an element for studying the vibration of unrestrained curved timoshenko beams abstract a timoshenko beam finite element is developed using trigonometric basis functions the properties and performance of these new elements as applied to the vibration of beams are explored through a series of illustrative problems that treat both straight and curved geometries the effect of using a consistent mass matrix vs a diagonal mass matrix has been investigated as has the effect of the wave content of the basis functions for specific wave parameter values trigonometric basis functions are shown to allow the recovery of rigid body modes in the case of straight and circular arc curved beams they are also shown to exhibit good convergence and accuracy characteristics in both geometries natural frequency results for a complete series of free free curved beams are reported

unicity of minimal rank completions for tri diagonal partial block matrices abstract in this paper we establish necessary and sufficient conditions for a minimal rank completion of a tri diagonal partial block matrix to be unique the case of 4 4 block matrices with four consecutive diagonals is also considered along with its application to the problem of completing a matrix and its inverse

a variational principle for finite planar deformation of straight slender elastic beams abstract the objective of this paper is to present a finite element formulation for finite deformation analysis of a slender straight elastic in plane beam the formulation is based on a modified hu washizu variational principle in which exact nonlinear kinematic equations are taken into account only the rotation in the interior of an element needs to be approximated thus the ambiguities concerning the order of polynomial approximations for different field variables e g longitudinal and lateral displacements are avoided the euler lagrange equations of this principle are among others exact kinematic and equilibrium conditions for the beam the solution capabilities are illustrated with numerical examples two different finite elements are examined with 3rd order interpolation polynomials for the rotation and with 5th order polynomials excellent convergence and accuracy of both elements is demonstrated the results indicate that the accuracy of the present elements is not notably influenced by the length of the element and the order of numerical integration if it is greater than a minimal value only one load step was sufficient in calculations to achieve the results

dispersion of low energy waves for the generalized benjamin bona mahony equation on considere les equations suivantes u t u x f u x u xxx 0 1 et u t u x f u x u xxt 0 2 avec f r r est c on donne le comportement des solutions ondes a basse energie pour certains f

rank preserving diagonal completions of a matrix abstract suppose a is an n by n matrix over a field f we prove that it is possible to complete the diagonal entries of a so that the resulting rank of a is as small as possible when n 3r where r is the off diagonal rank of a and n r 3 1

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

c 1 comonotone hermite interpolation via parametric cubics abstract we present a local scheme for c 1 hermite interpolation the interpolant constructed is a parametric piecewise cubic and it is comonotone with the data the approximation order is also investigated under appropriate assumptions the method is fourth order accurate

a mixed stress model for linear elastodynamics of arbitrarily curved beams this work presents a mixed stress finite element for linear elastodynamics of arbitrarily curved beams based on a modified hellinger reissner functional a rational approach to choose the stress approximation is proposed in particular the self equilibrated stress is augmented by some stress modes obtained from the lower order displacement approximation using the equilibrium equations in such a way that the total number of stress modes is equal to the number of strain modes the rationale is to preserve all the interactions among the stresses proper of a curved structure without compromising the flexibility of the element an arbitrarily curved geometry is described using a parametric hermitian interpolation scheme tuned by minimizing the initial curvature of the arch the effectiveness of the present approach is numerically demonstrated copyright 2007 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

a hybrid perturbation galerkin technique with applications to slender body theory a two step hybrid perturbation galerkin technique for solving a variety of applied mathematics problems involving a small parameter is presented the first step consists of using a regular or singular perturbation method to determine the asymptotic expansion of the solution in terms of the small parameter then the approximate solution is assumed to have the form of a sum of perturbation coefficient functions multiplied by unknown amplitudes gauge functions in the second step the classical bubnov galerkin method is used to determine these amplitudes the resulting hybrid method has the potential of overcoming some of the drawbacks of the perturbation and bubnov galerkin methods applied separately while combining some of the good features of both the proposed method is applied to some singular perturbation problems in slender body theory the results obtained from the hybrid method are compared with approximate solutions obtained by other methods and the applicability of the hybrid method to broade

equivalence between mixed models and displacement models sing reduced integration relations between displacement models using reduced integration and hybrid mixed models based on the hellinger reissner and the herrmann principles have been investigated five different isoparametric elements are considered for the case of plane stress plane strain and plate bending elasticity problems it is proved that the following relations hold for both models provided the number of terms of a polynomial expressing the stress or strain component in the formulation of the mixed models is equal to the number of integration points of the reduced integration if the stiffness matrix of the mixed model is evaluated by the identical integration rule with the reduced integration the stiffness matrices of both models are identical thus both models are equivalent if the stiffness matrix of the mixed model is integrated exactly by the identical integration rule with the reduced integration the displacement models using uniform reduced integration and selective reduced integration are equivalent to the mixed models based on the hellinger reissner and the herrmann principles except for the 6 node element respectively

non linear theory for elastic beams and rods and its finite element representation abstract in terms of displacement components and angular coordinates expressions are given for large curvatures of elastic beams and rods the application of these expressions to bifurcation and stability problems is discussed on the basis of the principle of virtual work this approach is compared with the finite element representation given in a previous paper 1 an error in this previous paper is corrected further it is shown that the condition of zero extension of the axis in the case of the euler column requires a division into a rather large number of finite elements to ensure an accurate result for the stability coefficient the conditions of zero extension of the axis and zero curvature about one principal axis of the cross section in the case of the lateral buckling of the end loaded cantilever have similar consequences

the curved beam deep arch finite ring element revisited an attempt is made to understand the errors arising in curved finite elements which undergo both flexural and membrane deformation it is shown that with elements of finite size i e a practical level of discretization at which reasonably accurate results can be expected there can be errors of a special nature that arise because the membrane strain fields are not consistently interpolated with terms from the two independent field functions that characterize such a problem these lead to errors described here as of the second kind and a physical phenomenon called membrane locking it seems possible to determine optimal integration rules that will allow the extensional deformation of a curved beam deep arch finite ring element to be modelled by independently chosen low order polynomial functions and which will recover the inextensional case in the penalty limit of extreme thinness without spurious locking constraints what is emphasized is that the choice of shape functions or subsequent operations to determine the discretized functionals must consistently model the physical requirements the problem imposes on the field variables

exact solution of in plane vibrations of circular arches with account taken of axial extension transverse shear and rotatory inertia effects exact solution of free in plane vibrations of circular arches of uniform cross section is given by considering axial extension transverse shear and rotatory inertia effects in contrast with kirchhoff s beam theory the restrictions of perpendicular cross section and inextensible arc length are removed the principal axes of the cross section are assumed to coincide with the principal normal and binormal vectors of the centerline of the beam a solution procedure is applied to obtain the fundamental matrix for various end conditions natural frequencies and mode shapes are given in figures and tables

effects of approximations in analyses of beams of open thin walled cross section part ii 3 d non linear behaviour in a companion paper the effects of approximations in the flexural torsional stability analysis of beams was studied and it was shown that a second order rotation matrix was sufficiently accurate for a flexural torsional stability analysis however the second order rotation matrix is not necessarily accurate in formulating finite element model for a 3 d non linear analysis of thin walled beams of open cross section the approximations in the second order rotation matrix may introduce self straining due to superimposed rigid body motions which may lead to physically incorrect predictions of the 3 d non linear behaviour of beams in a 3 d non linear elastic plastic analysis numerical integration over the cross section is usually used to check the yield criterion and to calculate the stress increments the stress resultants the elastic plastic stress strain matrix and the tangent modulus matrix a scheme of the arrangement of sampling points over the cross section that is not consistent with the strain distributions may lead to incorrect predictions of the 3 d non linear elastic plastic behaviour of beams this paper investigates the effects of approximations on the 3 d non linear analysis of beams it is found that a finite element model for 3 d non linear analysis based on the second order rotation matrix leads to over stiff predictions of the flexural torsional buckling and postbuckling response and to an overestimate of the maximum load carrying capacities of beams in some cases to perform a correct 3 d non linear analysis of beams an accurate model of the rotations must be used a scheme of the arrangement of sampling points over the cross section that is consistent with both the longitudinal normal and shear strain distributions is needed to predict the correct 3 d non linear elastic plastic behaviour of beams copyright 2001 john wiley sons ltd

continuum based stiffened composite shell element for geometrically nonlinear analysis a continuum based laminated stiffened shell element is used to investigate the static geometrically nonlinear response of composite shells the element is developed from a three dimensional continuum element based on the incremental total lagrangian formulation the newton raphson method or modified riks method is used to trace the nonlinear equilibrium path a number of sample problems of unstiffened and stiffened shells are presented to show the accuracy of the present element and to investigate the nonlinear response of laminated composite plates and shells inite element analyses of the large displacement the ories are based on the principle of virtual work or the associated principle of stationary potential energy horrigmoe and bergan1 presented classical variational principles for non linear problems by considering incremental deformations of a continuum a survey of various principles in incremental form in different reference configurations such as the total lagran gian and updated lagrangian formulations is presented by wiinderlich 2 in the total lagrangian description all static and kinematic variables are referred to the initial configura tion finite element models based on such formulations have been used in the analysis of arch and shell instability prob lems 3 6 a special numerical technique must be adopted to trace the path of the load deflection curve near the limit point i e critical buckling load and in the postbuckling region because the stiffness matrix in the vicinity of the limit point is nearly singular and the descending branch of the load deflec tion curve in the postbuckling region is characterized by a negative definite stiffness matrix many methods have been proposed to solve limit point problems among these are the simple methods of suppressing equilibrium iterations 7 8 the introduction of artificial spring 3 the displacement control method 7 8 and the constant arc length method of riks 11 12 reviews of these most commonly used techniques are con tained in refs 13 and 14 among these methods the modified riks method appears to be the most effective in conjunction with the finite element method many investigators11 15 have used this method in its original or modified form to determine the pre and postbuckling behavior of various types of struc tures such as arches shells and domes in most of these works only isotropic material was considered very few works of the nonlinear buckling analysis of laminated com posite structures are reported in the literature 16 when solving the problems of shells with stiffeners by the finite element method a beam element whose displacement pattern is compatible with that of the shell is required in analyzing eccentrically stiffened cylindrical shell kohnke and schnobrich17 proposed a 16 deg of free dom dof isotropic beam finite element that has displacements compatible with

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 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 new efficient approach to the formulation of mixed finite element models for structural analysis abstract a new finite element formulation is developed based on the hellinger reissner principle with independent strain by dividing the assumed strain into the lower order part and the higher order part the new formulation can be made much more efficient than the standard mixed formulation in addition the present new approach provides a rational way of introducing stabilization matrix to suppress undesirable kinematic modes author

stability and convergence of mixed methods for elastic rods of arbitrary geometry abstract a timoshenko s small strain model for elastic rods with arbitrary geometry is analyzed using mixed finite element methods based on the hellinger reissner principle after presenting the mathematical model and commenting on some drawbacks of standard finite element approximations a stabilized mixed formulation is derived by adding to the galerkin formulation least squares residual of the equilibrium equations stability uniform convergence and error estimates are proved and results of numerical experiments are presented illustrating the behavior of the finite element approximations confirming the predicted rates of convergence and attesting the robustness of the stabilized mixed formulation

decay of solutions of some nonlinear wave equations abstract we study the large time behaviour of solutions to the initial value problem for the korteweg de vries equation and for the regularized long wave equation with a dissipative term appended using energy estimates a maximum principle and a transformation of cole hopf type sharp rates of temporal decay of certain norms of the solution are obtained

a hybrid perturbation galerkin technique that combines multiple expansions a two step hybrid perturbation galerkin method for the solution of a variety of differential equations type problems is found to give better results when multiple perturbation expansions are employed the method assumes that there is a parameter in the problem formulation and that a perturbation method can be used to construct one or more expansions in this parameter an approximate solution is constructed in the form of a sum of perturbation coefficient functions multiplied by computed amplitudes in step one regular and or singular perturbation methods are used to determine the perturbation coefficient functions the results of step one are in the form of one or more expansions each expressed as a sum of perturbation coefficient functions multiplied by a priori known gauge functions in step two the classical bubnov galerkin method uses the perturbation coefficient functions computed in step one to determine a set of amplitudes that replace and improve upon the gauge functions the hybrid method has

minimal rank completions for block matrices abstract this paper deals with the following problem of completion of block matrices let aij i j be given matrices find additional matrices aij such that the completion a aij ni j 1 has lowest possible rank the structure of the set of all minimal rank completions is studied special attention is paid to the toeplitz case together with its connection to the partial realization problem

mixed models and reduced selective integration displacement models for nonlinear shell analysis simple mixed models are developed for the geometrically nonlinear analysis of shells a total lagrangian description of the shell deformation is used and the analytical formulation is based on a form of the nonlinear shallow shell theory with the effects of transverse shear deformation and bending extensional coupling included the fundamental unknowns consist of eight stress resultants and five generalized displacements of the shell and the element characteristic arrays are obtained by using the hellinger reissner mixed variational principle the polynomial interpolation or shape functions used in approximating the stress resultants are in general of different degree than those used for approximating the generalized displacements the stress resultants are discontinuous at the element boundaries and are eliminated on the element level the equivalence and near equivalence between the mixed models developed herein and displacement models based on reduced selective integration of both transverse shear and extensional energy terms is discussed the use of reduction methods in conjunction with the mixed models is outlined and the advantages of mixed models over displacement models are delineated analytic expressions are derived for the rigid body and spurious or zero energy models for the various mixed models and their equivalent displacement models also the advantages of mixed models over equivalent displacement models are outlined numerical results are presented to demonstrate the high accuracy and effectiveness of the mixed models developed and to compare their performance with other mixed models reported in the literature

computational aspects of vector like parametrization of three dimensional finite rotations theoretical and computational aspects of vector like parametrization of three dimensional finite rotations which uses only three rotation parameters are examined in detail in this work the relationship of the proposed parametrization with the intrinsic representation of finite rotations via an orthogonal matrix is clearly identified careful considerations of the consistent linearization procedure pertinent to the proposed parametrization of finite rotations are presented for the chosen model problem of reissner s non linear beam theory pertaining details of numerical implementation are discussed for the simplest choice of the finite element interpolations for a 2 node three dimensional beam element a number of numerical simulations in three dimensional finite rotation analysis are presented in order to illustrate the proposed approach

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

in plane stability of arches classical buckling theory is mostly used to investigate the in plane stability of arches which assumes that the pre buckling behaviour is linear and that the effects of pre buckling deformations on buckling can be ignored however the behaviour of shallow arches becomes non linear and the deformations are substantial prior to buckling so that their effects on the buckling of shallow arches need to be considered classical buckling theory which does not consider these effects cannot correctly predict the in plane buckling load of shallow arches this paper investigates the in plane buckling of circular arches with an arbitrary cross section and subjected to a radial load uniformly distributed around the arch axis an energy method is used to establish both non linear equilibrium equations and buckling equilibrium equations for shallow arches analytical solutions for the in plane buckling loads of shallow arches subjected to this loading regime are obtained approximations to the symmetric buckling of shallow arches and formulae for the in plane anti symmetric bifurcation buckling load of non shallow arches are proposed and criteria that define shallow and non shallow arches are also stated comparisons with finite element results demonstrate that the solutions and indeed approximations are accurate and that classical buckling theory can correctly predict the in plane anti symmetric bifurcation buckling load of non shallow arches but overestimates the in plane anti symmetric bifurcation buckling load of shallow arches significantly

singular nonsingular and bounded rank completions of aci matrices an affine column independent matrix is a matrix whose entries are polynomials of degree at most 1 in a number of indeterminates where no indeterminate appears with a nonzero coefficient in two different columns a completion is a matrix obtained by giving values to each of the indeterminates affine column independent matrices are more general than partial matrices where each entry is either a constant or a distinct indeterminate we determine when the rank of all completions of an affine column independent matrix is bounded by a given number generalizing known results for partial matrices we also characterize the square partial matrices over a field all of whose completions are nonsingular the maximum number of free entries in such matrices of a given order is determined as well as the partial matrices with this maximum number of free entries

error on the constitutive relation in the nonlinear formulation of elastic beams the plane case the error measure we propose in this study is based on the mechanics concept of error on the constitutive relation the error on the constitutive relation has already been used for the updating of elastic structures in dynamics under the small perturbation assumption here we extend this concept to the context of geometrically nonlinear elastic beams large rotations and finite strains it is shown that the proposed strategy which uses measured experimental data enables us to locate damaged substructures or if the quality of the model is to be checked the method provides the location of the zones of modelling errors the formulation and its derivation are developed and a complete algorithmic treatment is presented the implications of the formulation of the errors location discussed in this paper are illustrated through a set of numerical examples

a locking free finite element formulation and reduced models for geometrically exact kirchhoff rods in this work we suggest a locking free geometrically exact finite element formulation incorporating the modes of axial tension torsion and bending of thin kirchhoff beams with arbitrary initial curvatures and cross section shapes the proposed formulation has been designed in order to represent general load cases and three dimensional problem settings in the geometrically nonlinear regime of large deformations from this comprehensive theory we not only derive a general beam model but also several reduced formulations which deliver accurate solutions for special problem classes concerning the beam geometry and the external loads the advantages of these reduced models arise for example in terms of simplified finite element formulations less degrees of freedom per element and consequently a higher computational efficiency of the corresponding numerical models a second core topic of this publication is the treatment of membrane locking which is a locking phenomenon predominantly occurring in highly slender curved structures thus exactly in the prime field of application for kirchhoff theories in order to address the membrane locking effect we will propose a new interpolation strategy for the axial strain field and compare this method with common approaches such as assumed natural strains ans or reduced integration the effectiveness of this method as well as the consistency and accuracy of the general finite element formulation and the reduced beam models will be illustrated with selected numerical examples

field consistent strain interpolations for the quadratic shear flexible beam element a shear flexible i e timoshenko quadratic isoparametric beam element with two degrees of freedom per node is critically examined from the point of view of the consistency of the constrained strain fields that arise in the thin beam limits the errors in terms of convergence of displacement fields and violent oscillations of stress fields that emerge when exactly integrated elements are used are predicted a priori and confirmed with numerical experiments the rationale behind the use of optimal stress sampling at gaussian points is also derived directly from these arguments

wave motion analysis in arch structures via wavelet finite element method abstract the application of b spline wavelet on interval bswi finite element method for wave motion analysis in arch structures is presented in this paper instead of traditional polynomial interpolation scaling functions at certain scales have been adopted to form the shape functions and construct wavelet based elements different from other wavelet numerical methods adding wavelets directly the element displacement field represented by the coefficients of wavelets expansions is transformed from wavelet space to physical space via the corresponding transformation matrix the energy functional of the arch is obtained by the generalized shell theory and the finite element model for wave motion analysis is constructed according to hamilton s principle and the central difference method in time domain taking the practical application into account damaged arch waveguides are also investigated proper analysis of the responses from structure damages allows one to indicate the location very precisely this paper mainly focuses on the crack in structures based on castigliano s theorem and the pairs equation the local flexibility of crack is formulated for bswi element numerical experiments are performed to study the effect of wave propagations in arch waveguides that is frequency dispersion and mode spilt in the arch the responses of the arch with cracks are simulated under the broad band narrow band and chirp excitations in order to estimate the spatial time and frequency concentrations of responses the reciprocal length time frequency transform and correlation coefficient are introduced in this investigation

on completions of symmetric and antisymmetric block diagonal partial matrices abstract a partial matrix is a matrix where only some of the entries are given we determine the maximum rank of the symmetric completions of a symmetric partial matrix where only the diagonal blocks are given and the minimum rank and the maximum rank of the antisymmetric completions of an antisymmetric partial matrix where only the diagonal blocks are given

smooth postbuckling stresses by a modified finite element method exact postbuckling stresses usually vary fairly smoothly unfortunately finite element postbuckling stresses tend to be much less well behaved the result is that second order postbuckling constants determined by the finite element method may be highly inaccurate the reason is that in finite element solutions transverse displacements associated with the buckling fields furnish too rapidly varying postbuckling strain contributions while the postbuckling axial or membrane displacements contribute strain components that are sufficiently smooth thus creating an internal postbuckling strain and stress mismatch the present study suggests a modified finite element method that handles the problem which is a special example of membrane locking by introducing the postbuckling strains as independent variables in general the method provides rather complicated finite element expressions however by a suitable choice of interpolating functions the resulting finite element equations themselves may be found to be the usual ones and yet provide smooth postbuckling stresses and therefore good values of the postbuckling constants

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

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

elastic postbuckling with nonlinear constraints abstract koiter s asymptotic theory of initial postbuckling and imperfection sensitivity of elastic structures is expanded such that almost any auxiliary condition can be handled directly the theory which employs lagrange multiplier techniques is valid for strain measures that are quadratic in the displacements i e it is exact for lagrangian strain measures and auxiliary conditions and loadings that are quartic in their arguments although the general formulas are rather lengthy and complicated in the general case they simplify considerably in almost all cases of practical interest frequently the formulas are only slightly more complex than in the equivalent situation without auxiliary conditions the method is applied to the example of an elastic inextensible circular ring but is suitable for all elastic structures which comply with the above mentioned requirements regarding strain measure and auxiliary conditions

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

geodesic finite elements for cosserat rods we introduce geodesic finite elements as a new way to discretize the non linear configuration space of a geometrically exact cosserat rod these geodesic finite elements naturally generalize standard one dimensional finite elements to spaces of functions with values in a riemannian manifold for the special orthogonal group our approach reproduces the interpolation formulas of crisfield and jelenic geodesic finite elements are conforming and lead to objective and path independent problem formulations we introduce geodesic finite elements for general riemannian manifolds discuss the relationship between geodesic finite elements and coefficient vectors and estimate the interpolation error then we use them to find static equilibria of hyperelastic cosserat rods using the riemannian trust region algorithm of absil et al we show numerically that the discretization error depends optimally on the mesh size copyright 2009 john wiley sons ltd

corotational mixed finite element formulation for thin walled beams with generic cross section the corotational technique is adopted here for the analysis of three dimensional beams the technique exploits the technology that applies to a two noded element a coordinate system which continuously translates and rotates with the element in this way the rigid body motion is separated out from the deformational motion in this paper a mixed formulation are adopted for the derivation of the local element tangent stiffness matrix and nodal forces the mixed finite element formulation is based on an incremental form of the two field hellinger reissner variational principle to permit elasto plastic material behavior the local beam kinematics is based on a low order nonlinear strain expression using bernoulli assumption the present formulation captures both the saint venant and warping torsional effects of thin walled open cross sections shape functions that satisfy the nonlinear local equilibrium equations are selected for the interpolation of the stress resultants in particular for the torsional forces and the twist rotation degree of freedom a family of hyperbolic interpolation functions is adopted in lieu of conventional polynomials governing equations are expressed in a weak form and the constitutive equations are enforced at each integration cross section along the element length a consistent state determination algorithm is proposed this local element together with the corotational framework can be used to analyze the nonlinear buckling and postbuckling of thin walled beams with generic cross section the present corotational mixed element solution is compared against the results obtained from a corotational displacement based model having the same beam kinematics and corotational framework the superiority of the mixed formulation is clearly demonstrated

a legendre spectral element model for sloshing and acoustic analysis in nearly incompressible fluids a new spectral finite element formulation is presented for modeling the sloshing and the acoustic waves in nearly incompressible fluids the formulation makes use of the legendre polynomials in deriving the finite element interpolation shape functions in the lagrangian frame of reference the formulated element uses gauss lobatto legendre quadrature scheme for integrating the volumetric stiffness and the mass matrices while the conventional gauss legendre quadrature scheme is used on the rotational stiffness matrix to completely eliminate the zero energy modes which are normally associated with the lagrangian fe formulation the numerical performance of the spectral element formulated here is examined by doing the inf sup test on a standard rectangular rigid tank partially filled with liquid the eigenvalues obtained from the formulated spectral element are compared with the conventional equally spaced node locations of the h type lagrangian finite element and the predicted results show that these spectral elements are more accurate and give superior convergence the efficiency and robustness of the formulated elements are demonstrated by solving few standard problems involving free vibration and dynamic response analysis with undistorted and distorted spectral elements and the obtained results are compared with available results in the published literature

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

non linear buckling and postbuckling analysis of arches with unequal rotational end restraints under a central concentrated load abstract this paper presents an analytical study of the non linear elastic in plane buckling and postbuckling behaviour of pin ended shallow circular arches having unequal elastic rotational end restraints under a central concentrated radial load the principle of stationary potential energy is used to derive the differential equations of equilibrium based on which the analytical solution for the non linear equilibrium path is derived it is found that the non linear behaviour of an arch having unequal rotational end restraints is much more complicated than that of an arch with equal rotational end restraints the arch may have a non linear equilibrium path that consists of one or two unstable equilibrium paths and two or four limit points and it may even have a non linear looped equilibrium path in some cases the number of limit points on the non linear equilibrium path of an arch depends on its slenderness ratio and included angle and on the stiffnesses of the unequal rotational end restraints the switches in terms of an arch geometry parameter which is introduced in the paper are derived for distinguishing between arches with two limit points and those with four limit points as well as for distinguishing between arches and beams curved in elevation the principle of conservation of energy at neutral equilibrium is used to derive the differential equations of buckling equilibrium which are then used to investigate the buckling behaviour it is found that an arch with unequal rotational end restraints cannot buckle in a bifurcation mode comparisons with finite element results show that the analytical solutions can accurately predict the non linear buckling and postbuckling behaviour

partial matrices whose completions have ranks bounded below abstract a partial matrix over a field f is a matrix whose entries are either elements of f or independent indeterminates a completion of such a partial matrix is obtained by specifying values from f for the indeterminates we determine the maximum possible number of indeterminates in a partial m n matrix whose completions all have rank at least equal to a particular k and we fully describe those examples in which this maximum is attained our main theoretical tool which is developed in section 2 is a duality relationship between affine spaces of matrices in which ranks are bounded below and affine spaces of matrices in which the left or right nullspaces of elements possess a certain covering property

korteweg de vries burgers system in rn abstract we study cauchy problem in r n for the korteweg de vries burgers system parabolic regularization technique is used to prove its global in time solvability the regularization effect of the laplacian term is observed for the viscous solutions constructed in the paper asymptotic behavior of weak solutions is discussed next in the language of the global attractors

the inertia of a hermitian matrix having prescribed diagonal blocks abstract for i 1 m let h i be an n i n i hermitian matrix with inertia in h i i i i we characterize in terms of the i i i the range of in h where h varies over all hermitian matrices which have a block decomposition h x ij i j 1 m in which x ij is n i n j and x ii h i

a three noded shear flexible curved beam element based on coupled displacement field interpolations an efficient shear flexible three noded curved beam element is proposed herein the shear flexibility is based on timoshenko beam theory and the element has three degrees of freedom viz tangential displacement u radial displacement w and the section rotation a quartic polynomial interpolation for flexural rotation is assumed a priori making use of the physical composition of in terms of and u a novel way of deriving the polynomial interpolations for u and w is presented by solving force moment and moment shear equilibrium equations simultaneously the field interpolation for is then constructed from that of and u the procedure leads to high order polynomial field interpolations which share some of the generalized degrees of freedom by means of coefficients involving material and geometric properties of the element when applied to a straight euler bernoulli beam all the coupled coefficients vanish and the formulation reduces to classical quintic in w and quadratic in u element with u w and w x as degrees of freedom the element is totally devoid of membrane and shear locking phenomena the formulation presents an efficient utilization of the nine generalized degrees of freedom available for the polynomial interpolation of field variables for a three noded curved beam element numerical examples on static and free vibration analyses demonstrate the efficacy and locking free property of the element copyright 2001 john wiley sons ltd

penalty finite element models for nonlinear dynamic analysis a simple penalty finite element formulation is presented for the large rotation dynamic analysis of curved beams the analytical formulation is based on a form of reissner s large deformation theory with the effects of transverse shear deformation and the extensibility of the centerline constrained through the use of the penalty method reduced integration is used in evaluating the elemental stiffness arrays and the temporal integration is performed by using newmark s method numerical results are presented to demonstrate the effectiveness of the finite elements developed

decay estimates for the solutions of some nonlinear evolution equations abstract we study decay estimates for the solutions and their derivatives to the initial value problems for some generalized nonlinear evolution equations which have lower order diffusion terms

the null space and non conventional basis functions in the mixed finite element method the two matrix equations obtained from the mixed finite element method are uncoupled through the use of the qr decomposition the procedure obtains the basis for the null space which is composed of solutions to the homogeneous flux equilibrium equation the result is that even for statically indeterminate problems stresses can be obtained without solving for the displacement field furthermore the stress field can be decomposed directly into contributions from applied forces and from prescribed displacements the appearance of kinetic modes can easily be monitored the possibility of utilizing non conventional basis functions for use in conjunction with the qr decomposition is explored briefly simple examples are given to illustrate the theoretical results

mixed models and reduction techniques for large rotation nonlinear problems abstract mixed models and reduction techniques are presented for the large rotation nonlinear analysis of curved beams a total lagrangian description of the beam deformation is used and the analytical formulation is based on a form of reissner s large deformation theory with the effects of transverse shear deformation and extensibility of the centerline included only planar deformations are considered and the fundamental unknowns consist of the three internal forces and the three generalized displacements of the beam the element characteristic arrays are obtained by using the hellinger reissner mixed variational principle the polynomial interpolation or shape functions used in approximating the internal forces are in general of different degree than those used for approximating the generalized displacements and the forces are discontinuous at interelement boundaries the use of reduction methods in conjunction with the mixed models is outlined and the advantages of mixed models over displacement models are delineated numerical results are presented to demonstrate the high accuracy and effectiveness of both the mixed models and the reduction techniques

the inertia of a hermitian matrix having prescribed complementary principal submatrices abstract for i 1 2 let hi be a given ni ni hermitian matrix we characterize the set of inertias in h 1 x x h 2 x is n 1 n 2 in terms of in h1 and in h2

geometrically exact thin walled beam including warping formulated on the special euclidean group se 3 abstract based on a formulation on the special euclidean group s e 3 a geometrically exact thin walled beam with an arbitrary open cross section is proposed to deal with the finite deformation and rotation issues the beam strains are based on a kinematic assumption where warping deformation and wagner effects are included such that the nonlinear behavior of a thin walled beam is predicted accurately particular under large torsion to reduce the nonlinearity of rigid motion static and dynamic equations are derived in the s e 3 framework based on the local frame approach as the value of the iteration matrix including the jacobian matrix of inertial and internal forces is invariable under arbitrary rigid motion the number of updates required during the computation process decreases sharply which drastically improves the computational efficiency furthermore the isogeometric analysis iga based on the non uniform rational b splines nurbs basis functions which promotes the integration of computer aided design cad and computer aided engineering cae is adopted to interpolate the displacement rotation and warping fields separately the interpolated strain measures satisfy the objectivity by removing the rigid motion of the reference point to obtain the symmetric jacobian matrix of internal forces the linearization operation is conducted based on the previously converged configuration a lie group s e 3 extension of the generalized time integration method is utilized to solve the equations of motion for thin walled beams finally the proposed formulation is successfully tested and validated in several static and dynamic numerical examples

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

formulation and analysis of variational methods for time integration of linear elastodynamics abstract a general framework for variational formulations of linear elastodynamics is introduced variational formulations which originate from the principle of minimum potential energy the hellinger reissner principle and the hu washizu principle are considered they are characterized by the weak enforcement of the velocity displacement relation and of the initial displacement and velocity special cases are obtained by a priori enforcement of these conditions the time integration algorithms obtained discretizing in the space and time domains are studied with reference to accuracy stability and high frequency behaviour it is shown that some of these algorithms are very competitive with respect to classical finite difference methods the only drawback of variational methods seems to be the large size of the set of simultaneous equations which they generate effective solution procedures aiming at minimizing both the storage and the computational effort will be discussed in a separate paper

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

temporal asymptotic behavior of solutions of the benjamin ono burgers equation abstract we study certain similarity solutions of the benjamin ono burgers bob equation and their role in the asymptotic behavior of the general solution for small initial data in l 1 r we prove that a solution of the bob equation exists in bc r l 1 r and depends continuously on its initial data results about existence uniqueness regularity and spatial asymptotics of solutions of a similarity reduction of the bob equation are proved furthermore the solutions of the bob equation are proved to converge as t to appropriate similarity solutions faster than the typically sharp rate of decay of the solutions

rotation free isogeometric analysis of an arbitrarily curved plane bernoulli euler beam abstract the present study elucidates linear static analysis for plane beam structures using the isogeometric approach a novel methodology for rotation free analysis of an arbitrarily curved bernoulli euler beam in the convective frame of reference is derived in detail the full degeneration of a 3d continuum beam to a 1d line has been presented and a fully applicable isogeometric finite element has been obtained the driving force behind developing the present research has been the derivation of the nurbs based isogeometric analysis which will enable an elegant formulation of the plane bernoulli euler beams being a function only of the global rectangular cartesian coordinates the verification and accuracy of the research are obtained via a thorough comparison between theory finite element analyses and relevant examples from literature an excellent agreement of results is achieved and usefulness for academic and practical purposes alike are proved the effects of the hpk refinements are illuminated and it is observed that the convergences for the most variables and refinement techniques are not monotonic a special attention is paid to the influence of the product of maximum curvature and thickness of beam on the accuracy of the solution the limits of applicability of the present approach are defined for a few specific types of analyses the derived formulation is geometrically exact and appropriate for the analysis of strongly curved bernoulli euler beams

mixed models and reduced selective integration displacement models for nonlinear analysis of curved beams simple mixed models are developed for the geometrically nonlinear analysis of deep arches a total lagrangian description of the arch deformation is used and the analytical formulation is based on a form of the nonlinear deep arch theory with the effects of transverse shear deformation included the fundamental unknowns consist of the six internal forces and generalized displacements of the arch and the element characteristic arrays are obtained by using hellinger reissner mixed variational principle the polynomial interpolation functions used in approximating the forces are one degree lower than those used for approximating the displacements and the forces are discontinuous at the interelement boundaries the equivalence between the mixed models developed herein and displacement models based on reduced integration of both the transverse shear and extensional energy terms is discussed the advantages of mixed models over equivalent displacement models are outlined numerical results are presented to demonstrate the high accuracy and effectiveness of the mixed models developed and compare their performance with other mixed models reported in the literature

coexisting equilibria and stability of a shallow arch unilateral displacement control experiments and theory the equilibria and stability of a shallow prestressed arch beam column are investigated theoretically and experimentally the deflection of the arch is unilaterally constrained by a displacement control device both snap through and remote coexisting equilibria are observed force deflection curves for primary and secondary equilibrium branches are measured for varying constraint locations the effect of the constraint location on the critical condition at which stability is lost resulting in a jump to a remote equilibrium is investigated good agreement is attained between experimental data and theoretical results based on minimization of the constrained strain energy and an inextensibility assumption

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

maximum rank matrix completion abstract the maximum rank completion problem is the problem of given a partial matrix that is a matrix where we are only given some of the entries filling in the unknown entries in such a way as to maximize the rank applications include bipartite matching and matroid intersection for linearly represented matroids we describe an algorithm that finds a maximum rank completion by perturbing an arbitrary completion in a greedy way

locking and robustness in the finite element method for circular arch problem in this paper we discuss locking and robustness of the finite element method for a model circular arch problem it is shown that in the primal variable i e the standard displacement formulation the p version is free from locking and uniformly robust with order p k and hence exhibits optimal rate of convergence on the other hand the h version shows locking of order h 2 and is uniformly robust with order h p 2 for p 2 which explains the fact that the quadratic element for some circular arch problems suffers from locking for thin arches in computational experience if mixed method is used both the h version and the p version are free from locking furthermore the mixed method even converges uniformly with an optimal rate for the stress

completing a matrix when certain entries of its inverse are specified abstract suppose a a 11 a 12 a 21 a 22 is an n n matrix where a 11 has order p q if a is nonsingular let a 1 have the transposed partitioning b 11 b 12 b 21 b 22 first we show that the nullities of a 11 and b 22 are equal the remainder of our paper deals with the following completion problem suppose m 1 m 2 n 1 n 2 are nonnegative integers such that m 1 m 2 n 1 n 2 n 0

the role of non linear theories in transient dynamic analysis of flexible structures abstract transient dynamic analysis of flexible structures undergoing large motions is considered for rotating structures it is explicitly shown that appropriate account of the influence of centrifugal force on the bending stiffness requires the use of a geometrically non linear at least second order beam theory use of a first order linearized linear beam theory results in a spurious loss of bending stiffness for a rotating plane beam a set of linear partial differential equations of motion that includes all inertia effects coriolis centrifugal acceleration of revolution and coupling between extensional and flexural deformations is derived from the fully non linear beam theory by consistent linearization the analysis is subsequently extended to the more general case of a plate accomodating shear deformation and undergoing a general three dimensional rotating motion the discretization process of the resulting linear equations of motion for the beam and the plate is also discussed

invariant hermitian finite elements for thin kirchhoff rods i the linear plane case abstract the objective of this work is to develop finite elements for the simulation of thin kirchhoff rods that is rods where the influence of transverse shear strain can be neglected leading to governing equations that require c 1 continuous finite element interpolations we start in this paper with the formulation of new finite elements for the infinitesimal plane deformations of otherwise generally curved rods addressing in particular the lack of invariance observed in related traditional treatments of plane arches by non invariance we mean that the classical rigid body modes of translations and infinitesimal rotations of the underlying structural theory lead to non zero strains in the element an effect sometimes referred to as self straining this phenomenon occurs in curved configurations of the elements and in the infinitesimal setting of interest in this paper it results in the elements stiffness matrix not to exhibit the correct zero energy modes associated with these physical rigid body motions as a consequence the basic equilibrium relations of balance of forces and moments are not satisfied for the associated nodal forces and moments in non invariant finite elements in this context we propose a new framework for the formulation of c 1 continuous finite elements based on hermitian shape functions that do lead to the satisfaction of this fundamental invariance property accounting fully for the rod s curvature and associated kirchhoff s kinematics in contrast to common straight framework element approximations based on the simplified euler bernoulli beam kinematics the formulation relies crucially on a consistent treatment of the kinematics and the assumed interpolation of the rods geometry an additional assumed strain treatment is considered to avoid limitations of membrane locking complete rigorous analyses and several numerical simulations are presented to prove and illustrate the properties and performance of the newly proposed finite element methods