elastoplastic and contact analysis based on consistent dynamic formulation of co rotational planar elements abstract co rotational formulation is one of the approaches applicable to geometrically nonlinear problems in structural analysis based on the assumptions of small degree of strain and large displacement the cr formulation allows an accurate geometrically nonlinear structural analysis in this paper an improved cr formulation for dynamic nonlinear analysis is developed for the planar element also extension of the local formulation is conducted for an elastoplastic problem a simplified approach for planar elastoplastic analyses is proposed by maintaining the original feature of the cr formulation moreover a modified approach in the local formulation is suggested in order to improve the level of accuracy in the plastic state contact analysis using the lagrange multiplier is then implemented during the present validation procedure of the proposed approaches four different planar elements are employed those elements are investigated with regard to efficiency and accuracy in a general geometrically nonlinear and a geometrically nonlinear elastoplastic problem respectively it is found that the present dynamic formulations gives both accuracy and consistency for all types of the plane element also comparison regarding the choice of the local formulation for the elastoplastic problem is systematically conducted and demonstrated

a simplified finite element method for large deformation post buckling analyses of large frame structures using explicitly derived tangent stiffness matrices in this paper a simplified economical and highly accurate method for large deformation post buckling analyses of frame type structures is presented using the concepts of polar decomposition of deformation an explicit expression for the tangent stiffness matrix of a member element of the frame that accounts for the nonlinear bending stretching coupling is derived in closed form at any point in the load deformation path the ranges of validity of the present simplified approach are discussed several example problems to demonstrate the feasibility of the present approach over ranges of deformation that are well beyond those likely to occur in practical large frame structures are included

co rotational beam elements with warping effects in instability problems the present paper investigates the formulation of 3d co rotational beam elements for the buckling and post buckling analysis of frame structures following pacoste and eriksson comput methods app

the k property of n billiard balls ii computation of neutral linear spaces the present second part of the article the k property ofn billiard balls finishes the proof of the kolmogorov mixing property for hard ball systems in tori provided that the number of balls dos not exceed the dimension of the torus the missing computation of the linear spaces of non expanding and non contracting tangent vectors at phase points with combinatorially rich symbolic collision sequences is presented hereby

exact integration of constitutive equations in elasto plasticity a unified approach is presented for establishing exact integration of the constitutive equations in elastoplasticity assuming the total strain rate direction to be constant this unified approach includes all previous exact integration procedures as special cases and in addition some new closed form solutions are derived for combined kinematic and isotropic hardening special emphasis is laid on combined kinematic and isotropic hardening for von mises material and on isotropic hardening for mohr coulomb and tresca materials

a system of one dimensional balls with gravity we introduce a hamiltonian system with many degrees of freedom for which the nonvanishing of some lyapunov exponents almost everywhere can be established analytically

assumed stressc0 quadrilateral triangular plate elements by interrelated edge displacements this paper presents two simple quadrilateral c0 plate bending elements with explicit element stiffness matrix the element formulation is based on assumed element stress fields and the interrelated transverse displacement and rotation along element boundaries the interrelated edge displacements not only can result in higher order displacements interpolations for higher accuracy element and overcome the shear locking in thin plate analysis encountered by c0 plate elements but can also unify the four noded quadrilateral element and its corresponding three noded triangular element the latter cannot be achieved by the assumed displacement formulation the numerical examples demonstrate the accuracy and robustness of the present assumed stress c0 plate elements

a nonlinear hu washizu variational formulation and related finite element implementation for spatial beams with arbitrary moderate thick cross sections abstract a nonlinear three dimensional finite beam element based on a hu washizu variational formulation is presented in addition to the standard beam strains based on kinematic assumptions further deformation modes are introduced these additional modes allow for a the consideration of a complete three dimensional stress field providing an interface for arbitrary three dimensional material models and b the consideration of cross section warping whose shape might shift during the elastic or inelastic deformation beside the fact that the resulting finite element formulation is locking free full gauss integration in length direction and remarkable robustness even for very large load steps these additional degrees of freedom do not increase the total number of global unknowns of a beam structure each element node exhibits the common 3 translational and 3 rotational degrees of freedom the additional degrees of freedom are eliminated on element level via static condensation as a consequence e g a bi moment can not be applied at a free end this restricts the applicability of the formulation to a class of problems where the influence of the bi moment is negligible it is shown that global acting polynomial ansatz functions are not suitable to describe warping of cross sections with an arbitrary shape for this reason a new concept based on local ansatz functions is presented the general criteria to design the warping ansatz functions are discussed in detail several examples with moderate thick cross sections are investigated

on a non linear formulation for curved timoshenko beam elements considering large displacement rotation increments summary an incremental total lagrangian formulation for curved beam elements that includes the effect of large rotation increments is developed a complete and symmetric tangent stiffness matrix is obtained and the numerical results show in general an improvement over the standard formulation where the assumption of infinitesimal rotation increments is made in the derivation of the tangent stiffness matrix

analysis of flexible multibody systems with spatial beams using mixed variational principles a general nite element formulation is presented for dynamic analysis of spatial elastic beams for small strains in a multi body conguration the tangent maps associated to the nite rotation vector are used to compute the tangent matrices used to integrate implicitly the equations of motion in descriptor form a corotational method and a mixed variational method are used to compute the tangent stiness matrix the tangent constraint matrices are obtained using consistent linearization of the constraint equations the tangent inertia matrices including the gyroscopic and centrifugal terms are also obtained by using the tangent maps of rotation the numerical examples analyzed in this paper include dynamic analysis of exible beam structures and multi exible body systems with open and closed kinematic loops a comparison with the previous results in the literature shows a very good performance in terms of time integration step and number of elements used 1998 john wiley sons ltd this paper deals with the modelling of the spatial dynamic behaviour of homogeneous isotropic and linear elastic one dimensional deformable bodies such as beams or rods undergoing arbitrarily large rotations and translations and small strains the beam may be connected to other beams by means of kinematic constraints to form an open or closed loop multi exible body mechanical system a previous paper 1 of which the present one is a sequel has investigated two dierent methods to compute the tangent stiness of the elastic beam in particular the elastic tangent stiness matrix and residual vector for a space beam were derived with two dierent approaches a primal total lagrangean tl corotational approach and a mixed updated lagrangean ul approach the idea of that derivation was to formally adopt a variational setting to derive the eld equations of motion of constrained exible bodies previously a consistent variational approach for multi exible body dynamics with beams has only been presented by cardona and geradin 2 in their work they used the principle of virtual work and methods of non linear structural

an analysis of a new class of integration algorithms for elastoplastic constitutive relations an accuracy analysis of a new class of integration algorithms for finite deformation elastoplastic constitutive relations recently proposed by the authors is carried out in this paper for simplicity attention is confined to infinitesimal deformations the integration rules under consideration fall within the category of return mapping algorithms and follow in a straightforward manner from the theory of operator splitting applied to elastoplastic constitutive relations general rate independent and rate dependent behaviour with plastic hardening or softening associated or non associated flow rules and nonlinear elastic response can be efficiently treated within the present framework isoerror maps are presented which demonstrate the good accuracy properties of the algorithm even for strain increments much larger than the characteristic strains at yielding

quasiperiodic motions in dynamical systems review of a renormalization group approach power series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory in the case of quasiperiodic solutions the issue of convergence of the series is bedeviled by the so called small divisor problem in this paper we review a method recently introduced to deal with such a problem based on renormalization group ideas and multiscale techniques applications to both quasi integrable hamiltonian systems kolmogorov arnold moser kam theory and non hamiltonian dissipative systems are discussed the method is also suited to situations in which the perturbation series diverges and a resummation procedure can be envisaged leading to a solution which is not analytic in the perturbation parameter we consider explicitly examples of solutions which are only c in the perturbation parameter or even defined on a cantor set

a consistent co rotational finite element formulation for geometrically nonlinear dynamic analysis of 3 d beams abstract a co rotational total lagrangian finite element formulation for the geometrically nonlinear dynamic analysis of spatial euler beam with large rotations but small strain is presented the nodal coordinates displacements rotations velocities accelerations and the equations of motion of the structure are defined in a fixed global set of coordinates the beam element has two nodes with six degrees of freedom per node the element nodal forces are conventional forces and moments the kinematics of beam element are defined in terms of element coordinates which are constructed at the current configuration of the beam element both the element deformation nodal forces and inertia nodal forces are systematically derived by consistent linearization of the fully geometrically nonlinear beam theory using the d alembert principle and the virtual work principle in the current element coordinates an incremental iterative method based on the newmark direct integration method and the newton raphson method is employed here for the solution of the nonlinear equations of motion numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method

the characteristic exponents of the falling ball model we study the characteristic exponents of the hamiltonian system ofn 2 point massesm 1 m n freely falling in the vertical half line q q 0 under constant gravitation and colliding with each other and the solid floorq 0 elastically this model was introduced and first studied by m wojtkowski hereby we prove his conjecture all relevant characteristic lyapunov exponents of the above dynamical system are nonzero provided thatm 1 m n i e the masses do not increase as we go up andm 1 m 2

a consistent co rotational formulation for non linear three dimensional beam elements abstract this paper describes a co rotational formulation for three dimensional beams in which both the internal force vector and tangent stiffness matrix are consistently derived from the adopted strain measures the latter relate to standard beam theory but are embedded in a continuously rotating frame a set of numerical examples show that the element provides an excellent numerical performance

a kinematically exact finite element formulation of planar elastic plastic frames a finite element formulation of finite deformation static analysis of plane elastic plastic frames subjected to static loads is presented in which the only function to be interpolated is the rotation of the centroid axis of the beam one of the advantages of such a formulation is that the problem of the field consistency does not arise exact non linear kinematic relationships of the finite strain beam theory are used which assume the bernoulli hypothesis of plane cross sections finite displacements and rotations as well as finite extensional and bending strains are accounted for the effects of shear strains and non conservative loads are at present neglected yet they can simply be incorporated in the formulation because the potential energy of internal forces does not exist with elastic plastic material the principle of virtual work is introduced as the basis of the finite element formulation a generalized principle of virtual work is proposed in which the displacements rotation extensional and bending strains and the lagrangian multipliers are independent variables by exploiting the special structure of the equations of the problem the displacements the strains and the multipliers are eliminated from the generalized principle of virtual work a novel principle is obtained in which the rotation becomes the only function to be approximated in its finite element implementation it is shown that n 1 point numerical integration must be employed in conjunction with n node interpolation polynomials for the rotation and the lobatto rule is recommended regarding the integration over the cross section it is demonstrated by numerical examples that due to discontinuous integrands no integration order defined as computationally efficient yet accurate enough could be suggested the theoretical findings and a nice performance of the derived finite elements are illustrated by numerical examples

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

generalized mixed finite element model for pre and post quasistatic buckling response of thin walled framed structures adopting an updated lagrangian approach the general framework for the fully non linear analysis of thin walled framed structures is developed using a simple two node c0 model hmb2 the governing equations are derived based on a consistent linearization of an incremental mixed variational principle of modified hellinger reissner type with independent assumptions for displacement and strain fields all coupled significant modes of deformations i e stretching bending shear torsion and warping are accounted for in the generalized beam theory employed emphasis is placed on devising effective solution procedures to deal with finite rotations in space particularly with regard to their effect on the derivations of load correction matrices corresponding to configuration dependent externally applied forces moments the effectiveness and practical usefulness of present model are demonstrated through a number of test problems involving beam assemblages undergoing large displacements and rotations in space

geometrically non linear and elastoplastic three dimensional shear flexible beam element of von mises type hardening material a three dimensional elastoplastic beam element being capable of incorporating large displacement and large rotation is developed and examined elastoplastic constitutive equations are applied to the beam element based upon the assumption of small deformational strain leading to a material formulation which is completely objective for the application of stress update procedures the continuum type equations of plastic model of j2 mixed hardening are transformed into the beam equations by satisfying beam hypotheses an effective stress update algorithm is proposed to integrate elastoplastic rate equations by means of the so called multistep method which is a method of successive control of residuals on yield surfaces it avoids severe divergence when the displacement increments become large which is usual for the continuation methods material tangent stiffness matrix is derived by using consistent elastoplastic modulus resulting from the integration algorithm and is combined with geometric tangent stiffness matrix different from other elements the present element is shear flexible and can satisfy the plasticity condition in a pointwise fashion a great number of numerical examples are analysed and compared with the literature the proposed beam element is verified to be not only quite accurate but also very effective for the analyses of pre buckling and large deflection collapse of spatial framed structures

static and dynamic analysis of space frames using simple timoshenko type elements in this paper a finite element method for geometrically and materially non linear analyses of space frames is described beams with both solid and thin walled open cross sections are considered the equations of equilibrium are formulated using an updated incremental lagrangian description the elements developed can undergo large displacements and rotations but the incremental rotations are assumed to be small the material behaviour is described by elastoplastic temperature dependent elastoplastic and viscoplastic models with special reference to metals computationally more economical formulations based on the relationship between stress resultants and generalized strain quantities are also presented in the case of thin walled beams the torsional behaviour is modelled using a two parameter warping model where the angle of twist and the axial variation of warping have independent approximations this approach yields average warping shear strains directly from the displacement assumptions and no discrepancy between stress and strain fields exists

a nonlinear beam finite element for the post buckling analysis of plane frames by koiter s perturbation approach abstract a nonlinear finite element of plane beam oriented to koiter s perturbation analysis of frames is presented the locking problem arising in the calculation of the post critical curvature is thoroughly examined and solved at the continuum level by means of a suitable definition of the kinematic settings the proposed strain measures are lagrangian and rationally founded geometrically exact the exact solutions of the critical problem have been chosen as shape functions of the finite element so obtaining an immediately convergent finite element in the calculation of the critical load and of the post critical slope

static and dynamic analysis of space frames with non linear flexible connections this paper deals with the effect of non linearly flexible hysteritic joints on the static and dynamic response of space frames it is shown that a complementary energy approach based on a weak form of the compatibility condition as a whole of a frame member and of the joint equilibrium conditions for the frame is best suited for the analysis of flexibly jointed frames the present methodology represents an extension of the authors earlier work14 on rigidly connected frames in the present case also an explicit expression for the tangent stiffness matrix is given when i each frame member along with the flexible connections at its ends is represented by a single finite element ii each member can undergo arbitrarily large rigid rotations and only moderate relative rotations and iii the non linear bending stretching coupling is accounted for in each member several examples with both quasi static and dynamic loading are included to illustrate the accuracy and efficiency of the developed methodology

a return mapping algorithm for plane stress elastoplasticity an unconditionally stable algorithm for plane stress elastoplasticity is developed based upon the notion of elastic predictor return mapping plastic corrector enforcement of the consistency condition is shown to reduce to the solution of a simple nonlinear equation consistent elastoplastic tangent moduli are obtained by exact linearization of the algorithm use of these moduli is essential in order to preserve the asymptotic rate of quadratic convergence of newton methods the accuracy of the algorithm is assessed by means of iso error maps the excellent performance of the algorithm for large time steps is illustrated in numerical experiments

a consistent model for lazzaro winner take all circuit with invariant subthreshold behavior this paper considers the basic lazzaro winner take all analog computing circuit with n current inputs n voltage outputs and a bias current motivated by low power applications we find a mathematical accurate model of the network when all mos devices remain inside the subthreshold region all along the transient this involves analytical inequalities relating the range of admissible input currents to transistor parameters to supply voltage and to bias current the restrictions are sufficiently weak to allow extra demands of functionality or performance the technical novelty here is that by a slight cut of the maximal subthreshold domain and by choosing proper coordinates we get a ordinary differential equation invariance problem on a rectangle of re n 1 space which is analytically tractable a more precise localization of the asymptotically stable steady state inside the region of interest is also inferred although the work is mainly theoretical an effort to infer simple interpretable formulas useful for synthesis has been made the results are numerically verified and their feasibility is discussed in detail the subject matter is new and it is worth extending it to larger classes of circuits with regional behavior

on the differentiation of the rodrigues formula and its significance for the vector like parameterization of reissner simo beam theory in this paper we present a systematic way of differentiating up to the second directional derivative i the rodrigues formula and ii the spin rotation vector variation relationship to achieve this goal several trigonometric functions are grouped into a family of scalar quantities which can be expressed in terms of a single power series these results are then applied to the vector like parameterization of reissner simo beam theory enabling a straightforward derivation and leading to a clearer formulation in particular and in contrast with previous formulations a relatively compact and obviously symmetric form of the tangent operator is obtained the paper also discusses several relevant issues concerning a beam finite element implementation and concludes with the presentation of a few selected illustrative examples copyright 2002 john wiley sons ltd

on steplength algorithms for a class of continuation methods the continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form fx b f d subset r n 1 to r n for given b in r n while these methods are similar in structure to those used for ode solvers their errors are independent of the history of the process and are solely determined by the termination criterion of the corrector at the current step this suggests the use of a posteriors estimates of the convergence radii of the corrector it is proved here that such estimates cannot be obtained from the sequence of corrector iterates alone but that they require some global information about f however it is shown that a finite sequence of corrector iterates does allow for the computation of effective estimates of the convergence quality of certain types of correctors this is used for the design of various step algorithms for continuation processes two of them are based on a newton corrector while the third o

finite element asymptotic analysis of slender elastic structures a simple approach an asymptotic method directly derived from koiter s theory and suitable for the solution of elastic buckling problems and its natural adaptation to a numerical solution by means of a finite element technique are presented here the order of the extrapolation of the equilibrium equations has been intentionally kept very low because attention has been entirely devoted to all those features theoretical definitions eigenproblem numerical techniques suitable fem implementation which make such an approach competitive with respect to the classic step by step methods for plane frames and 3d pin jointed trusses the performances of the algorithm numerical accuracy and computational cost are compared with those of riks are length method

an arc length method including line searches and accelerations this paper describes a method for introducing line searches into the arc length solution procedure such line searches may be used at each iteration to calculate an optimum scalar step length which scales the normal iterative vector in practice a loose tolerance is provided so that on many iterations the line searches are avoided however on difficult iterations the line searches are shown to lead to a substantial improvement in the convergence characteristics a simple single parameter acceleration is also developed using line search concepts the new arc length method is applied to both the geometrically nonlinear analysis of shallow shells and the materially nonlinear analysis of reinforced concrete beams and slabs significant improvements are demonstrated in relation to the standard arc length method

geometrically nonlinear dynamic formulation for three dimensional co rotational solid elements abstract this study presents a three dimensional nonlinear dynamic formulation based on a co rotational cr approach for solid elements the cr formulation is relatively efficient but it is based on the assumption of small strains during large displacement the novel idea of the present formulation involves the use of the cr formulation through a three dimensional solid element for inertial quantities in addition to an internal force vector and a stiffness matrix the present dynamic formulation is derived from lagrange s equation of motion in this procedure the cr formulation i e element independent cr is one of the most attractive features that is strongly manifested in an efficient manner consequently this obtains the governing equation of motion including motion driven inertial components physical quantities induced by the prescribed motion four examples are presented to demonstrate the accuracy of the present dynamic formulation finally the results are compared with those obtained by abaqus and the findings reveal that the proposed dynamic formulation is in good agreement with existing predictions

the system of one dimensional balls in an external field ii we modify the system introduced in w1 so that we can establish the nonvanishing ofall lyapunov exponents easily

stress integration schemes for novel homogeneous anisotropic hardening model abstract numerical formulations and implementation of stress integration algorithms in the elasto plastic finite element method are provided for the homogeneous yield function based anisotropic hardening hah model this model is able to describe complex material behavior under non monotonic loading conditions two numerical algorithms based on the semi explicit and fully implicit schemes are compared in terms of accuracy to efficiently treat the yield locus distortion when the strain path changes a multi step newton raphson method is proposed to calculate the first and second derivatives of the hah yield surface for the validation of the developed numerical algorithms the r value anisotropy is compared for the conventional yield model with classical isotropic hardening and for the hah model moreover detailed error analysis is presented using iso error maps the results show that the fully implicit stress integration algorithm based on the closet point projection method leads to better accuracy in general however the semi explicit algorithm also provides comparable accuracy if an appropriate time increment is chosen furthermore in spite of the yield surface distortion the developed numerical algorithms can successfully update stress with the equivalent level of the error for the conventional yield model

accuracy and stability of integration algorithms for elastoplastic constitutive relations an analysis of accuracy and stability of algorithms for the integration of elastoplastic constitutive relations is carried out in this paper reference is made to a very general internal variable formulation of plasticity and to two families of algorithms that generalize the well known trapezoidal and midpoint rules to fit the present context other integration schemes such as the radial return mean normal and closest point procedures are particular cases of this general formulation the meaning of first and second order accuracy in the presence of the plastic consistency condition is examined in detail and the criteria derived are used to identify two second order accurate members of the proposed algorithms a general methodology is also derived whereby the numerical stability properties of integration schemes can be systematically assessed with the aid of this methodology the generalized midpoint rule is seen to have far better stability properties than the generalized trapezoidal rule finally numerical examples are presented that illustrate the performance of the algorithms

time integration scheme for elastoplastic models based on anisotropic strain rate potentials modelling of plastic anisotropy requires the definition of stress potentials coinciding with the yield criteria in case of the associated flow rules or alternatively plastic strain rate potentials the latter approach has several advantages whenever material parameters are determined by means of texture measurements and crystal plasticity simulations this paper deals with a phenomenological description of anisotropy in elastoplastic rate insensitive models by using strain rate potentials a fully implicit time integration algorithm is developed in this framework and implemented in a static implicit finite element code algorithmic details are discussed including the derivation of the consistent algorithmic tangent modulus and the numerical treatment of the yield condition typical sheet forming applications are simulated with the proposed implementation using the recent non quadratic strain rate potential srp2004 18p numerical simulations are carried out for materials that exhibit strong plastic anisotropy the numerical results confirm that the presented algorithm exhibits the same generality robustness accuracy and time efficiency as state of the art yield criterion based algorithms

on finite element implementation of geometrically nonlinear reissner s beam theory three dimensional curved beam elements abstract finite element implementation of the three dimensional finite strain beam theory of reissner is considered in this work in contrast with some earlier works on the subject discussed here are the beam elements whose reference axes are arbitrary space curved lines we have shown that an improved representation of curved reference geometry significantly increases the accuracy of the results however it also makes the choice of non locking finite element interpolations somewhat more delicate a hierarchical displacement interpolation proposed here is proved to be capable of eliminating both shear and membrane locking phenomena a very satisfying non locking performance is demonstrated for a set of problems in nonlinear elasto statics

elasto plastic finite element method based on incremental deformation theory and continuum based shell elements for planar anisotropic sheet materials abstract an implicit approach for the incremental analysis of planar anisotropic sheet forming processes is developed based on the incremental deformation theory the incremental deformation theory based on the minimum plastic work path enables convenient decoupling of deformation and rotation by the polar decomposition the mathematical description of a constitutive law for the incremental deformation theory is obtained from the flow theory along the minimum plastic work path the resulting constitutive law is then incorporated in an elasto plastic finite element analysis code in the elasto plastic formulation continuum based resultant cbr shell element is employed the cbr shell allows large rotation and large membrane bending strain the laminar coordinate system is taken to coincide with planar anisotropic material axes then planar anisotropic axes during deformation are updated using a newly developed algorithm based on the polar decomposition an iterative solving method based on the incremental deformation theory is also developed for an accurate and stable stress integration the planar anisotropy is incorporated into the formulation for sheet forming by introducing non quadratic barlat s yield function for verification purposes two examples have been simulated and compared with experimental results the good verification results show that the present elasto plastic formulation for planar anisotropic sheet materials can provide a good theoretical basis for more extended analyses of sheet forming processes

a mixed finite element for three dimensional nonlinear analysis of steel frames abstract this paper develops and demonstrates a mixed beam finite element that can represent the inelastic three dimensional stability behavior of frames composed of open walled cross section members the kinematics of deformation of the element include finite rotation and warping of the cross section due to torsion the material inelasticity is based on a two space model that includes the effect of shear stresses due to uniform torsion in addition to normal stresses due to axial force biaxial bending and bimoment initial geometric imperfections and residual stresses are accommodated the formulation is based on a two field displacement and generalized stress hellinger reissner hr variational principle the interpolation of the generalized stresses i e cross section stress resultants along the element length is based on the geometrically exact nonlinear governing differential equations of equilibrium this leads to significant improvements in the accuracy over conventional displacement based elements in problems involving highly nonlinear variations in the curvature along the member lengths due to geometric nonlinearity and or distributed plasticity effects several numerical examples involving two and three dimensional elastic and inelastic stability behavior are presented to illustrate the capabilities of the formulation

dynamic analysis of beam structures considering geometric and constitutive nonlinearity abstract a fully geometric and constitutive nonlinear model for the description of the dynamic behavior of beam structures is developed the proposed formulation is based on the geometrically exact formulation for beams due to simo but in this article an intermediate curved reference configuration is considered the resulting deformation map belongs to a nonlinear differential manifold and therefore an appropriated version of newmark s scheme is used in updating the kinematics variables each material point of the cross section is assumed to be composed of several simple materials with their own constitutive laws the mixing rule is used to describe the resulting composite an explicit expression for the objective measure of the strain rate acting on each material point is deduced in this article details about its numerical implementation in the time stepping scheme are also addressed viscosity is included at the constitutive level by means of a thermodynamically consistent visco damage model developed in terms of the material description of the first piola kirchhoff stress vector the constitutive part of the tangent tensor is deduced including the effect of rate dependent inelasticity finally several numerical examples validating the proposed formulation are given

a formulation and solution procedure for post buckling of thin walled structures abstract transformations of nodal quantities between spacially fixed global coordinate system and element fixed local one are presented to give governing equilibrium equations for a post buckling large displacement analysis of thin walled three dimensional structures the transformations are established primarily on the basis of the description of the nodal locations by their initial coordinates as well as their displacements an effective corrective process for constituting and solving structural incremental nonlinear equilibrium equations is developed by clear separation of the rigid body rotation from the displacements numerical examples show that the method can satisfactorily pursue nonlinear load displacement behaviour including snap through and bifurcation buckling only with incremental calculations and without any special techniques as eigenmode analysis

a study on the instability problem for 3d frames instead of the traditional load following process this paper presents large deflection post buckling analysis of the three dimensional elastic frame from a dynamic point of view a co rotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted the pre buckling analysis employs the constant arc length method together with the modified newton raphson iteration method and the extrapolation technique to improve the convergence behaviour as the limit state being examined is passed the previous converged solution is adopted to start the nonlinear dynamic analysis based on the average acceleration of the newmark algorithm with a slow rate of load incrementation to trace the load deflection path beyond the limit point as a result the snap through problem is overcome without down loading the efficiency of the damping coefficients on the control of displacement overshooting and vibration which are the common phenomena using this procedure under conventional structural damping is also studied and applied to all the numerical examples this study is demonstrated by application to typical three dimensional frames which show snap through behaviour in static analysis with any increment of load after the limit point

dynamic response and instability of frame structures abstract this paper presents a geometrically non linear dynamic instability analysis for both two and three dimensional frames which may be subjected to finite rotations the finite element displacement method based on the beam column approach is employed to derive the non linear equations governing the behaviour of plane and spatial frames a co rotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted the governing dynamic equilibrium equations are obtained from the static equations by adding the inertia and damping terms the implicit newmark time integration with the newton raphson nr iteration method is employed dynamic critical loads are defined by the budiansky roth criterion several numerical examples are illustrated to demonstrate the effectiveness of the present method

symmetry preserving return mapping algorithms and incrementally extremal paths a unification of concepts in this work we seek to characterize the conditions under which an elastic plastic stress update algorithm preserves the symmetries inherent to the material response from a numerical standpoint the aim is to determine under what conditions a stress update algorithm produces symmetric consistent tangents when applied to materials obeying normality for the ideally plastic solid we show that only the fully implicit or closest point return mapping algorithm is symmetry preserving for hardening plasticity symmetry cannot be preserved in general unless suitable restrictions are imposed on the constitutive equations we show that these restrictions amount to the existence of a pseudo internal energy function acting as a joint potential for both the direction of plastic flow and the hardening moduli in view of the fact that holonomic methods based on incrementally extremal paths also result in update rules possessing a potential structure and hence in symmetric tangents we address the question of whether any connections exist between the two approaches we show that holonomic methods and the fully implicit algorithm may indeed be brought into correspondence

summation of divergent series and borel summability for strongly dissipative differential equations with periodic or quasiperiodic forcing terms we consider a class of second order ordinary differential equations describing one dimensional systems with a quasiperiodic analytic forcing term and in the presence of damping as a physical application one can think of a resistor inductor varactor circuit with a periodic or quasiperiodic forcing function even if the range of applicability of the theory is much wider in the limit of large damping we look for quasiperiodic solutions which have the same frequency vector of the forcing term and we study their analyticity properties in the inverse of the damping coefficient we find that even the case of periodic forcing terms is nontrivial as the solution is not analytic in a neighborhood of the origin it turns out to be borel summable in the case of quasiperiodic forcing terms we need renormalization group techniques in order to control the small divisors arising in the perturbation series we show the existence of a summation criterion of the series in this case also however this cannot be interpreted as borel summability

the k property of four billiard balls a further step is achieved toward establishing the celebrated boltzmann sinai ergodic hypothesis for systems of four hard balls on the torus 2 it is shown that on the submanifold of the phase specified by the trivial conservation laws the system is ak flow all parts of our previous demonstration providing the analogous result for three hard balls are simplified and strengthened the main novelties are i a refinement of the geometric algebraic methods used earlier helps us to bound the codimension of the arising implicitly given set of degeneracies even if we can not calculate their exact dimension that was possible for three billiards as a matter of fact it is this part of our arguments where further understanding and new ideas are necessary before attacking the general ergodic problem ii in the pasting part of the proof which is a sophisticated version of hopf s classical device the arguments are so general that it is hoped they work in the general case too this is achieved for four balls in particular by a version of the transversal fundamental theorem which on one hand is simpler and more suitable for applications than the previous one and on the other hand as we have discovered earlier is the main tool to prove global ergodicity of semi dispersing billiards iii the verification of the chernov sinai ansatz is essentially simplified and the new idea of the proof also promises to work in the general case

finite element analysis of two and three dimensional elasto plastic frames the natural approach abstract the paper presents a finite element analysis of large deformations in elasto plastic frames attention is focused on the step by step analysis of the nonlinear behaviour of such structures advantage is taken of the so called natural formulation in terms of the generalised cross sectional forces limit state functions applying to different loading conditions and frame geometries are implemented into a computer program a number of test examples illustrate practical applications

a consistent 3d corotational beam element for nonlinear dynamic analysis of flexible structures abstract the purpose of the paper is to present a corotational beam element for the nonlinear dynamic analysis of 3d flexible frames the novelty of the formulation lies in the use of the corotational framework i e the decomposition into rigid body motion and pure deformation to derive not only the internal force vector and the tangent stiffness matrix but also the inertia force vector and the tangent dynamic matrix as a consequence cubic interpolations are adopted to formulate both inertia and internal local terms in the derivation of the dynamic terms an approximation for the local rotations is introduced and a concise expression for the global inertia force vector is obtained to enhance the efficiency of the iterative procedure an approximate expression of the tangent dynamic matrix is adopted four numerical examples are considered to assess the performance of the new formulation against the one suggested by simo and vu quoc 1988 48 it was observed that the proposed formulation proves to combine accuracy with efficiency in particular the present approach achieves the same level of accuracy as the formulation of simo and vu quoc but with a significantly smaller number of elements

studies in anisotropic plasticity with reference to the hill criterion algorithms based upon the notion of return mapping have been developed for the hill yield function of anisotropic plasticity the relative accuracy of two algorithms is assessed by means of iso error maps the choice of the algorithm turns out to be much more critical for the orthotropic hill criterion than for the underlying isotropic von mises plasticity model a tangent operator that is consistent with the developed integration algorithm is formulated and its efficiency is assessed compared with the classical continuum tangent operator the model has been applied to three shell plate structures

a geometrically exact rod model incorporating shear and torsion warping deformation abstract a fully nonlinear three dimensional rod model is developed that incorporates transverse shear and torsion warping deformation the geometric setting is that of a constrained body model with configuration space modeled on r3 so 3 r a differentiable manifold the proposed model incorporates the classical notion of bi moment and bi shear in a fully nonlinear geometrically exact context explicit properly invariant constitutive equations that generalize those of the linearized theory are developed the underlying variational formulation of the model is discussed and computational procedures employing a galerkin projection are addressed numerical examples are presented that illustrate the performance of the formulation

bifurcation phenomena and attractive periodic solutions in the saturating inductor circuit in this paper we investigate bifurcation phenomena such as those of the periodic solutions for the unperturbed nonlinear system g x cbxz 0 with g x z ac x 2 1c x 2 and ao1 bo0 when we add the two competing termskf t cg x with f t a time periodic analytic forcing function and go0 the dissipative parameter the resulting differential equation g x xcbxcg xkf t z 0 describes approximately an electronic system known as the saturating inductor circuit for any periodic orbit of the unperturbed system we provide conditions which give rise to the appearance of subharmonic solutions furthermore we show that other bifurcation phenomena arise as there is a periodic solution with the same period as the forcing function f t which branches off the origin when the perturbation is switched on we also show that such a solution which encircles the origin attracts the entire phase space when the dissipative parameter becomes large enough we then compute numerically the basins of attraction of the attractive periodic solutions by choosing specific examples of the forcing function f t which are dictated by experience we provide evidence showing that all the dynamics of the saturating inductor circuit is organized by the persistent subharmonic solutions and by the periodic solution around the origin

the finite deformation theory for beam plate and shell part iii the three dimensional beam theory and the fe formulation a finite deformation model based on the timoshenko beam theory is proposed for the three dimensional beam structures the exact green lagrangian strains are derived the finite element formulation and the corresponding algorithm are presented for the model numerical examples are given to illustrate its validity and effectiveness

the k property of three billiard balls sinai s strengthened version of the ergodic hypothesis is proved for three billiard balls on the v dimensional torus on connected components of the submanifold of the phase space specified by the trivial conservation laws of the energy and of the trajectory of the center of mass the system is a k flow to cope with the difficulty that in the isomorphic one particle billiard system the scatterers are not strictly convex geometric algebraic ergodic theoretic and topological methods are elaborated

primal and mixed variational principles for dynamics of spatial beams a general formulation is presented for static and dynamic analysis of spatial elastic beams capable of undergoing finite rotations and small strains the tangent maps associated to the finite rotation vector are used to compute the tangent iteration matrices used to integrate implicitly the equations of motion in descriptor form a total lagrangian primal corotational method and an updated lagrangian mixed variational method are proposed to compute the tangent stiffness matrix the tangent inertia matrices including the gyroscopic and centrifugal terms are also obtained by using the tangent maps of rotation the numerical examples analyzed in this paper include static pre and postbuckling and dynamic analysis of flexible beams structures the new finite elements show a very good performance in terms of fewer number of elements used and accuracy during the simulation both for static and dynamic problems

on the relationship between the eulerian and lagrangian descriptions of finite rigid plasticity the motivation for the present paper is to clarify certain unresolved issues pertaining to the relation between the lagrangian or referential and eulerian or spatial strain space formulations of finite plasticity for conceptual simplicity attention is confined to rigid plastic materials it is shown first that for constitutive equations in which the hardening parameter is a scalar the lagrangian and eulerian descriptions are equivalent and that additionally the choice of objective stress rate is immaterial in the light of these developments the role of objective rates is further explored in connection with more general anisotropic hardening laws which contain a shift tensor a form of the constitutive equation for the rate of the shift tensor is motivated in which the choice of objective rate is arbitrary it is then demonstrated that the structure of the constitutive equations of the theory in both the eulerian and lagrangian descriptions is form invariant under arbitrary transformations of objective rate the approach taken here contrasts with that adopted in a number of recent papers in which preference is given to one particular objective rate or another

a study on the instability problem for 2d frames this paper presents large deflection post buckling analysis of the two dimensional elastic frame from a dynamic point of view a co rotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted an increment iterative method based on the modified newton raphson method and extrapolation techniques for improving the convergence behaviour combined with a constant arc length control method is employed for the solution of the non linear equilibrium equations before the limit point a non linear dynamic analysis based on the average acceleration of the newmark algorithm with a slow rate of load incrementation is carried out to trace the load deflection path beyond the limit point as a result the snap through problem is overcome without down loading the efficiency of the method proposed in this study is demonstrated by typical two dimensional frames which show snap through behaviour in static analysis with any increment of load after the limit point

quasiperiodic attractors borel summability and the bryuno condition for strongly dissipative systems we consider a class of ordinary differential equations describing one dimensional analytic systems with a quasiperiodic forcing term and in the presence of damping in the limit of large damping under some generic nondegeneracy condition on the force there are quasiperiodic solutions which have the same frequency vector as the forcing term we prove that such solutions are borel summable at the origin when the frequency vector is either any one dimensional number or a two dimensional vector such that the ratio of its components is an irrational number of constant type in the first case the proof given simplifies that provided in a previous work of ours we also show that in any dimension d for the existence of a quasiperiodic solution with the same frequency vector as the forcing term the standard diophantine condition can be weakened into the bryuno condition in all cases under a suitable positivity condition the quasiperiodic solution is proved to describe a local attractor

nonlinear postbuckling analysis of plates and shells by four noded strain element this paper deals with the nonlinear postbuckling analysis of plates and shells by the finite element method the four noded c strain element for nonlinear plate anaysis developed by the authors is first extended to the nonlinear shell analysis and then used for the nonlinear postbuckling analysis of plates and shallow shells the element tangent stiffness matrix presented here is given explicitly i e without any numerical integration consequently this c strain element is much more computationally economical and efficient than the widely used elements obtained from numerical integration the efficiency and accuracy of the present four noded strain element for postbuckling analysis are demonstrated by five numerical examples of various plates and shells

a mixed hybrid finite beam element with an interface to arbitrary three dimensional material models abstract a theory for spatial beams and an associated mixed finite element formulation are presented the basic kinematic assumption allows for finite rotations and transverse shear strains the relation of the green lagrangean strains to the beam strains is derived a three field variational formulation with independent displacements rotations stress resultants and beam strains is developed using the kinematic and static field equations and the constitutive equations higher order stress resultants are set to zero within the euler lagrange equations the formulation in this paper is restricted to rectangular cross sections the interpolation of the beam strains consists of two parts the first part is identical to the stress resultant interpolation the second part describes cross section warping and allows for transverse normal strains within this approach an interface to arbitrary three dimensional constitutive laws can easily be realized due to the mixed interpolation technique full gauss integration in length direction of the beam element can be applied to obtain a locking free element formulation the effectiveness of the method is demonstrated by means of several numerical examples we present results for elasticity inelasticity and stability problems and evaluate the load carrying capacities of spatial beam structures the essential feature of the new element is the robustness in nonlinear applications it allows very large load steps in comparison to other element formulations

an excursion into large rotations the present discourse develops an enlarged exploration of the matrix formulation of finite rotations in space initiated in 1 it is shown how a consistent but subtle matrix calculus inevitably leads to a number of elegant expressions for the transformation or rotation matrix t appertaining to a rotation about an arbitrary axis also analysed is the case of multiple rotations about fixed or follower axes particular attention is paid to an explicit derivation of a single compound rotation vector equivalent to two consecutive arbitrary rotations this theme is discussed in some detail for a number of cases semitangential rotations for which commutativity holds first proposed in 2 3 are also considered furthermore an elementary geometrical analysis of large rotations is also given finally we deduce in an appendix using a judicious reformulation of quarternions the compound pseudovector representing the combined effect of n rotations in the author s opinion the present approach appears preferable to a pure vectorial scheme and even more so to an indicial formulation and is computationally more convenient

finite rotation analysis and consistent linearization using projectors abstract a systematic procedure is presented that may be used to extend the capabilities of an existing linear finite element to accomodate finite rotation analysis this procedure is a generalization of the work presented in computers and structures volume 30 pp 257 267 the basis of our approach is the element independent co rotational algorithm where the element rigid body motion translations and rotations is separated from the deformational part of its total motion the variation of this co rotational relation results in a projector matrix with the property that a consistent internal force vector is invariant under its action the consistent tangent stiffness matrix is shown to depend on this invariance condition through the derivative of the projector this results in an unsymmetric tangent matrix whose anti symmetric part depends on the out of balance force vector in this paper we prove that using the symmetric part of the tangent matrix the newton iteration retains its quadratic rate of convergence this approach has been used to solve a number of large rotation test example problems the results demonstrate that it is possible to analyze structures undergoing large rotations within a general co rotational framework using simple and economical finite elements the resulting improvements in the performance of these simple elements are brought about through the use of convenient software utilities as pre and post processors to the element routines

element behavior in post critical plane frame analysis depending on the type of uni modal cuspoid catastrophe which governs the local behavior of a structure at a simple critical point correct prediction of the post critical path requires an accurat

elasto plastic large deformation analysis of space frames a plastic hinge and stress based explicit derivation of tangent stiffnesses summary this paper deals with elasto plastic large deformation analysis of space frames it is based on a complementary energy approach a methodology is presented wherein i each member of the frame modelled as an initially straight space beam is sought to be represented by a single finite element ii each member can undergo arbitrarily large rigid rotations but only moderately large relative rotations iii a plastic hinge method with arbitrary locations of the hinges along the beam is used to account for plasticity iv the non linear bending stretching coupling is accounted for in each member v the applied loading may be non conservative and vi an explicit expression for the tangent stiffness matrix of each element is given under conditions i to v several examples with both quasi static and dynamic loading are given to illustrate the accuracy and efficiency of the approaches presented

non linear b stability and symmetry preserving return mapping algorithms for plasticity and viscoplasticity a class of second order accurate return mapping algorithms is presented which lead to symmetric algorithmic tangent moduli and contain the classical backward euler return maps as a particular case more importantly it is shown that this class of return maps is contractive relative to the natural norm defined by the complementary helmholz free energy function b stability since the equations of classical plasticity and viscoplasticity are shown to be contractive relative to this natural norm the requirement of b stability furnishes the appropriate notion of unconditionally stable algorithms for plasticity and viscoplasticity the analysis that follows depends critically on the assumption of convexity in particular the models of plasticity and viscoplasticity considered obey the principle of maximum plastic dissipation the proposed algorithms obey the discrete counterpart of this classical principle

invariant sets for the varactor equation xc x m z f t with f t positive periodic and continuous is studied after describing some physical applications of this equation we construct a variety of invariant sets for it thereby partitioning the phase plane into regions in which solutions grow without bound and also those in which bounded periodic solutions exist

simplified lattice beam elements for geometrically nonlinear static dynamic and postbuckling analysis a simple beam finite element for modeling the geo mjnetrically nonlinear behavior of flexible space lattice beams with repetitive geometry is developed the element is formulated using expressions for the strain and kinetic energies of an equivalent beam continuum to account for the nonlinear effects the incremental stiffness matrix of the equivalent continuum is used to evaluate the present nonlinear formulation and assess its accuracy a series of three nonlinear beam problems are solved using this simple element 1 static large deflection analysis 2 large amplitude free vibration analysis and 3 buckling and postbuckling analysis alternative solutions for the three examples show that the present formulation may be used as an efficient substitute for a detailed truss model in static and dynamic geometrically nonlinear analysis of beamlike lattice structures

globally and locally attractive solutions for quasi periodically forced systems abstract we consider a class of differential equations x x g x f t with r d describing one dimensional dissipative systems subject to a periodic or quasi periodic diophantine forcing we study existence and properties of trajectories with the same quasi periodicity as the forcing for g x x 2 p 1 p n we show that when the dissipation coefficient is large enough there is only one such trajectory and that it describes a global attractor in the case of more general nonlinearities including g x x 2 describing the varactor equation we find that there is at least one trajectory which describes a local attractor

beam p version finite element for geometrically non linear vibrations in space abstract a beam p version hierarchical finite element for geometrically non linear vibrations in space is presented stresses are obtained by adding the effects of traction torsion and bending a simplified version of green s strain tensor and the generalised hooke s law are used the mass and stiffness matrices are obtained by the principle of the virtual work and the ensuing non linear equations of motion are solved by newmark s method some examples of linear spacial motion and non linear planar motion are tested in order to verify the element derived and it is demonstrated that this element requires a small number of degrees of freedom for accuracy non linear vibrations in space are then studied

numerical integration of rate constitutive equations in finite deformation analysis abstract in analysis of finite deformation problems the use of constitutive equations in rate form is often required in a spatial setting these equations may express a relationship between some objective rate of spatial stress tensor and the rate of deformation constitutive equations of this type characterize a variety of material models including hyperelasticity hypoelasticity and elastoplasticity employing geometrical concepts a family of unconditionally stable and incrementally objective algorithms is proposed for the integration of such rate constitutive equations these algorithms which are appropriate for finite deformation analysis are applicable to any choice of stress rate and in most cases employ quantities that arise naturally in the context of finite element analysis examples illustrate the objectivity and accuracy of the algorithms

finite element theory for curved and twisted beams based on exact solutions for three dimensional solids part 1 beam concept and geometrically exact nonlinear formulation a geometrically exact and completely consistent finite element theory for curved and twisted beams is proposed it is based on the kinematical hypothesis generally formulated for large deformation and accounts for all kinds of deformation of a three dimensional solid translational and rotational displacements of the cross sections warping of their plane and distortion of their contours the principle of virtual work is applied in a straightforward manner to all non zero six components of the strain and stress tensors expressions are given for tangent matrices of elastic inertia and external forces and specific techniques for discretization and updating are developed for the analysis of beams in inertial and non inertial frames finally the numerical properties of the finite element models are demonstrated through examples c 1998 elsevier science s a all rights reserved