a consistent finite element formulation for linear buckling analysis of spatial beams a consistent co rotational finite element formulation and numerical procedure for the linear buckling analysis of three dimensional elastic euler beam is presented a mechanism for generating configuration dependent conservative moment is proposed and the corresponding load stiffness matrix is derived it is assumed that the prebuckling displacements and rotations of the structure and the corresponding deformations of the elements are linearly proportional to the external loading the prebuckling rotations of the structure are fixed axis rotations or small rotations and the effect of the prebuckling displacement on transformation matrix for the coordinates transformation can be ignored all coupling among bending twisting and stretching deformations for beam element is considered by consistent linearization of the fully geometrically nonlinear beam theory an inverse power method for the solution of the generalized eigenvalue problem is used to find the buckling load and buckling mode numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method

the finite deformation theory for beam plate and shell part v the shell element with drilling degree of freedom based on biot strain due to the facts that the spatial moment force is non conservative and the three components of the finite rotation pseudo vector are not independent the ordinary variational principle based on a functional achieving its stationary value is not suitable for the problem subject to non conservative loading the finite element method proposed in this paper is based on the virtual work principle rigid normal assumption polar decomposition and total lagrangian formulation

a continuation method for nonlinear complementarity problems over symmetric cones in this paper we introduce a new p type property for nonlinear functions defined over euclidean jordan algebras and study a continuation method for nonlinear complementarity problems over symmetric cones this new p type property represents a new class of nonmonotone nonlinear complementarity problems that can be solved numerically

h sub infinity optimal control with state feedback a h sub infinity optimal control problem in which the measured outputs are the states of the plant is considered the main result shows that the infimum of the norm of the closed loop transfer function using linear static state feedback equals the infimum of the norm of the closed loop transfer function over all stabilizing dynamic even nonlinear time varying state feedback controllers

stress resultant geometrically nonlinear shell theory with drilling rotations part ii computational aspects abstract in this work we discuss some details of the numerical implementation of the geometrically nonlinear shell theory presented in part i two possibilities to represent finite rotations with an orthogonal matrix and with a rotation vector are examined in detail along with their mutual relationship in both spatial and material description the issues pertinent to the consistent linearization procedure corresponding to these rotation parameterizations are also carefully considered the geometrically nonlinear method of incompatible modes is extended to the nonlinear shell theory under consideration and used to provide a 4 node shell element with enhanced performance a rather extensive set of numerical examples in nonlinear elastostatics is solved in order to corroborate the non locking performance of the incompatible mode based shell elements the examples include not only analyses of simple shell structures undergoing very large displacements and rotations but also cases of strong practical interest such as non smooth shell structures and shell structures with stiffeners

shear flexible models for spatial buckling of thin walled curved beams the governing non linear finite element equations for the spatial stability analysis of curved beams using a simple two noded model are derived based on the incremental form of a mixed variational principle with independent discretization for its generalized strain field and the reference line displacements as well as cross sectional warping and bending twisting rotations the formulation is valid for both open and closed type thin walled sections and this is accomplished by the use of a kinematic description based on a generalized beam theory in which shear deformation due to both flexural and warping torsional actions is accounted for the effect of finite rotations in space is included resulting in a second order accurate geometric stiffness matrix and ensuring that all significant instability modes can be predicted finally the results obtained in a number of numerical simulations for lateral torsional bifurcation buckling of circular arches are presented to illustrate the model effectiveness and practical usefulness and to provide explanations for the source of discrepancies noted in the results obtained in previous investigations

conservative integration of rigid body motion by quaternion parameters with implicit constraints summary an angular momentum and energy conserving time integration algorithm for rigid body rotation is formulated in terms of the quaternion parameters and the corresponding four component conjugate momentum vector via hamilton s equations the introduction of an extended mass matrix leads to a symmetric set of eight state space equations of motion the extra inertial parameter serves as a multiplier on the kinematic constraint and it is demonstrated that convergence characteristics are improved by selecting this parameter somewhat larger than the inertial moments external loads enter these equations via the set of momentum equations initially the normalization of the quaternion array is introduced via a lagrange multiplier however this lagrange multiplier can be expressed explicitly in terms of the gradient of the external load potential and elimination of the lagrange multiplier from the final format leaves only an explicit projection applied to the external load potential gradient an algorithm is developed by forming a finite increment of the hamiltonian this procedure identifies the proper selection of increments and mean values and leads to an algorithm with conservation of momentum and energy implementation conservation properties and accuracy of the algorithm are illustrated by examples with a flying box and a spinning top copyright 2012 john wiley sons ltd

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

on the parametrization of finite rotations in computational mechanics a classification of concepts with application to smooth shells this paper concerns the computational treatment of large rotations with application to the finite element discretisation of smooth shells formulations based on various rotational parametrizations are reviewed and classified with respect to their update structure category ii is based on total rotational degrees of freedom leading to an additive update structure whereas category i relies on linearized rotational degrees of freedom leading to a multiplicative update structure based on this classification new formulations are developed among them are the rodrigues formula applied within category ii with additive update structure for the two components of the rotation vector and two successive elementary rotations in the sense of euler angles applied within category i yielding a singularity free formulation the numerical examples confirm that every rotational parametrization when applied within category ii leads to singularities whereas the application within category i enables the calculation of overall rotations unrestricted in size without any singularity

explicit isogeometric collocation for the dynamics of three dimensional beams undergoing finite motions abstract we initiate the study of three dimensional shear deformable geometrically exact beam dynamics through explicit isogeometric collocation methods the formulation we propose is based on a natural combination of the chosen finite rotations representation with an explicit geometrically consistent lie group time integrator we focus on extending the integration scheme originally proposed for rigid body dynamics to our nonlinear initial boundary value problem where special attention is required by neumann boundary conditions the overall formulation is simple and only relies on a geometrically consistent procedure to compute the internal forces once control angular and linear accelerations of the beam cross sections are obtained from the previous time step the capabilities of the method are shown through numerical applications involving very large displacements and rotations and different boundary conditions

integrating finite rotations abstract we study the integration of problems of evolution in the rotation group instead of attacking the problem in the nonlinear differential manifold so 3 pure rotational dynamics as is usually done we derive equations for the complete problem of motion translational and rotational dynamics on an extended manifold we develop a generalization of runge kutta methods that by design ensures that the solution will remain on the manifold for any choice of the tableau this is obtained through configuration updates performed via the exponential map we show how certain terms can be approximated while retaining the order of accuracy of the scheme and how the method conserves the total momentum of the system within this framework we develop two nonlinearly unconditionally stable time integration schemes that are associated with discrete laws of conservation dissipation of the total energy the dissipating algorithm generalizes to the nonlinear case the high frequency damping characteristics provided by some well known conventional methods we present numerical results to support our analysis and we develop a complete application of this methodology to the nonlinear dynamics of three dimensional rods undergoing large displacements and finite rotations under the assumption of small strains

finite element theory for curved and twisted beams based on exact solutions for three dimensional solids part 2 anisotropic and advanced beam models consistent finite element formulations for beams made of anisotropic materials and taking into account non classic inhomogeneous torsion have been developed the formulations are based on a kinematical hypothesis that includes exact solutions for three dimensional solids under terminal loading they describe warping of the cross sections in and out of their planes as well as their rigid displacements and rotations their large deformation and geometrically exact description by finite rotations are considered for the cases of monoclinic orthotropic and transversely isotropic materials exact solutions for the solid made from a monoclinic material have been deduced c 1998 elsevier science s a all rights reserved

correction of the fi result in h sub infinity control and parameterization of h sub infinity state feedback controllers the authors correct the parameterization of the h sub infinity controller of the full information fi problem derived by j c doyle et al 1989 then they parameterize the h sub m0 state feedback controller and explain how dynamical free parameters implied in it are related to constant feedback gains different from the central solution f sub infinity

an efficient mixed variational reduced order model formulation for nonlinear analyses of elastic shells summary the koiter newton method had recently demonstrated a superior performance for non linear analyses of structures compared to traditional path following strategies the method follows a predictor corrector scheme to trace the entire equilibrium path during a predictor step a reduced order model is constructed based on koiter s asymptotic post buckling theory which is followed by a newton iteration in the corrector phase to regain the equilibrium of forces in this manuscript we introduce a robust mixed solid shell formulation to further enhance the efficiency of stability analyses in various aspects we show that a hellinger reissner variational formulation facilitates the reduced order model construction omitting an expensive evaluation of the inherent fourth order derivatives of the strain energy we demonstrate that extremely large step sizes with a reasonable out of balance residual can be obtained with substantial impact on the total number of steps needed to trace the complete equilibrium path more importantly the numerical effort of the corrector phase involving a newton iteration of the full order model is drastically reduced thus revealing the true strength of the proposed formulation we study a number of problems from engineering and compare the results to the conventional approach in order to highlight the gain in numerical efficiency for stability problems this article is protected by copyright all rights reserved

a hybrid mixed model for non linear shell analysis and its applications to large rotation problems adopting an updated lagrange approach the general framework for the fully non linear analysis of curved shells is developed using a simple quadrilateral c0 model hmsh5 the governing equations are derived based on a consistent linearization of an incremental mixed variational principle of the modified hellinger reissner type with independent assumptions for displacement and strain fields emphasis is placed on devising effective solution procedures to deal with large rotations in space finite stretches and generalized rate type material models in particular a geometrically exact scheme for configuration update is developed by making use of the so called exponential mapping algorithm and the resulting element was shown to exhibit a quadratic rate of asymptotic convergence in solving practical shell problems with newton raphson type iterative schemes for the purpose of updating the spatial stress field of the element an objective generalized midpoint integration rule is utilized which relies crucially on the concept of polar decomposition for the deformation gradient and is in keeping with the underlying mixed method finally the effectiveness and practical usefulness of the hmsh5 element are demonstrated through a number of test cases involving beams plates and shells undergoing very large displacements and rotations

a spatially curved beam element with warping and wagner effects this paper presents a new spatially curved beam element with warping and wagner effects that can be used for the non linear large displacement analysis of members that are curved in space the non linear behaviour of members curved in space shows that the wagner effects are substantial in the large twist rotation analysis most existing finite beam element models such as abaqus and ansys cannot predict the non linear large displacement response of members curved in space correctly because the wagner effects viz the wagner moment and the corresponding finite strain terms have not been considered in these finite beam elements as a consequence these finite beam elements do not provide correct predictions for the out of plane buckling and postbuckling behaviour of arches as well in this paper the symmetric tangent stiffness matrix has been derived based on the finite rotations parameterized by the conventional displacements the warping and wagner effects both the wagner moment and the corresponding finite strain terms and their constitutive relationship are included in the spatially curved beam element two components of the initial curvature the initial twist and their interactions with the displacements are also considered in the spatially curved beam element this ensures that the large twist rotation analysis for the members curved in space is accurate comparisons with existing experimental analytical and numerical results show that the spatially curved beam element is accurate and efficient for the non linear elastic analysis of curved members buckling and postbuckling analysis of arches and in its ability to predict large deflections and twist rotations in more arbitrarily curved members copyright 2005 john wiley sons ltd

a co rotational finite element formulation for buckling and postbuckling analyses of spatial beams abstract a consistent co rotational finite element formulation and numerical procedure for the buckling and postbuckling analyses of three dimensional elastic euler beam is presented all coupling among bending twisting and stretching deformations for a beam element is considered by consistent second order linearization of the fully geometrically nonlinear beam theory however the third order terms which are relevant to the twist rate and curvature of the beam axis are also considered an incremental iterative method based on the newton raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations the zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state a bisection method of the arc length is proposed to find the buckling load numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method and to investigate the effect of third order terms on the buckling load and postbuckling behavior of three dimensional beams

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

a co rotational quasi conforming 4 node resultant shell element for large deformation elasto plastic analysis abstract a co rotational quasi conforming formulation of a 4 node stress resultant shell element is presented for non linear analysis of plate and shell structures the tangent stiffness matrix in this quasi conforming formulation is explicitly integrated this makes the element computationally efficient in incremental non linear analysis it includes drilling degrees of freedom which improves membrane behavior and allows the modeling of stiffened plates and shells it is also free of shear locking behavior the formulation of the geometrical stiffness is derived using the full definition of green strain tensor the inclusion of the bending moment and transverse shear resultant forces in the geometric stiffness allows effective analysis of stability problems of moderately thick plates and shells the stresses are accurately taken at the nodal points without extrapolation the plasticity is traced by applying the von mises yield condition and prandtl reuss flow rule to discrete points through the thickness the multi layered approach is based on equally spaced stations including extreme fibers a modified trapezoidal rule is used for the numerical integration of the constitutive relation in the plasticity part numerous tests are carried out for the non linear validation of present 4 node shell element and the results are in good agreement with references

parametrization of state feedback h controllers the state feedback h control problem is considered in this paper it is shown that under mild conditions this problem can be solved simply by using the conjugation approach proposed by kimura and kawatani with some extended treatment to the case that t3 s is tall and strictly proper a partial parametrization of state feedback controllers achieving the prescribed norm bound is obtained this is an eminent difference from previous work on the state feedback h control problem where only a static feedback control law the central solution of our result is provided both four block and one block problems are solved in this paper the results of this paper embrace those of doyle et al and petersen

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

synthesis of reduced order h controllers based on conjugation a new state space characterization of solutions of a class of model matching problem the so called model matching problem of the first kind by limebeer has recently been derived by kimura based o

a formulation of the mitc4 shell element for finite strain elasto plastic analysis abstract a finite strain elasto plastic formulation is developed for the mitc4 shell element the developed formulation is based on lee s multiplicative decomposition of the deformation gradient and on the hyperelastic expression of von mises flow rule expressed in terms of hencky s strain tensor the formulation incorporates thickness stretching degrees of freedom that are condensed at the element level by imposing the classical in layer plane stress condition a symmetric and consistent tangent stiffness matrix is also developed

statics of thin walled pretwisted beams the displacement and strain fields of thin walled pretwisted beams are prescribed in terms of generalized displacements for extension bending torsion and warping differential equations and boundary conditions are obtained from the elastic potential energy functional without assuming coincidence of the beam axis with any of the structural axes this procedure gives a unique consistent definition of sectional moments and generalized forces some simple explicit formulae are derived for homogeneous tension torsion for the general case a computer code is developed on the basis of discretized generalized displacements and a modified energy functional devised to obtain consistent lengthwise variation of the stresses and a unique decomposition of the torsional moment examples show agreement with analytical results for cylindrical beams and illustrate the various coupling effects for beams with pretwist they also demonstrate the usefulness of the explicit formulae for homogeneous tension torsion

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

complementarity forms of theorems of lyapunov and stein and related results abstract the well known lyapunov s theorem in matrix theory continuous dynamical systems asserts that a complex square matrix a is positive stable i e all eigenvalues lie in the open right half plane if and only if there exists a positive definite matrix x such that ax xa is positive definite in this paper we prove a complementarity form of this theorem a is positive stable if and only if for any hermitian matrix q there exists a positive semidefinite matrix x such that ax xa q is positive semidefinite and x ax xa q 0 by considering cone complementarity problems corresponding to linear transformations of the form i s we show that a complex matrix a has all eigenvalues in the open unit disk of the complex plane if and only if for every hermitian matrix q there exists a positive semidefinite matrix x such that x axa q is positive semidefinite and x x axa q 0 by specializing q to i we deduce the well known stein s theorem in discrete linear dynamical systems a has all eigenvalues in the open unit disk if and only if there exists a positive definite matrix x such that x axa is positive definite

evaluation of simple bifurcation points and post critical path in large finite rotation problems the main aspects of an algorithm for the exact evaluation of single bifurcation points along the nonlinear equilibrium trajectory of flexible mechanisms with large 3 d rotations are discussed a first version of this algorithm was presented elsewhere 1 in this work we extend its range of application to problems involving large finite rotations several flexible mechanisms and structures presenting bifurcation points are analyzed solutions of perfect bifurcation cases are compared to solutions of the literature which were computed by geometric perturbation

a simple element for static and dynamic response of beams with material and geometric nonlinearities a unified simple 6 degrees of freedom beam finite element and the associated computational procedures have been developed for the fast and efficient solution of a wide class of static and dynamic response problems of the beam type with material and or geometrical nonlinearities the material nonlinearity is treated by including its effect in the governing equations by forming the stiffness matrix of each element using a two dimensional grid of gauss points and using the material properties at each point corresponding to the uniaxial strain at that point examples are provided for metal and reinforced concrete beams a powerful yet straightforward method for the solution of elastica problems of beams and frames using the beam element developed by the senior author has been extended for determining the dynamic response of beams undergoing large displacements including large rotations the solution procedure involves piecewise linearization of response equations and iterations at each incremental step to achieve equilibrium the solution procedure is simple and easy to apply a variety of problems is solved to determine the applicability of the proposed simple formulations excellent agreement with existing analytical solutions which employ higher order elements demonstrates the efficiency and versatility of the present simple beam element in nonlinear analysis

a kinematically exact space finite strain beam model finite element formulation by generalized virtual work principle the present paper presents a novel finite element formulation for static analysis of linear elastic spatial frame structures extending the formulation given by simo and vu quoc a geometrically exact rod model incorporating shear and torsion warping deformation int j solids structures 27 3 1991 371 3931 along the lines of the work on the planar beam theory presented by saje a variational principle for finite planar deformation of straight slender elastic beams internat j solids and structures 26 1990 887 9001 we apply exact non linear kinematic relationships of the space finite strain beam theory assuming the bernoulli hypothesis and neglecting the warping deformations of the cross section finite displacements and rotations as well as finite extensional shear torsional and bending strains are accounted for in the formulation a deformed configuration of the beam is described by the displacement vector of the deformed centroid axis and an orthonormal moving frame rigidly attached to the cross section of the beam the position of the moving frame relative to a fixed reference frame is specified by an orthogonal matrix parametrized by the rotational vector which rotates the moving frame from an arbitrary position into the deformed configuration in one step also the incremental rotational vector is introduced which rotates the moving frame from the configuration obtained at the previous iteration step into the current configuration of the beam its components relative to the fixed global coordinate system are taken to be the rotational degrees of freedom at nodal points because in 3 d space both the axial and the follower moments are non conservative not the variational principle but the principle of virtual work has been introduced as a basis for the finite element discretization here we have proposed the generalized form of the principle of virtual work by including exact kinematic equations by means of a procedure similar to that of lagrangian multipliers this makes possible the elimination of the displacement vector field from the principle so that the three components of the incremental rotational vector field remain the only functions to be approximated in the finite element implementation of the principle other researchers on the other hand employ the three components of the incremental rotational vector field and the three components of the incremental displacement vector field as a result more accurate and efficient family of beam finite elements for the non linear analysis of space frames has been obtained a one field formulation results in the fact that in the present finite elements the locking never occurs any combination of deformation states is described equally precisely this is in contrast with the elements developed in literature where in order to avoid the locking a reduced numerical integration has to be applied which unfortunately diminishes the accuracy of the solution polynomials have been chosen for the approximation of the components of the rotational vector in this case the order of the numerical integration can rationally be estimated and the computer program can be coded in such a way that the degree of polynomials need not be limited to a particular value the newton method is used for the iterative solution of the non linear equilibrium equations in an non equilibrium configuration the tangent stiffness matrix obtained by the linearization of governing equations using the directional derivative is non symmetric even for conservative loadings only upon achieving an equilibrium state the tangent stiffness matrix becomes symmetric thus obtained tangent stiffness matrix can be symmetrized without affecting the rate of convergence of the newton method for non conservative loadings however the tangent stiffness matrix is always non symmetric the numerical examples demonstrate capability of the present formulation to determine accurately the non linear behaviour of space frames in numerical examples the out of plane buckling loads are determined and the whole pre and post critical load displacement paths of a cantilever and a right angle frame are traced these in the analysis of space beams standard verification example problems show excellent accuracy of the solution even when employing only one element to describe the displacements of the size of the structure itself the rotations of 2n and extensional strains much beyond the realistic values of linear elastic material

warping and in plane twist parameters in kinematics of finite rotation shells abstract the finite rotation shell equations are derived from a 3d mixed formulation with explicit rotations constitutive equations are written in terms of the relaxed biot stress and right stretch strain the shell kinematics is based on a generalised reissner hypothesis and rotations depending on the thickness coordinate the generalized reissner hypothesis includes a bubble like term for the warping of a cross section the rotations linearly vary over the thickness and are parametrised by a canonical rotation vector the components of the rotation vector can be identified as the in plane twist parameter and two warping parameters consequences of this new rotation vector for shell equations especially for rotation constraints and strain measures are examined and a comparison with the formulation which includes the in plane twist parameter is made the new rotation vector is treated as an implicit kinematical parameter and recovered from the rotation constraint hence a 6 parameter shell formulation admitting c 0 approximations is retained

matrix tensor notation part ii skew and curved coordinates in part i a notation called matrix tensor notation was introduced for rectilinear orthogonal coordinates part ii discusses how the same notation is equally efficient for vectors and 2nd order tensors in skew bases and skew curved coordinates without considering coordinates first vectors and 2nd order tensors are described in skew bases covariant and contravariant descriptions being distinguished similar to tensor notation a consistent interpretation of deformation and strain in terms of skew bases is given the use of the transpose symbol complicates the matrix algebra but integrates it with tensor algebra which makes it possible to interpret customary tensor equations as relations in space with all the advantages that an isomorphism with euclidean space has it is demonstrated how different metrics can be assigned arbitrarily to define higher dimensional orthogonal vectors in abstract higher dimensional space a consistent notation is given to distinguish between subspace and subbase to write higher order tensors as matrices or 2nd order tensors as vectors partial transpose is introduced which is a formal stretching operation to write any higher order tensor product operation as matrix vector product the description of the variable bases in generally curved coordinates is given in this notation together with the corresponding notation for the partial differentiation rules the vector operations grad div and curl are discussed and an almost physical notation for the christoffel symbols is given however the replacement of tensor notation for higher than second order tensors by matrix tensor notation is not feasible a version of matrix tensor notation that merges with the customary printed form is presented in an appendix further applications are deferred to part iii applying the new notation to curved coordinates in function space and the equations of mechanics projected on finite dimensional subspaces

spatial stability analysis of thin walled space frames a clearly consistent finite element formulation for spatial stability analysis of thin walled space frames is presented by applying linearized virtual work principle and introducing vlasov s assumption the improved displacement field for unsymmetric thin walled cross sections is introduced based on inclusion of second order terms of finite rotations and the potential energy corresponding to the semitangential moments is consistently derived in the present formulation displacement parameters of axial and bending deformations are defined at the centroid axis and parameters of lateral and torsional deformations at the shear centre axis and all bending torsional coupling effects due to unsymmetric cross sections are taken into account for finite element analysis cubic hermitian polynomials for the flexural beam with four types of end conditions are utilized as shape functions of hermitian space frame element also load correction stiffness matrices for off axis point loadings are derived based on the second order rotation terms finite element solutions for the spatial buckling analysis of thin walled space frames are compared with available solutions and other researcher s results

co rotational formulation for geometric nonlinear analysis of doubly symmetric thin walled beams abstract a doubly symmetric thin walled beam element with open section is derived using co rotational cr total lagrangian tl formulation the effects of deformation dependent third order terms of element nodal forces on the buckling load and post buckling behavior are investigated all coupling among bending twisting and stretching deformations for beam element is considered by consistent second order linearization of the fully geometrically nonlinear beam theory however all third order terms of nodal forces which are relevant to the twist rate rate of twist rate and curvature of the beam axis are also considered an incremental iterative method based on the newton raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations the zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state a parabolic interpolation method of the arc length is used to find the buckling load numerical examples are presented to demonstrate the accuracy and efficiency of the proposed element and to investigate the effect of third order terms of element nodal forces on the buckling load and post buckling behavior of doubly symmetric thin walled beams

objectivity of strain measures in the geometrically exact three dimensional beam theory and its finite element implementation the paper discusses the issue of discretization of the strainconfiguration relationships in the geometrically exact theory of threedimensional 3d beams which has been at the heart of most recent

nonlinear finite element analysis of elastic systems under nonconservative loading natural formulation part ii dynamic problems abstract the paper presents a nonlinear finite element analysis of elastic structures subject to nonconservative forces attention is focused on the stability behaviour of such systems this leads mathematically to non self adjoint boundary value problems which are of great theoretical and practical interest in particular in connection with alternative modes of instability like divergence of flutter only quasistatic effects are however considered in the present part the methodology of our theory is general but the specific thrust of the present research is directed towards the analysis of structures acted upon by displacement dependent nonconservative follower forces in a finite element formulation the analysis of geometrically nonlinear elastic systems subject to such forces gives in general rise to a contributory nonsymmetric stiffness matrix known as the load correction matrix as a result the total tangent stiffness matrix becomes unsymmetric an indication of the non self adjoint character of the problem our theory is based on the natural mode technique 1 2 3 and permits i a a simple but elegant derivation of the load correction matrix the application of the general theory as evolved in this paper is demonstrated on the beam element in space a number of numerical examples are considered including divergence and flutter types of instability for which exact analytic solutions are known the problems demonstrate the efficiency of the present finite element formulation the paper furnishes also a novel and concise formulation of finite rotations in space which may be considered as a conceptual generalization of the theory presented in 2 3

z transformations on proper and symmetric cones z transformations motivated by the similarities between the properties of z matrices on r n and lyapunov and stein transformations on the semidefinite cone mathcal s n we introduce and study z transformations on proper cones we show that many properties of z matrices extend to z transformations we describe the diagonal stability of such a transformation on a symmetric cone by means of quadratic representations finally we study the equivalence of q and p properties of z transformations on symmetric cones in particular we prove such an equivalence on the lorentz cone

nonlinear thin shell analysis using a curved triangular element abstract this paper describes a simple but robust curved triangular thin shell element the element has twelve degrees of freedom namely the displacement components in the corner nodes and the rotations about the element sides the element is based upon the well known facet element with constant membrane strains and constant stress couples and is augmented to take into account the influence of the changes of curvature on the membrane deformations due to the quadratic character of this influence curved geometries can be modelled in order to permit arbitrarily large rotational increments reference vectors are attached to each element side these reference vectors are updated by means of orthogonal transformations the latter are determined uniquely by the subsequent orientations of the associated element sides the changes of curvature are evaluated using these reference vectors and a corotated reference configuration to describe time independent elastoplasticity a layered shell model is applied while besseling s fraction model is adopted for each layer a number of numerical examples illustrate the performance of the element

efficient return algorithms for associated plasticity with multiple yield planes a new return method for implicit integration of linear isotropic yield criteria is presented the basic idea is to perform all the manipulations in the principal stress space and thereby achieve very simple formulae for calculating the plastic corrector stresses based on the constant gradient of such criteria the return formulae are in closed form and no iteration is required the method accounts for three types of stress return return to a single yield plane to a discontinuity line at the intersection of two yield planes and to a discontinuity point at the intersection between three or more yield planes the infinitesimal and the consistent elastoplastic constitutive matrix are calculated for each type of stress return as are the conditions to ascertain which type of return is required the method is exemplified with the mohr coulomb yield criterion

kinematically exact curved and twisted strain based beam the paper presents a formulation of the geometrically exact three dimensional beam theory where the shape functions of three dimensional rotations are obtained from strains by the analytical solution of kinematic equations in general it is very demanding to obtain rotations from known rotational strains in the paper we limit our studies to the constant strain field along the element the relation between the total three dimensional rotations and the rotational strains is complicated even when a constant strain field is assumed the analytical solution for the rotation matrix is for constant rotational strains expressed by the matrix exponential despite the analytical relationship between rotations and rotational strains the governing equations of the beam are in general too demanding to be solved analytically a finite element strain based formulation is presented in which numerical integration in governing equations and their variations is completely omitted and replaced by analytical integrals some interesting connections between quantities and non linear expressions of the beam are revealed these relations can also serve as useful guidelines in the development of new finite elements especially in the choice of suitable shape functions

finite element analysis of linear and non linear planar deformations of elastic initially curved beams reference epfl article 102719view record in web of science record created on 2007 04 23 modified on 2016 08 08

the helicoidal modeling in computational finite elasticity part i variational formulation the finite elasticity mechanics of continua capable of a polar description is formulated by an alternative modeling to keeping position and orientation as uncoupled fields the rototranslation between two material particles can be described by a single complex tensorial quantity which is recognized to be orthogonal its linearization gives the characteristic curvature and differential vectors underlying the helicoidal modeling in both the sense of the body geometric description and the evolution of a deforming body after due introduction to dual tensors and rototranslations the polar description of the continuum is addressed with particular care to mixed differentiations of the rototranslation field then a thorough variational framework is established for the most general polar continuum under hyperelasticity hypothesis and the three field two field and one field principles are drawn and linearized the proposed modeling is expected to be profitably exploited in non linear finite element analyses of solids undergoing finite displacements rotations and strains

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

effective modelling of spatial buckling of beam assemblages accounting for warping constraints and rotation dependency of moments a simple two noded finite element model for the three dimensional buckling analysis of beam assemblages is developed the underlying generalized beam theory employed accounts for all coupled significant modes of deformations including stretching bending shear and torsion as well as warping its mixed formulation is of the two field type utilizing linear and constant interpolants for displacements and strain fields due considerations are given to the effect off inite rotations in space on the non linear kinematic descriptions as well as the configuration dependent behaviour of externally applied moment vectors of the conservative and non conservative types the performance of the model and particularly the accuracy of its geometric and load correction stiffnesses is assessed in a fairly complete set of numerical simulations

shape optimal design of elastic space frames with non linear response this paper describes an approach to shape optimal design of elastic planar frames with non linear response the foundation of the proposed approach forms an appropriate strategy of shape representation of the structure based on the design element technique a frame structure is treated as to be assembled from several frame design elements which in turn may consist of several appropriately joined beam finite elements the shape of each frame design element is defined by convenient functions involving bezier blending polynomials the original formulation of the beam finite element proposed by saje is modified in order to fit nicely into the context of the frame design element technique the formulation of the shape optimal design problem in a form of a problem of non linear mathematical programming and its solution by employing gradient based methods of mathematical programming are discussed briefly the theory is illustrated with two numerical examples

conjugation interpolation and model matching in h abstract the notion of conjugation is introduced to give a unified framework of algebraic operations connected to the analytic aspects of transfer functions two classes of conjugations are shown to be important one is the lossless conjugation that performs inner outer factorization and the other is the j lossless conjugation that solves the model matching problem related to h 8 control the solution of the model matching problem obtained by j lossless conjugation is regarded as the state space version of the directional interpolation theory and is much simpler than those obtained via the nehari approximation

kinematics of finite rotation shells with in plane twist parameter abstract the paper concerns shell equations derived from a 3d mixed formulation with explicit rotations the constitutive equation is postulated in terms of the relaxed biot stress and right stretch strain the shell kinematics is based on the reissner hypothesis and on a constant linear approximation of the rotation tensor with respect to the thickness coordinate the rotation around the shell director is parametrized by the drill and in plane twist parameters with the latter one associated with the linear approximation the in plane twist parameter is new and hence consequences of its presence for the shell rotation constrains and strain measures are examined and compared to the formulation with the drill rotation only in the presented formulation the drill rotation is retained explicitly and can be recovered from the drill rotation constraint while the in plane twist parameter is an internal parameter of the theory in this way the enhanced shell kinematics does not increase the number of degrees of freedom of a discrete model

a unified co rotational framework for solids shells and beams abstract the paper describes a unified framework for applying the co rotational method to the analysis of solids shells and beams the general method stems from an unusual application of the technique which involves solid elements the proposed framework allows a formulation that is simpler than many of the earlier procedures and in addition gives a direct indication of the terms in the tangent stiffness which may be ignored

decay rate estimates for the quasi linear timoshenko system with nonlinear control and damping terms in this work we consider a class of quasi linear timoshenko systems with nonlinear control and damping terms we investigate the stabilization of weak solutions to the initial boundary value problem and obtain explicit decay rate estimation depending on control and damping terms

application of ring elements in the nonlinear analysis of shells of revolution under nonaxisymmetric loading abstract a problem oriented method for the nonlinear elastic plastic analysis of nonsymmetrically loaded shells of revolution is described it is based on an approximation of the circumferential distribution of the loads and variables by fourier series ring finite elements are used in the axial direction their stiffness matrices are obtained by accurate numerical integration rather than by trial functions to be able to take full advantage of the fourier decomposition and the one dimensional discretization all nonlinearities are treated iteratively in the form of pseudo load vectors thus the various fourier harmonics are governed by uncoupled algebraic equations and no explicit knowledge of the nonlinear global tangent stiffness matrix is required convergence problems of the iterative procedure are avoided by employing a sort of conjugate gradient method with a special preconditioning within each increment numerical examples involving strongly nonlinear behavior and highly nonsymmetric states of stress and deformation are given which demonstrate the effectiveness of the method

generalized energy momentum method for non linear adaptive shell dynamics in the present study a generalization of the energy momentum method denoted by generalized energy momentum method applied to the non linear dynamics of shells will be developed within the framework of the generalized a method this algorithmic environment contains the unconditionally stable energy momentum method and its numerically damped version as well as the classical newmark and a methods as special cases in order to control the size of the time steps of the integration scheme with respect to accuracy and eaciency an adaptive time stepping procedure based on local a posteriori error estimation will be improved for non linear dynamical systems and applied to the proposed class of algorithms the spatial discretization is realized by an eight noded finite shell element of reissner mindlin type including an extensible shell director field permitting the application of three dimensional material laws the original formulation of this finite element will be developed for non linear dynamic analysis and adapted for the employment within the introduced energy conserving decaying time integration scheme 1999 elsevier science s a all rights reserved

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

on sigma lossless transfer functions and related questions abstract this paper is concerned with a systematic approach to the properties of lossless rational transfer functions in the discrete as well as in the continuous time case as a result a unifying framework is revealed where several known results fit naturally special attention is given to the embedding problem of the lyapunov equation in view of its direct application to generalized levinson algorithms

on a stress resultant geometrically exact shell model part vi conserving algorithms for non linear dynamics in parts i to v of the present work the formulation and finite element implementation of a non linear stress resultant shell model are considered in detail this paper is concerned with the extension of these results to incorporate completely general non linear dynamic response of special interest here is the dynamics of very flexible shells undergoing large overall motion which conserves the total linear and angular momentum and for the hamiltonian case the total energy a main goal of this paper is the design of non linear time stepping algorithms and the construction of finite element interpolations which preserve exactly these fundamental constants of motion it is shown that only a very special class of algorithms namely a formulation of the mid point rule in conservation form exactly preserves the total linear and angular momentum for the hamiltonian case a somewhat surprising result is proved regardless of the degree of non linearity in the stored energy function a generalized mid point rule algorithm always exists which exactly conserves energy the conservation properties of a time stepping algorithm need not and in general will not be preserved by the spatial discretization precise conditions which ensure preservation of these conservation properties are derived a number of numerical simulations are presented which illustrate the exact conservation properties of the proposed methodology

finite element formulation of geometrically exact three dimensional beam theories based on interpolation of strain measures this paper presents a new finite element formulation of the geometrically exact finite strain beam theory the governing equations of the beam element are derived in which the strain vectors are the only unknown functions the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal is enforced to be satisfied at chosen points the solution is found by a collocation algorithm the linearity of the strain space not only simplifies the application of newton s method on the non linear configuration space but also leads to the strain objectivity of the proposed method the accuracy and the efficiency of the derived numerical algorithm are demonstrated by several examples c 2003 elsevier b v all rights reserved

minimum eigenvalue inequalities for z transformations on proper and symmetric cones abstract a linear transformation l defined on a finite dimensional real hilbert space is said to be a z transformation on a proper cone k if x k y k and x y 0 l x y 0 where k is the dual of k examples of such transformations include z matrices on r n lyapunov and stein transformations on the semidefinite cone for a z transformation l l min l is an eigenvalue of l with a corresponding eigenvector in k in this article when k is a product cone we relate the z property positive stable property minimum real eigenvalue of l with those of a subtransformation of l and its schur complement

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

on rigid components and joint constraints in nonlinear dynamics of flexible multibody systems employing 3d geometrically exact beam model abstract in this work we discuss the finite element implementation of the internal constraints in a three dimensional 3d geometrically exact beam model which can be formulated as holonomic constraint relationships model problems chosen for a more detailed consideration include a general joint constraint between beams and the beam connected to a rigid component appropriate modifications of the standard form of the geometrically exact beam element arrays are carried out in order to impose explicitly these kinds of constraints into time integration schemes for nonlinear dynamics consequently only the minimum number of unknowns is retained for the global set of nonlinear equations to be solved thus avoiding the use of the extra variables the lagrange multipliers and the pertinent difficulties in integrating the system of differential algebraic equations a number of numerical simulations considering dynamic analysis of multibody systems with rigid flexible components and joint constraints are presented in order to illustrate versatility of the proposed procedure

on the block norm p property a real n n matrix m is said to be a p matrix if all its principal minors are positive in a recent paper chua and yi 2 described this property in terms of norm there exists a 0 such that for all nonnegative diagonal matrices d and vectors x mx dx x in this paper we introduce a block version of this property for a linear transformation defined on a product of normed or inner product spaces in addition to relating this to real positive stability and positive principal minor properties we study the invariance of this property by principal subtransformations and schur complements we also specialize this property to z transformations and to euclidean jordan algebras

a simplified co rotational method for quadrilateral shell elements in geometrically nonlinear analysis summary this paper presents a simplified co rotational formulation for quadrilateral shell elements inheriting the merit of element independence from the traditional co rotational approach in literature with the objective of application to nonlinear analysis of civil engineering structures the authors further simply the formulation of the geometrical stiffness using the small strain assumption which is valid in the co rotational approach with the warping effects considered as eccentricities compared with the traditional element independent co rotational method the projector is neglected both in the tangent stiffness matrix and the internal force vector for simplicity in formulation meanwhile a quadrilateral flat shell element allowing for drilling rotations is adopted and incorporated into this simplified co rotational algorithm for geometrically nonlinear analysis involved with large displacements and large rotations several benchmark problems are presented to confirm the efficiency and accuracy of the proposed method for practical applications

analyses of large quasistatic deformations of inelastic bodies by a new hybrid stress finite element algorithm abstract a new hybrid stress finite element algorithm suitable for analyses of large quasistatic inelastic deformations is presented the algorithm is based upon a generalization of de veubeke s complementary energy principle the principal variables in the formulation are the nominal stress rate and spin and the resulting finite element equations are discrete versions of the equations of compatibility and angular momentum balance the algorithm produces true rates time derivatives as opposed to increments there results a complete separation of the boundary value problem for stress rate and velocity and the initial value problem for total stress and deformation hence their numerical treatments are essentially independent after a fairly comprehensive discussion of the numerical treatment of the boundary value problem we launch into a detailed examination of the numerical treatment of the initial value problem covering the topics of efficiency stability and objectivity the paper is closed with a set of examples finite homogeneous deformation problems which serve to bring out important aspects of the algorithm

nonlinear analysis and elastic plastic load carrying behaviour of thin walled spatial beam structures with warping constraints an analysis of the elastic plastic load carrying behaviour of thin walled spatial beam structures is presented it is based on a beam theory valid for large displacements and rotations which admits arbitrary cross sections curved axes initial imperfections a general material description and which fully accounts for the influence of warping constraints as well as the stress history dependence of the elastic plastic shear moduli an incremental updated lagrangian viewpoint is adopted in the derivation of the basic beam equations from a generalized variational principle and in the numerical solution procedure the displacement finite element approach is followed the associated tangential stiffness matrices are obtained by direct numerical integration of the governing incremental differential equations rather than through the use of shape functions in connection with a virtual work principle applications of the theory are given in which the influence of the loading configuration material parameters geometric nonlinearities and warping constraints on the load carrying behaviour and on the bifurcation and ultimate loads of thin walled beam structures is explored

disturbance attenuation and h optimization a design method based on the algebraic riccati equation this note presents a method for designing a state feedback control law to reduce the effect of disturbances on the output of a given linear system the problem under consideration involves attenuating the disturbances to a prespecified level the construction of the state feedback control law requires the solution of a certain algebraic riccati equation by applying the procedure with successively smaller values of the prespecifled disturbance level the result also provides an approach to the h infty optimization problem for the state feedback case

work conjugacy between rotation dependent moments and finite rotations in this paper we investigate the work conjugacy between rotation dependent moments and finite rotation measures the methodology adopted consists of writing all the relevant quantities in terms of the rotation vector using expressions that remain exact in the finite rotation range through this procedure we show how i to identify work conjugate rotation dependent moments and rotation measures ii to derive a necessary condition for moment conservativeness and iii to obtain the general form of an isotropic conservative rotation dependent moment several moment and finite rotation definitions that have been used in the past are investigated and in particular it is shown that the various existing definitions for the so called semi tangential moments are distinct in the finite rotation range and that not all of them are conservative the tangent operator symmetry is discussed in the context of finite element analysis of conservative systems with rotational degrees of freedom adopting either an additive or a multiplicative update