators. Their explicit expressions depend on one of the element types — isoparametric, subparametric or superparametric. The versatility of SFE is demonstrated in practical applications involving well-known solutions. The classical Euler-Bernoulli and Timoshenko theories are derived using a single one-dimensional element. The beams with a thin-walled cross-section are described as an assemblage of semi-analytical elements. On the basis of a higher-order theory, the conventional semi-analytically based finite elements (SABFE) are derived. Their basic relations are also presented. Some numerical examples of the applications to thin-walled beams illustrate the basic features. Possible extensions of SFE to dynamics, geometric non-linearity and shells are briefly discussed.

Instability of Tapered Thin‐Walled Beams of Generic Section

Buckling loads and natural frequencies and the corresponding modal shapes and forms for thin‐walled tapered beams of open sections are examined using the finite‐element method. A tapered thin‐walled bar finite element with seven degrees of freedom at each node is adopted. In the virtual work formulation, the updated Lagrangian approach is adopted in which the effect of geometric nonlinearity is considered. A rigorous expression for strains based on membrane theory of shells is considered. The flexural stiffness matrix, geometric stiffness matrix, and consistent mass matrices are derived in a companion paper. The convergence and accuracy of the method is tested based on other numerical results. Using the present theory, one is able to investigate various torsional and flexural static and dynamic instability problems. Examples are presented and comparisons are made with the existing solutions.