a higher order beam finite element for bending and vibration problems abstract the finite element equations for a variationally consistent higher order beam theory are presented for the static and dynamic behavior of rectangular beams the higher order theory correctly accounts for the stress free conditions on the upper and lower surfaces of the beam while retaining the parabolic shear strain distribution the need for a shear correction coefficient is therefore eliminated full integration of the shear stiffness terms is shown to result in the recovery of the kirchhoff constraint for thin beams without introducing spurious locking constraints the accuracy of this formulation is demonstrated by using several numerical examples for the cases of small and large displacements for a hinged hinged beam the linear thickness shear mode frequency can be matched with the timoshenko frequency to yield a shear coefficient of 0 824 matching the bending frequencies between the two theories indicates a shear coefficient for the timoshenko theory that changes with mode number and slenderness ratio the influence of in plane inertia and slenderness ratio on the non linear frequency is examined for beams with a number of different support conditions

dynamics of timoshenko beam on linear and nonlinear foundation phase relations significance of the second spectrum stability abstract some controversial and unclear issues in the timoshenko beam tb theory are analyzed with emphasis on features of two branches of frequencies taking into account the elastic foundation creates additional opportunities for comparative analysis of both the branches relationship between the second branch modes and so called thickness shear mode is discussed phase relations between the deformation and force factors in beam for both the branches are established which generalize the previously established relations it is shown that the elastic foundation can change these relations for the first branch based on this analysis the significance of the second spectrum is assessed and peculiarities of dispersion relations for frequency phase and group velocities are explained free transverse waves in infinitely long beam on nonlinear cubic foundation are studied using the multiple scales method the obtained nonlinear schrodinger equation enables us to check stability of the harmonic waves in infinite beam and to determine diapasons of their modulational instability

on shear deformable beam theories the frequency and normal mode equations of the homogeneous orthotropic bickford beam abstract this paper provides a step forwards the construction and documentation of the frequency equations and the characteristic functions of a general three degrees of freedom theory that describes the plane motion of shear deformable elastic beams the governing equations of this shear deformable beam theory g3dofbt involve a general shape function of the transverse beam co ordinate parameter the a posteriori choice of which specifies the distribution of the transverse shear strain or stress along the beam thickness different choices of this shape function produce as particular cases the corresponding governing equations of different beam theories these include the differential equations of the euler bernoulli beam theory as well as the corresponding equations of the shear deformable theories due to timoshenko and bickford other examples can also be found by considering the shear deformable beam theories produced as one dimensional versions of relevant refined plate theories since corresponding developments of the timoshenko beam theory are already available in the literature the bickford theory is considered as the pilot beam theory in this study the frequency equations the characteristic functions and the orthogonality conditions of this theory which assumes a through thickness parabolic distribution of the transverse shear strain or stress are constructed analytically for all the classical sets of boundary conditions applied at the beam ends some preliminary numerical results are also presented and discussed for beams having both their ends simply supported or clamped

characteristic varieties and vanishing cycles on presente une etude systematique des varietes caracteristiques des systemes holonomes a singularites regulieres ce sont des sous varietes lagrangiennes coniques dans le fibre cotangent t x

experimental study of the timoshenko beam theory predictions abstract the theory of flexural vibrations proposed by timoshenko almost 90 years ago has been the subject of several recent papers in the timoshenko beam theory a critical frequency f c is expected and for frequencies f larger than f c some authors argue that a second spectrum exists this is still the matter of much debate in order to settle this question we have performed experimental measurements of flexural normal mode frequencies and amplitudes which are much more precise than what has been obtained up to now for some purposes cylindrical rods are more convenient whereas in other cases rectangular beams are easier to deal with so we analyze rods and beams of different shapes and sizes we show that below f c all observed resonant peaks can be associated with one of the compressional torsional or timoshenko resonances and that no resonant peaks are left undetected the validity of timoshenko beam theory below f c and that only one spectrum appears in this regime is therefore proved the comparison between theory and experiment for rectangular beams also shows that our experimental setup is appropriate so we can use it confidently for f f c the measurements we report here in this regime show the existence of a second spectrum at least for free free boundary conditions

selective and reduced numerical integrations for nurbs based isogeometric analysis we propose a new approach to construct selective and reduced integration rules for isogeometric analysis based on nurbs elements the notion of an approximation space that approximates the target space is introduced we explore the use of various approximation spaces associated with optimal patch wise numerical quadratures that exactly integrate the polynomials in approximation spaces with the minimum number of quadrature points patch rules exploit the higher continuity of spline basis functions the tendency of smooth spline functions to exhibit numerical locking in nearly incompressible problems when using a full gauss legendre quadrature is alleviated with selective or reduced integration stability and accuracy of the schemes are examined analyzing the discrete spectrum in a generalized eigenvalue problem we propose a local algorithm which is robust and computationally efficient to compute element by element the quadrature points and weights in patch rules the performance of the methods is assessed on several numerical examples in two dimensional elasticity and reissner mindlin shell structures

convergence properties and derivative extraction of the superconvergent timoshenko beam finite element abstract the superconvergent timoshenko beam element of reddy comput methods appl mech engrg 149 1997 113 132 is derived here using an assumed strain approach a method of reference field formulation is proposed from which pointwise higher order derivatives moments and shears can be derived for this element and theoretical proofs of the existence of such points are presented it is shown that the order of convergence may be at least two orders higher than the fe derivative field in the cases where uniform discretization is adopted finally a smoothed derivative field is derived by using a least square polynomial fit over these points several simple numerical examples are presented to demonstrate the ability of the method to extract exact derivative fields from the fe results

extrapolated fields in the formulation of the assumed strain elements part ii three dimensional problems abstract a new assumed strain quadrilateral element with accurate in plane bending behavior is presented the basic idea consists in introduction of a parallelogram domain associated with the given quadrilateral element development of the related assumed strain field for the parallelogram and subsequent extrapolation of this field to the domain of the original quadrilateral the assumed strain field for the parallelogram is constructed by identification of various modes of its deformation and then by proper modification of the strain field in some of these modes in particular the strain operator with respect to the so called in plane bending modes is modified so as to simulate the strain field resulting from the assumptions usually made in structural mechanics the modification of the strain fields leads to the assumed strain operator the proposed new assumed strain quadrilateral element exhibits remarkably accurate performance for both membrane and bending dominated problems even when severely distorted and high aspect ratio meshes are considered and when materials are nearly incompressible

improvements in 3 node triangular shell elements low order triangular finite shell elements are computationally economical and easy to implement but often exhibit very slow convergence two new membrane formulations for triangular shell elements are examined which rectify these drawbacks the first element is based on the marguerre shallow shell theory and a strain projection method that eliminates spurious membrane strain energy resulting expressions are provided in an explicit form for easy implementation of the element the second element is based on a linear membrane field governed by normal rotations and reduced quadrature the difficulties with shell normal rotations are analysed and a method for omitting these rotations while preserving rigid body motion is presented and tested finally a set of test problems are examined which show the importance of mesh patterns and degrees of freedom per node on triangular element performance

variational derivation of governing differential equations for truncated version of bresse timoshenko beams abstract in this paper we provide a variational derivation of modified bresse timoshenko equations this process leads to the additional term to so called truncated set of bresse timoshenko equations this new term is associated with the modified slope inertia the two sets of truncated versions of the bresse timoshenko equations are contrasted with each other as well as with the original bresse timoshenko equations on the example of the a beam that is simply supported at its both ends and b the cantilever beam with end mass it is concluded that application of either truncated sets of equations is advantageous over the original bresse timoshenko equations as far as two truncated sets of equations are concerned the variationally derived set appears to be preferable

weak grain like structures a grain like structure is a cascade connection of a primitive lfsr into an nfsr it is well known that such structures generate sequences with periods multiples of the period of primitive lfsr sequences in this paper we study weak grain like structures i e grain like structures generating at least one sequence with minimum period assume the orders of lfsr and nfsr are n and m respectively we prove that weak grain like structures always exist when m n for m n we give three classes of weak grain like structures then we extend the method in the second class to prove that weak grain like structures exist when m ge n lg n 4 moreover our experimental data shows that the ratio of weak grain like structures approximates a value which is a little more than 63 for small m n and weak grain like structures exist with m 3 or 4 when n le 18

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

improved numerical integration for locking treatment in isogeometric structural elements part i beams a general mathematical framework is proposed in this paper to define new quadrature rules in the context of bb spline nurbs based isogeometric analysis high order continuity across the elements within a patch turned out to have higher accuracy than c0c0 finite elements as well as a better time efficiency unfortunately a maximum regularity accentuates the shear and membrane locking in thick structural elements the improved selective reduced integration schemes are given for uni dimensional beam problems with basis functions of order two and three and can be easily extended to higher orders the resulting bb spline nurbs finite elements are free from membrane and transverse shear locking moreover no zero energy modes are generated the performance of the approach is evaluated on the classical test of a cantilever beam subjected to a distributed moment and compared to lagrange under integrated finite elements

the second spectrum of timoshenko beam theory further assessment a review of contributions and views on the second spectrum of timoshenko beam theory tbt over the past two decades together with some new results are presented it is shown that the timoshenko frequency equation factorises not solely for hinged hinged end conditions as is often claimed but also for guided guided and guided hinged these new cases may be regarded as portions of a multi span hinged hinged beam a higher derivative lagrangian that leads directly to the well known fourth order timoshenko beam equation is reviewed a simple relationship between the so called ostrogradski energy and the mechanical energy is derived for hinged hinged end conditions it is shown that the ostrogradski energy is positive for the first spectrum but negative for the second within some branches of physics this would be sufficient to conclude that the second spectrum is unphysical a numerical example presented by levinson and cooke is re examined using both tbt and exact plane stress elastodynamic theory agreement is excellent for the first spectrum however the second spectrum predictions are not in consistent agreement with any single mode of vibration for long wavelength it is very close to the second asymmetric mode but as wavelength shortens it becomes closer to the second symmetric then the third asymmetric modes the conclusion remains unchanged the second spectrum predictions of tbt should be disregarded

reduced integration and the shear flexible beam element the timoshenko beam element is studied in an investigation of shear locking in the development of c 0 continuous elements using shear flexible or penalty function type formulations a penalty type argument is used to degenerate thick elements to thin elements and the various approximations of the shear related energy terms act as different types of constraints depending on the formulation two types of constraints may emerge which are classified as true or spurious formulations that ensure only true constraints in the extreme penalty limit cases display superior performance in the thick element situation the argument is extended to a shallow curved timoshenko beam and another mechanism inplane locking arises if all energy terms corresponding to the membrane energy are exactly integrated

an efficient curved beam finite element the plane two node curved beam finite element with six degrees of freedom is considered knowing the set of 18 exact shape functions their approximation is derived using the expansion of the trigonometric functions in the power series unlike the ones commonly used in the fem analysis the functions suggested by the authors have the coefficients dependent on the geometrical and physical properties of the element from the strain energy formula the stiffness matrix of the element is determined it is very simple and can be split into components responsible for bending shear and axial forces influences on the displacements the proposed element is totally free of the shear and membrane locking effects it can be referred to the shear flexible parameter d and compressible parameter e systems neglecting d or e yields the finite elements in all necessary combinations i e curved euler bernoulli beam or curved timoshenko beam with or without the membrane effect applying the elaborated element in the calculations a very good convergence to the analytical results can be obtained even with a very coarse mesh without the commonly adopted corrections as reduced or selective integration or introduction of the stabilization matrices additional constraints etc for the small depth length ratio 1997 john wiley sons ltd

timoshenko beam finite elements using trigonometric basis functions a timoshenko beam finite element is developed using trigonometric basis functions the properties and performance of these new elements are explored through a series of illustrative problems that treat both straight and curved geometries both linear and nonlinear strain displacement relations are employed in the formulation and comparison is made to results obtained from full completely reduced and partially reduced integration of polynomial basis functions of degree one through three the results obtained in this work indicate that the trigonometric basis functions are a competitive alternative to polynomial basis functions with regard to accuracy and convergence and most importantly they possess a freedom from shear and membrane locking the trigonometric basis functions also allow the recovery of rigid body motions in the case of straight and circular arc curved beams

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

explanation and elimination of shear locking and membrane locking with field consistence approach abstract in this paper in the context of beam elements shear locking and membrane locking are explained and cured in a unified way with a field consistence approach in the view of field consistence approach when transverse displacements are coupled with section rotations in euler lagrangian equations and low order interpolations are adopted shear locking will occur similarly when in plane displacements are coupled with section rotations in euler lagrangian equations and low order interpolations are used membrane locking will happen the beam elements derived by field consistence approach are not only free of shear locking and membrane locking but also highly efficient compared with their isoparametric counterparts although theoretically the internal forces generalized strains in mixed variational principles can vary independently they are neither independent to each other nor independent of displacement fields from the point of view of the field consistence approach

behaviour of isoparametric quadrilateral family of lagrangian fluid finite elements this paper presents the formulation of both the consistent and inconsistent four eight and nine noded isoparametric quadrilateral fluid finite elements that are based on lagrangian frame of reference the mesh locking phenomenon due to simultaneous enforcement of twin constraints namely the incompressibility and irrotationality constraints is studied in detail the study shows that the characteristic of the locked fluid elements is that it always generates numerous spurious acoustic volume change modes upon the enforcement of rotational constraints that is the rotational constraints change the character of certain volume change modes the study further reinforces the necessity of rotational constraints in not only identifying the spurious pressure modes but also in reducing the computational effort for determining the eigenvalues and eigenvectors it is found that all fully integrated inconsistent models exhibit locking behaviour however the inconsistent eight and nine noded elements integrated with full integration of volumetric stiffness and one point integration of the rotational stiffness matrices gives excellent performance although they do not pass the inf sup test the four and nine noded consistent models are found to give locking free performance while their eight noded counterpart exhibited locking behaviour the study shows that only consistent nine noded element models pass the inf sup test the utility of these elements in coupled fluid structure interaction problem is also demonstrated

the second frequency spectrum of timoshenko beams abstract a second spectrum of frequencies was reported in early analytical work on the vibrations of timoshenko beams however in subsequent finite element modelling this phenomenon was either ignored or not definitively classified and recorded in fact from a recent finite element analysis with a high precision element it was even concluded that there is no separate second spectrum of frequencies except for the special case of hinged hinged beams and it was asserted that previous investigators had misinterpreted some frequencies thus introducing the notion of second frequencies in this paper a simple linear beam element with independent displacement fields and reduced integration to eliminate shear locking is used and enables one to detect the second spectrum accurately guidelines are provided which help to identify and classify the frequencies into two separate spectra

variational correctness and timoshenko beam finite element elastodynamics the finite element discretisation of the two noded timoshenko beam element for elastodynamics offers very interesting insights into the error analysis aspects of the formulation in this paper the relatively different order of convergence of the two spectra of the timoshenko beam theory and the extra variational aspect of the use of reduced integration to free the element of locking are investigated the correct variational basis for finite element analysis of elastodynamic problems is presumed to originate from the principle of virtual work with a simultaneous consideration of errors in both displacement and strains a variationally correct element would lock to make an element free of locking some degree of variational incorrectness must be brought in the present paper also demonstrates that reduced integration violates the virtual work principle which in turn causes the loss of boundedness of the finite element eigenvalues with the exact solution

a method to solve the nonlinear eigenvalue problem of timoshenko plane frames with rigid offsets and end releases abstract the study reported here makes use of the dynamic stiffness matrix dsm for timoshenko beams hence considering rotatory inertia and shear deflection to obtain the exact solution of the associated free vibration problem the natural frequencies and vibration modes in the presence of end releases and rigid offsets are all exactly obtained for different combinations of boundary conditions and in plane arrangements of members this is possible by solving the resulted nonlinear and transcendental eigenvalue problem using two general methods one based on the wittrick williams algorithm adapted to handle end release and rigid offset and the other on the power deflation secant method the latter offered here as a novelty is proven to be an efficient technique judging from the examples given using both algorithms and the finite element method it is concluded that the power deflation method is a powerful tool to solve the nonlinear eigenvalue problem leading to exact solutions of the free vibration of complex skeleton like structures

dynamics of transversely vibrating beams using four engineering theories abstract in this paper the full development and analysis of four models for the transversely vibrating uniform beam are presented the four theories are the euler bernoulli rayleigh shear and timoshenko first a brief history of the development of each beam model is presented second the equation of motion for each model and the expressions for boundary conditions are obtained using hamilton s variational principle third the frequency equations are obtained for four sets of end conditions free free clamped clamped hinged hinged and clamped free the roots of the frequency equations are presented in terms of normalized wave numbers the normalized wave numbers for the other six sets of end conditions are obtained using the analysis of symmetric and antisymmetric modes fourth the orthogonality conditions of the eigenfunctions or mode shape and the procedure to obtain the forced response using the method of eigenfunction expansion is presented finally a numerical example is shown for a non slender beam to signify the differences among the four beam models

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

on minimum period of nonlinear feedback shift registers in grain like structure grain is one of three hardware oriented finalists of the estream project a nonlinear feedback shift register nfsr in grain like structure is a cascade connection of a linear feedback shift register lfsr into an nfsr in which the characteristic polynomial of the lfsr is primitive and the feedback function of the nfsr is nonsingular in 2011 hu and gong pointed out that the period of the sequence generated by an nfsr in grain like structure is a multiple of the period of the sequence generated by its lfsr if the initial state of the lfsr is nonzero meanwhile they proposed an open problem for fixed feedback functions of an nfsr and an lfsr determine whether the sequences generated by the nfsr in grain like structure can achieve the minimum period i e the period of the lfsr when the initial state of the lfsr is nonzero and if they can achieve provide at least one pair of the initial states of the nfsr and lfsr clearly from a security point of view it is not preferable if the sequences generated by an nfsr in grain like structure achieve the minimum period this paper converts the open problem into a problem of solving an integer equation with respect to two unknown integers that uniquely correspond to the initial states of the nfsr and lfsr by viewing the nfsr as a boolean control network based on the integer equation this paper shows that for any given initial state of an n stage nfsr and any given nonzero initial state of an m stage lfsr the probability that the sequence generated by the nfsr in grain like structure achieves the minimum period 2 m 1 is at most 2 n this implies that the probability of the cascade connection used in grain achieving the minimum period is very small

a series expansion approach to interpreting the spectra of the timoshenko beam abstract the timoshenko beam model results in two fourth order partial differential equations in time and space consequently solving the boundary value problem yields two independent sequences of natural frequencies and two corresponding sequences of mode shapes a particular natural frequency and its corresponding mode shape describe one particular solution to the boundary value problem of the timoshenko beam from an eigenfunction expansion sense all these possible solutions have to be considered in the complete series expansion of the solution however the question of whether these two independent sequences of natural frequencies implies the existence of two distinct spectra of frequencies has been a long standing topic of debate and hitherto has not been resolved completely the object of this paper is to provide answers to some of the issues raised by this debate in this context the complete solution in a series form to the timoshenko beam is investigated and it is shown for the first time that a particular mode shape of the solution is naturally expressed by an ordered pair of characteristic values rather than a single characteristic value this representation facilitates the progressive ordering of all the natural frequencies of the system and their respective mode shapes in a single set and eliminates the remaining argument for the two spectra interpretation

field consistency and violent stress oscillations in the finite element method the fact that finite element models can give rise to13 violent stress oscillations has been known for some time it is also well known that in many problems there are optimal locations where stresses can be correctly sampled in spite of the presence of these violent stress fluctuations however it is less well known that these oscillations arise in a specific class of problems where there are multiple strain fields arising from one or more field variables and where one or more of these strain fields must be constrained in particular physical13 limits in this paper we show that unless the interpolations for these constrained strain fields are x2018 field consistentx2019 violent oscillations would set in these oscillations represent spurious self equilibrating stress fields generating spurious energy terms that lead to x2018 lockingx2019

vibration of a rectangular beam with a deforming cross section abstract levinson recently proposed a new theory for rectangular beam vibrations in which an inherent contradiction in conventional timoshenko beam theory is avoided by relaxing the assumption that plane sections remain plane in this paper this theory is reformulated in energy minimization terms it is found that the energy expressions are similar to those found for a timoshenko beam with a shear coefficient of 5 6 but they are not identical the two theories are therefore not equivalent at variance with levinson s conclusion it is shown that considerable care is needed to deal with a clamped boundary

consistency aspects of out of plane bending torsion and shear in a quadratic curved beam element curved beams in civil engineering applications call for out of plane bending and torsion under the action of out of plane transverse shear loads the design of a quadratic displacement type curved beam element capable of representing shear deformation as in the timoshenko beam theory will call for special attention to be paid to the manner in which the shear strain is to be represented field inconsistent representations of the out of plane transverse shear strain will result in a loss gf efficiency and introduce spuriou oscillations in the bending moment torsional moment and shear force the optimal field consistent assumed strain interpolation for the shear is derived here and it is demonstrated that it has very high accuracy and is free from spurious force and moment oscillations

on the two frequency spectra of timoshenko beams abstract this paper is concerned with the question of whether there are indeed two distinct spectra of frequencies for the transverse vibrations of timoshenko beams as has been claimed by a number of prior authors for the case of the simply supported beam and more recently for beams supported in an arbitrary manner elementary analysis leads to the conclusion that there is only a single frequency spectrum in the particular case of the simply supported beam the two frequency spectra viewpoint may be expedient as a device to compute frequencies but does not serve otherwise to explain the complex dynamical behavior of timoshenko beams

experimental study of the timoshenko beam theory predictions further results abstract in a previous paper 2012 we presented experimental results proving that the critical frequency f c predicted by timoshenko beam theory indeed exists we also showed that for frequencies f smaller than f c the spectrum is formed by almost equally spaced levels whereas for f f c the spectrum consists of pairs of eigenvalues very close to each other as predicted by numerical solutions of timoshenko s equation we shall refer to them as timoshenko doublets in this work we measure for the first time experimental dispersion relations for this purpose it was necessary to obtain normal mode amplitudes with a high precision which was done with a new experimental setup developed by us we found that experimental dispersion relations coincide very well with theoretical predictions furthermore we provide an explanation of timoshenko doublets

a little microlocal morse theory if a complex analytic function f has a stratified isolated critical point then it is known that the cohomology of the milnor fibre of f has a direct sum decomposition in terms of the normal morse data to the strata we use microlocal morse theory to obtain the same result under the weakened hypothesis that the vanishing cycles along f have isolated support we also investigate an index theoretic proof of this fact

reduced minimization in lagrangian mindlin plate element with arbitrary orientation under uniform isoparametric mapping for the displacement based lagrangian mindlin plate elements oriented arbitrarily under uniform isoparametric mapping without internal distortion a theoretical interpretation on the conventional shear reduced integration is presented by introducing the concept of reduced minimization it reveals that the conventional shear reduced integration such as uri and sri prevents the unmatched coefficients of shear strains from becoming spurious constraints thus the shear reduced integration eliminates spurious constraints and prevents locking