an asymptotic analysis of composite beams with kinematically corrected end effects a finite element based beam analysis for anisotropic beams with arbitrary shaped cross sections is developed with the aid of a formal asymptotic expansion method from the equilibrium equations of the linear three dimensional 3d elasticity a set of the microscopic 2d and macroscopic 1d equations are systematically derived by introducing the virtual work concept displacements at each order are split into two parts such as fundamental and warping solutions first we seek the warping solutions via the microscopic 2d cross sectional analyses that will be smeared into the macroscopic 1d beam equations the variations of fundamental solutions enable us to formulate the macroscopic 1d beam problems by introducing the orthogonality of asymptotic displacements to six beam fundamental solutions the end effects of a clamped boundary are kinematically corrected without applying the sophisticated decay analysis method the boundary conditions obtained herein are applied to composite beams with solid and thin walled cross sections in order to demonstrate the efficiency and accuracy of the formal asymptotic method based beam analysis famba presented in this paper the numerical results are compared to those reported in literature as well as 3d fem solutions

out of plane vibrations of thick rings abstract depending on its cross sectional geometry a ring can be modelled by at least three different theories kirkhope developed a simple ring theory which is applicable to out of plane vibrations of rings with small cross sectional aspect ratio i e compact rings and shell like rings for plate like rings the present analysis uses a levinson type plate theory to analyze the out of plane vibrations also a three dimensional finite element analysis using nastran shows better agreement than results of kirkhope s ring theory and mindlin s thick plate theory

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

on the asymptotic boundary conditions of an anisotropic beam via virtual work principle in this research an efficient and effective method is proposed to derive the boundary conditions of an anisotropic beam in the asymptotic sense we first set up the constrained virtual work by introducing the lagrange multiplier on the displacement prescribed boundary the macroscopic beam and microscopic cross section equations with the boundary conditions are simultaneously obtained by taking the asymptotic expansion on the displacement vector in this way the three dimensional characteristics of the beam are asymptotically smeared into the macroscopic beam equations and the beam boundary conditions the boundary conditions obtained are then compared to those from the decay analysis method the beam bending slope boundary condition obtained in the frame work of variational principle is different from the well known average condition this new boundary condition is more accurate than the average one for a sandwich beam this is further demonstrated and discussed via the examples of a cantilever beam loaded at the end

exact theory for a linearly elastic interior beam abstract in this paper an elasticity solution for a two dimensional 2d plane beam is derived and it is shown that the solution provides a complete framework for exact one dimensional 1d presentations of plane beams first an interior solution representing a general state of any 2d linearly elastic isotropic plane beam under a uniform distributed load is obtained by employing a stress function approach the solution excludes the end effects of the beam and is valid sufficiently far away from the beam boundaries then three kinematic variables defined at the central axis of the plane beam are formed from the 2d displacement field using these central axis variables the 2d interior elasticity solution is presented in a novel manner in the form of a 1d beam theory by applying the clapeyron s theorem it is shown that the stresses acting as surface tractions on the lateral end surfaces of the interior beam need to be taken into account in all energy based considerations related to the interior beam finally exact 1d rod and beam finite elements are developed by the aid of the axis variables from the 2d solution

much ado about shear correction factors in timoshenko beam theory abstract many shear correction factors have appeared since the inception of timoshenko beam theory in 1921 while rational bases for them have been offered there continues to be some reluctance to their full acceptance because the explanations are not totally convincing and their efficacies have not been comprehensively evaluated over a range of application herein three dimensional static and dynamic information and results for a beam of general both symmetric and non symmetric cross section are brought to bear on these issues only homogeneous isotropic beams are considered semi analytical finite element safe computer codes provide static and dynamic response data for our purposes greater clarification of issues relating to the bases for shear correction factors can be seen also comparisons of numerical results with timoshenko beam data will show the effectiveness of these factors beyond the range of application of elementary bernoulli euler theory an issue concerning principal shear axes arose in the definition of shear correction factors for non symmetric cross sections in this method expressions for the shear energies of two transverse forces applied on the cross section by beam and three dimensional elasticity theories are equated to determine the shear correction factors this led to the necessity for principal shear axes we will argue against this concept and show that when two forces are applied simultaneously to a cross section it leads to an inconsistency only one force should be used at a time and two sets of calculations are needed to establish the shear correction factors for a non symmetrical cross section

an intrinsic beam model based on a helicoidal approximation part i formulation the focus of the present work is directed towards the one dimensional non linear analysis of space curved and twisted beams undergoing large displacements and finite rotations according to cosserat s model a beam is intended here as a continuum generated by the rigid motion of a cross section along a curve the novelty of the proposed methodology is implied in the assumption that the reference line of the beam both in the undeformed and in the deformed configurations is a helicoid in space this formulation radically departs from the classical beam models based on polynomial interpolations of the independent fields in fact within the framework of helicoidal geometry a powerful description of space curved beam kinematics with large displacements and finite rotations can be developed furthermore the beam model developed herein can be proved to enjoy several invariance properties that make it particularly attractive for non linear analysis the formulation discussed in this work adheres to a mixed variational principle the independent fields being the generalized displacements and the sectional stress resultants in a companion paper we address the linearization of the governing weak form the finite element implementation of the proposed methodology with a novel incremental treatment of finite rotations and the development of a helicoidal model for rigid body dynamics using finite elements in time

another second order beam vibration theory explicit bending warping flexibility and restraint abstract a new symmetric beam bending vibration theory which explicitly includes bending warping is presented the governing differential equations are of sixth order and the boundary conditions allow enforcement of total fixity of the end of a beam the motivation for the study was the desire to determine if consideration of bending warping restraint in bending theory is as important as is torsion warping restraint in torsion theory natural frequency predictions are given and compared with other theories for constant cross section cantilevered beams of two types short deep rectangular beams and short thin shallow circular cylindrical shells the accuracy of the new theory is comparable with murty s levinson s bickford s and rehfield s theories for beams with rectangular cross sections it is also comparable with cowper s theory for general symmetric cross sections

eilenberg moore models for fibrations e m model is a new invariant in rational homotopy theory which gives us both a kiunneth object and a tate josefiak resolution with the e m model we study relations between formality of base total space and fibre of a serre fibration obstructions to k realizability of a cohomology equivalence between two continuous maps and formalizable maps

analogy between refined beam theories and the bernoulli euler theory abstract some refinements of the classical bernoulli euler theory of the bending of beams are shown to be completely analogous to the effects produced by sources of self stress in the bernoulli euler beam this is true for deflections bending moments and shearing forces they determine rotations stresses and strains according to the specific refined theory under consideration thus by analogy differences between the results of various refined theories having been the object of discussions in recent literature become accessible to a systematic classification in an appendix this strategy of treating refined theories from the point of view of the classical one is put into the more general context of the theory of science

what is the effective impact of the explosive orbital growth in discrete time one dimensional polynomial dynamical systems abstract we study the distribution of periodic orbits in one dimensional two parameter maps specifically we report an exact expression to quantify the growth of the number of periodic orbits for discrete time dynamical systems governed by polynomial equations of motion of arbitrary degree in addition we compute high resolution phase diagrams for quartic and for both normal forms of cubic dynamics and show that their stability phases emerge all distributed in a similar way preserving a characteristic invariant ordering such coincidences are remarkable since our exact expression shows the total number of orbits of these systems to differ dramatically by more than several millions even for quite low periods all this seems to indicate that surprisingly the total number and the distribution of stable phases is not significantly affected by the specific nature of the nonlinearity present in the equations of motion

on the analysis of prismatic beams using first order warping functions abstract a one dimensional theory for isotropic homogeneous beams with a general cross section is developed that utilizes both deformations in the cross section plane and warping out of the cross section plane where these deformation functions are known from saint venant s elasticity solutions for bending and flexure of a prismatic beam first order warping functions in contrast to existing technical beam theories which assume that local cross section deformations and some stresses can be neglected the present theory makes no assumption the finite element model is developed using the resulting differential equations along with a partially weak form formulation full three dimensional constitutive equations are used solutions are found using the finite element method numerical results which are presented for a wide variety of loadings and boundary conditions show that the calculated stresses and displacements are in excellent agreement with existing elasticity solutions

on locking free shear deformable beam finite elements abstract this is an elementary exposition of beam finite element models an overview of existing and new displacement finite element models of the euler bernoulli and shear deformation beam theories is presented a locking free finite element model using the form of the exact solution of the timoshenko beam theory is developed the element contains the euler bernoulli beam element as a special case and yields exact nodal values for the generalized displacements for constant material and geometric properties of beams the procedure is extended to develop a locking free finite element model of the simplified third order theory of beams the element contains the euler bernoulli and the locking free timoshenko beam finite elements as special cases the element stiffness matrix of the unified beam bending element is 4 4 and yet has the superconvergence character the advantage of this element is that it can be used to obtain exact nodal displacements and forces from the equilibrium equations of the element in a frame structural analysis using one finite element per a structural member this cannot be achieved with the traditional equal interpolation reduced integration element because the element exhibits shear locking for thin members unless two or more elements per a structural member are used

lagrange type formulation for finite element analysis of non linear beam vibrations abstract a lagrange type formulation for finite element analysis of non linear vibrations of immovably supported beams is presented two equations of motion coupled in axial and transverse displacements are derived by using lagrange s equations by neglecting the in plane inertial effects these equations are written in terms of the transverse displacement alone upon defining certain properties for the non linear oscillatory behaviour of the transverse displacement the governing equation is reduced to an equation in space alone from which the eigenvalue like quantity is computed the governing equation is solved in two ways a direct iteration technique is used in the first method to compute a numerically exact mode shape and the corresponding frequency a rayleigh quotient type of formulation similar to linear vibration analysis is used in the second approach to evaluate the frequency of vibration for a fundamental mode which is determined from a linear fem model and is maintained constant at all amplitudes numerical results are compared with available results and they corroborate the observations of earlier research workers

moving least square ritz method for vibration analysis of plates this paper presents a novel numerical method the moving least square ritz method mls ritz for free vibration analysis of classical thin plates the proposed method utilizes the strength of the moving least square approach to define the ritz trial function for the transverse displacement of the plates a set of points is pre selected on the calculation domain of a plate that forms the basis for the mls ritz trial function the edge support conditions of the plate are satisfied by forcing the boundary points to meet the geometric boundary conditions of the plate via a point substitution technique virtual points points outside the plate domain are introduced for clamped edges to improve the convergence and accuracy of the calculations square and right angled isosceles triangular plates of various combinations of edge support conditions are selected to examine the validity and accuracy of the mls ritz method extensive convergence studies are carried out to investigate the influence of the mls mesh size the mls support radius the number of gaussian integration points and the shape of the mls weight function on the proposed method comparing with the existing ritz methods the mls ritz method is highly stable and accurate and is extremely flexible for dealing with plates of arbitrary shapes and boundary conditions

relationships between bending solutions of classical and shear deformation beam theories the exact relationships between the deflections slopes rotations shear forces and bending moments of a third order beam theory and those of the euler bernoulli theory and the timoshenko beam theory are developed the relationships enable one to obtain the solutions of the third order beam theory from any known euler bernoulli or timoshenko beam theory solutions of beams for any set of boundary conditions and transverse loads the relationships can also be used to develop finite element models of the timoshenko and third order beam theories and determine numerical solutions from the finite element model of the euler bernoulli beam theory the finite element models are free of the shear locking that is found in the conventional shear deformable finite elements

accurate response of wing structures to free vibration load factors and nonstructural masses based on the carrera unified formulation this work extends variable kinematic finite beam elements to include load factors and nonstructural masses for the static and vibration analyses of complex metallic wing structures according to the carrera unified formulation variable kinematic beam theories are formulated in an automatic and hierarchical manner by expressing the displacement field as an arbitrary expansion through generic cross sectional functions both taylor like and lagrange polynomials are used in this paper to develop refined beam kinematics and the related theories are referred to as taylor expansion and lagrange expansion respectively the generalized unknowns of taylor expansion models are the beam axis displacements and the n order displacement derivatives with n being a free parameter of the analysis classical beam theories are clearly particular cases of the linear n 1 taylor expansion model on the other hand lagrange expansion models have only pure translational displacemen

rbf based meshless method for the free vibration of beams on elastic foundations in this paper solution is obtained for the free vibration differential equations of motion of an axially loaded beam on elastic foundation using a meshless method use is made of the multiquadrics radial basis function rbf in obtaining the numerical solution for four different cases 1 one end clamped the other end simply supported 2 both ends clamped 3 both ends simply supported and 4 a simple beam on elastic foundation with end rotational springs the approach is easier to implement and program as compared to grid mesh based methods such as the finite difference method fdm and the finite element method fem accuracy of the results obtained using the proposed method was verified using the analytical results available in the literature for the first three cases considered numerical results of the fourth case were aimed at justifying the use of the numerical scheme for a problem whose analytical solution is not readily available and to show the high accuracy of the rbf method the results prove that the method require much less number of nodes to converge to the correct solution as compared to fdm

free vibration analysis of civil engineering structures by component wise models abstract higher order beam models are used in this paper to carry out free vibration analysis of civil engineering structures refined kinematic fields are developed using the carrera unified formulation cuf which allows for the implementation of any order theory without the need for ad hoc formulations the principle of virtual displacements in conjunction with the finite element method fem is used to formulate stiffness and mass matrices in terms of fundamental nuclei the nuclei depend neither on the adopted class of beam theory nor on the fem approximation along the beam axis this paper focuses on a particular class of cuf models that makes use of lagrange polynomials to discretize cross sectional displacement variables this class of models are referred to as component wise cw in recent works according to the cw approach each structural component e g columns walls frame members and floors can be modeled by means of the same 1d formulation a number of typical civil engineering structures e g simple beams arches truss structures and complete industrial and civil buildings are analyzed and cw results are compared to classical beam theories euler bernoulli and timoshenko refined beam models based on taylor like expansions of the displacements on the cross section and classical solid shell fem solutions from the commercial code msc nastran the results highlight the enhanced capabilities of the proposed formulation it is in fact demonstrated that cw models are able to replicate 3d solid results with very low computational efforts

automatic energy momentum conserving time integrators for hyperelastic waves an energy momentum conserving time integrator coupled with an automatic finite element algorithm is developed to study longitudinal wave propagation in hyperelastic layers the murnaghan strain energy function is used to model material nonlinearity and full geometric nonlinearity is considered an automatic assembly algorithm using algorithmic differentiation is developed within a discrete hamiltonian framework to directly formulate the finite element matrices without recourse to an explicit derivation of their algebraic form or the governing equations the algorithm is illustrated with applications to longitudinal wave propagation in a thin hyperelastic layer modeled with a two mode kinematic model solution obtained using a standard nonlinear finite element model with newmark time stepping is provided for comparison

advanced 3d beam element of arbitrary composite cross section including generalized warping effects summary in this paper an advanced 20 20 stiffness matrix and the corresponding nodal load vector of a member of arbitrary composite cross section is developed taking into account shear lag effects due to both flexure and torsion the case of the three dimensional beam element of arbitrary homogeneous cross section is treated as a special one the composite member consists of materials in contact each of which can surround a finite number of inclusions nonuniform warping distributions are taken into account by employing four independent warping parameters multiplying a shear warping function in each direction and two torsional warping functions ten boundary value problems with respect to the kinematical components are formulated and solved employing the analog equation method a bem based technique the aforementioned boundary value problems are formulated employing either an improved stress field arising from the correction of shear stress components or the stress field arising directly from displacement considerations the warping functions and the geometric constants including the additional ones due to warping are evaluated employing a pure bem approach that is only boundary discretization of the cross section is used numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy the deviations arising from the use of the advanced 20 20 stiffness matrix and the classical 12 12 or 14 14 ones employed in commercial software packages are illustrated through examples of great practical interest moreover the influence of nonuniform warping effects necessitating the use of the aforementioned advanced stiffness matrix is also demonstrated copyright 2015 john wiley sons ltd

nondestructive determination of buckling for plates and bars including shear it will be shown that the generic integral equation for nondestructive determination of the buckling loads obtained previously baruch m integral equations for nondestructive determination of buckling loads for elastic bars and plates israel journal of technology vol 11 no 1 2 1973 pp 1 8 includes the influence of the transverse shear strains on the buckling load this can not be said for the integral equations obtained by differentiation of the generic integral equation

generalized warping effect in the dynamic analysis of beams of arbitrary cross section abstract in this paper a general formulation for the nonuniform warping dynamic analysis of beams of arbitrary simply or multiply connected cross section under arbitrary external loading and general boundary conditions is presented taking into account the effects of rotary and warping inertia the nonuniform warping distributions are taken into account by employing four independent warping parameters multiplying a shear warping function in each direction and two torsional warping functions respectively which are obtained by solving the corresponding boundary value problems formulated exploiting the longitudinal local equilibrium equation a shear stress correction is also performed in order to improve the stress field arising from the employed kinematical considerations ten initial boundary value problems are formulated with respect to the displacement and rotation components as well as to the independent warping parameters and solved using the analog equation method a boundary element method based technique in combination with an appropriate time integration scheme the warping functions and the geometric constants including the additional ones due to warping are evaluated employing a pure bem approach

modele minimal equivariant et formalite we study the rational equivariant homotopy type of a topological space x equipped with an action of the group of integers modulo n for n pk p prime k a positive integer we build an algebraic model which gives the rational equivariant homotopy type of x the homotopical fixed point set appears in the construction of a model of the fixed points set in general this model is different from g triantafillou s model ti for n p p prime we then give a notion of equivariant formality we prove that this notion is equivalent to the formalizability of the inclusion of fixed points set i xzp x examples and counterexamples of zp formal spaces are given

saint venant solutions for prismatic anisotropic beams abstract an analytical model is presented for determining the displacement and stress distributions of the saint venant extension bending torsion and flexure problems for a homogeneous prismatic beam of arbitrary section and rectilinear anisotropy the determination of the complete displacement field requires solving a coupled two dimensional boundary value problem for the local in plane deformations and warping out of the section plane the principle of minimum potential energy is applied to a discretized representation of the cross section ritz method to calculate solutions to this problem the behavior of an anisolropic beam is studied in detail using the resulting displacement and stress solutions where definitions are presented for the shear center center of twist torsion constant and a new geometric parameter the line of extension bending centers two sets of numerical results are presented to illustrate how section geometry beam length and material properties affect the behavior of a homogeneous anisotropic cantilever beam

analytical formulation and evaluation for free vibration of naturally curved and twisted beams abstract an analytical study for free vibration of naturally curved and twisted beams with uniform cross sectional shapes is carried out using spatial curved beam theory based on the washizu s static model in the governing equations of motion of the beams all displacement functions and the generalized warping coordinate are defined at the centroid axis and also the effects of rotary inertia transverse shear deformations and torsion related warping are included in the proposed model explicit analytical expressions are derived for the vibrating mode shapes of a curved bending torsional shearing coupled beam under clamped clamped boundary condition with the help of symbolic computing package mathematica and a process of searching is used to determine the natural frequencies comparisons of the present results with the fem results using beam elements in ansys code show good accuracy in computation and validity of the model further the present model is used for cylindrical helical springs with circular cross section fixed at both ends and the results indicate that the natural frequencies agree well with the theoretical and experimental results available

a modified fourier series method for the torsion analysis of bars with multiply connected cross sections a modified fourier series method is proposed for the torsion analysis of prismatic bars with multiply connected cross sections the key feature in the present approach is the combined use of polynomials and fourier series solutions unlike in the existing approaches which use the fourier series only the replacement of the zeroth harmonic terms in the fourier series solutions by carefully selected polynomials resolves the major problem of functional dependence which the direct fourier series method may pose the polynomials and fourier series solutions are selected to satisfy the governing equation exactly so that the numerical calculation is minimal the effectiveness and generality of the present method is verified through numerical examples

a new rectangular beam theory abstract a new theory for beams of rectangular cross section which includes warping of the cross sections is presented in the present work by satisfying the shear free conditions on the lateral surfaces of the beam a pair of coupled equations of motion are obtained such that no arbitrary shear coefficient is required it is shown that the uncoupled equation for the transverse displacement is the same as the corresponding equation in timoshenko beam theory provided that for the timoshenko equation the shear coefficient is taken to be 5 6 this value lies within the range of values 0 822 0 870 appearing in the literature for the beam of rectangular cross section results for two typical static examples are given for both the new theory and timoshenko beam theory these results are compared with the solutions of the comparable problems in the linear theory of elasticity for the end loaded cantilever beam the new theory predicts the same result for the neutral surface deflection as does the linear theory of elasticity while timoshenko beam theory underestimates the shear correction term by 20 for the uniformly loaded and simply supported case both beam theories provide the same overestimate of the central deflection when compared with the theory of elasticity solution

generalized beam theory applied to shear stiffness there is no consensus of opinion on the correct expressions for shear stiffness even in the apparently simple case of rectangular sections made from homogeneous isotropic material a general beam theory has been proposed which is applicable to all regular prismatic systems this has been used to find the appropriate beam like flexibilities for trusses the same approach can be used for normal beams giving values for the shear stiffnesses of various cross sections as particular results of a general theory embracing torsion binding extension and shear of regular prismatic systems

large deformations of planar extensible beams and pantographic lattices heuristic homogenization experimental and numerical examples of equilibrium the aim of this paper is to find a computationally efficient and predictive model for the class of systems that we call pantographic structures the interest in these materials was increased by the possibilities opened by the diffusion of technology of three dimensional printing they can be regarded once choosing a suitable length scale as families of beams also called fibres interconnected to each other by pivots and undergoing large displacements and large deformations there are however relatively few ready to use results in the literature of nonlinear beam theory in this paper we consider a discrete spring model for extensible beams and propose a heuristic homogenization technique of the kind first used by piola to formulate a continuum fully nonlinear beam model the homogenized energy which we obtain has some peculiar and interesting features which we start to describe by solving numerically some exemplary deformation problems furthermore we consider pantographic structures find the corresponding homogenized second gradient deformation energies and study some planar problems numerical solutions for these two dimensional problems are obtained via minimization of energy and are compared with some experimental measurements in which elongation phenomena cannot be neglected

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

high order theory of homogeneous plate flexure the need for higher order models to obtain accurate estimates to transverse shear and normal stresses and strains is well known in this paper a higher order shear deformation theory with provision for cubic variation of in plane displacements and parabolic variation of the normal displacement has been considered for detailed study numerical results have been obtained for infinite plate strip simply supported on opposite edges and infinitely long in the other direction subjected to sinusoidal static load displacements strains and stresses are compared with the exact solution results by the classical plate theory levinson s theory and a higher order theory based on partial deflections are also included this study shows that the present model can predict all displacements strains and stresses reasonably accurately further it is also noted that the statically equivalent estimates to transverse stresses from classical theory and the corresponding strains also agree closely with the exact solution

buckling and vibration of laminated composite plates using various plate theories analytical and finite element solutions of the classical first order and third order laminate theories are developed to study the buckling and free vibration behavior of cross ply rectangular composite laminates under various boundary conditions

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

warping displacement of timoshenko beam model abstract a refined timoshenko beam model which takes into account warping of the cross sections is presented the model extends saint venant s theory of uniform torsion to the generic loading of a beam in a coherent manner the use of kinetic and kinematic assumptions virtual work expression of full elasticity problem and the principle of virtual work in derivation brings the presentation to the usual context of engineering models a new definition of the warping displacement in terms of a variational problem is one of the outcomes boundary value problems for the stretch shear bending and torsion loading modes are also given analytic and numeric solutions to warping displacements refined constitutive equations and correction factors for circular rectangle triangle and open annular cross sections of isotropic material are compared with expressions in literature as an interesting detail shear correction factors are found to be purely geometrical quantities which contradicts many findings in literature

a generalization of shear deformation theories for axisymmetric multilayered shells a generalization of geometrically linear shear deformation theories for small elastic strains is presented for multilayered axisymmetric shells of general shape without any assumption other than neglecting the transverse normal strain the shear is taken into account by using a function f which is introduced in the assumed kinematics the boundary value problem is derived from the principle of virtual power with the shear function f in the kinematics all equations are directly applicable to kirchhoff love first order shear deformation third order shear deformation theories and obviously the proposed generalized shear deformation theory by using a certain sine shear function no shear correction factors are needed with the proposed generalization of shear deformation theories a numerical evaluation of the new theory is presented and compared with the above classic theories for a simply supported thick laminated cylindrical shell under an internal pressure

on fibrations with formal elliptic fibers abstract we prove that for a fibration of simply connected spaces of finite type f e b with f being positively elliptic and h f q not possessing non trivial derivations of negative degree the base b is formal if and only if the total space e is formal moreover in this case the fibration map is a formal map as a geometric application we show that positive quaternion kahler manifolds are formal and so are their associated twistor fibration maps

refined one dimensional formulations for laminated structure analysis this paper proposes one dimensional formulations based on hierarchical expansions of the unknown displacement variables for the analysis of multilayered structures made of anisotropic composite layers the hierarchical technique shows variable kinematic properties and it is based on the carrera unified formulation two different classes of refined theories are proposed the first expands the unknown variables in terms of power polynomials of the cross sectional coordinates it consists of a taylor like expansion the second class of onedimensional theories uses lagrange polynomials lagrange expansion and subdomain discretizations of the cross section and it leads to only pure displacements as the unknown variables taylor like expansion is used to develop equivalent single layer formulations and lagrange expansion is used to construct both equivalent single layer and layerwise descriptions the finite element method is employed to develop numerical applications using the carrera unified formulation finite element matrices are obtained in terms of a few fundamental nuclei that are formally independent of all the considered one dimensional formulations a number of numerical examples are given concerning on beams plates and more complex structures comparisons with results from plate and solid models are provided the following has been concluded 1 the proposed formulation represents a reliable compact and accurate method to develop refined one dimensional models 2 the present one dimensional models are very effective at detecting both global and local responses of composite structures 3 shell and solidlike results are obtained with a significant reduction in the computational costs

dynamic response of thin walled structures by variable kinematic one dimensional models abstract this paper investigates the accuracy capabilities of using variable kinematic modeling in compact and thin walled beam like structures with dynamic loadings carrera unified formulation cuf is employed to introduce refined one dimensional 1d models with a variable order of expansion for the displacement unknowns over the beam cross section classical euler bernoulli and timoshenko beam theories are obtained as particular cases of these variable kinematic models while a higher order expansion permits the detection of in plane cross section deformation since it leads to shell like solutions finite element fe method is used to provide numerical results and the newmark method is implemented as a time integration scheme some assessments with closed form solutions are discussed and comparisons with shell type results obtained with commercial fe software are made further analyses address both compact and thin walled cross sections in particular the case of a deformable thin walled cylinder loaded by time dependent internal forces is discussed the results clearly show that finite elements which are formulated in the cuf framework do not introduce additional numerical problems with respect to classical beam theories comparisons with elasticity solutions prove that the present 1d cuf model offers an accuracy in analyzing thin walled structures which is typical of shell or three dimensional models with a remarkable reduction in the computational cost required

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

toward a new engineering theory of bending fundamentals the classical engineering theory of bending due to bernoulli and euler serves as a cornerstone for structural analysis and design limitations of this theory however become apparent in flexural wave propagation studies it predicts infinite phase velocity as the wavelength becomes shorter this theoretical deficiency is corrected by timoshenko theory which accounts for transverse shear deformation a thorough study of exact elasticity solutions reveals that there are two additional effects that are the same order as transverse shear in bending behavior a new theory accounting for them is presented along with several applications the new equations are no more complicated than those of timoshenko ty pe theory yet they yield solutions which are exact or indistinguishable from exact in the examples studied

dynamic stiffness matrix of thin walled composite i beam with symmetric and arbitrary laminations abstract for the spatially coupled free vibration analysis of thin walled composite i beam with symmetric and arbitrary laminations the exact dynamic stiffness matrix based on the solution of the simultaneous ordinary differential equations is presented for this a general theory for the vibration analysis of composite beam with arbitrary lamination including the restrained warping torsion is developed by introducing vlasov s assumption next the equations of motion and force displacement relationships are derived from the energy principle and the first order of transformed simultaneous differential equations are constructed by using the displacement state vector consisting of 14 displacement parameters then explicit expressions for displacement parameters are derived and the exact dynamic stiffness matrix is determined using force displacement relationships in addition the finite element fe procedure based on hermitian interpolation polynomials is developed to verify the validity and the accuracy of this study the numerical solutions are presented and compared with analytical solutions the results from available references and the fe analysis using the thin walled hermitian beam elements particular emphasis is given in showing the phenomenon of vibrational mode change the effects of increase of the modulus and the bending twisting coupling stiffness for beams with various boundary conditions

a simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method a size dependent finite element model is developed nonlocal elasticity theory is used finite element solution is presented for buckling of microtubules the effect of elastic matrix surrounded by microtubules is modeled a simple nonlocal beam model is proposed to study buckling response of protein microtubules the size effect for buckling model of microtubules is considered by using the nonlocal continuum theory finite element procedure is used for solution of nonlocal differential equation of microtubules for elastic stability the influence of the small length scale on the buckling value is examined for different geometric parameters the effect of elastic matrix surrounded of microtubules is also examined and some benchmark results are presented

on the accuracy of one dimensional models for multilayered composite beams abstract the accuracy of 1 d models for composite beams made of linearly elastic orthotropic layers is estimated by means of the prager synge hypercircle method statically admissible and kinematically admissible stress fields are derived and asymptotic forms for the error bounds of o h l and o h l 2 for h l approaching zero are obtained for displacement based laminated beam theories where the axial displacement is represented in terms of a given set of coordinate functions defined over the beam height the condition of vanishing of relative mean square error for h l 0 is used to derive the constitutive law for the 1 d model explicit forms for the error bounds are given for the classical lamination theory first order shear deformation theory and two higher order theories quantitative error bounds are calculated for simply supported multilayered beams under sinusoidal transverse loading it is shown that starting from a wise choice of coordinate functions the accuracy of higher order theories can be almost independent of the beam lamination and up to 150 and 80 times higher than clt and fsdt respectively

general behavior and shear center location of prismatic anisotropic beams via power series abstract the behavior of a tip loaded anisotropic cantilever beam with an arbitrary cross section is studied using saint venant s semi inverse method along with a power series solution for the local in plane and out of plane deformation warping functions the power series coefficients are determined by solving a set of variationally derived linear algebraic equations using the resulting three dimensional displacement solutions the shear deformation associated with applied tip loads is investigated as well as the shear center location two different definitions of the shear center are presented for anisotropic beams by extending existing approaches developed for isotropic sections both of the extended definitions reveal the linear dependency of the shear center location on beam length numerical results are presented for three different cross sections ellipse triangle naca 0012 airfoil and two different materials al 6061 t6 off angle high strength graphite epoxy fibers

bending solutions of levinson beams and plates in terms of the classical theories using the mathematical similarity of the governing equations of the classical beam and plate theories and the levinson beam and plate theories and the basis of load equivalence exact relationships between the bending solutions of the two theories for beams and plates are derived these relationships enable the conversion of the well known classical euler bernoulli beam and kirchhoff plate solutions to their shear deformable counterparts using the levinson beam and plate theories examples are given to illustrate the use of these relationships

vibration and buckling of deep beam columns on two parameter elastic foundations natural frequencies and buckling stresses of a deep beam column on two parameter elastic foundations are analyzed by taking into account the effect of shear deformation depth change the transverse displacement w can vary in the depth direction of beam columns and rotatory inertia by using the method of power series expansion of displacement components a set of fundamental dynamic equations of a one dimensional higher order theory for thin rectangular beam columns subjected to axial stress is derived through hamilton s principle several sets of truncated approximate theories are applied to solve the eigenvalue problems of a simply supported deep elastic beam column in order to assure the accuracy of the present theory convergence properties of the minimum natural frequency and the buckling stress are examined in detail it is noted that the present approximate theories can predict the natural frequencies and buckling stress of deep beam columns on elastic foundations accurately compared with the timoshenko beam theory and the classical beam theory

relationships between classical and shear deformation theories of axisymmetric circular plates the exact relationships between the deflections slopes rotations shear forces and bending moments of a third order theory and those of the classical theory and the first order shear deformation theory for axisymmetric bending of isotropic circular plates are developed the relationships enable one to obtain the solutions of the third order theory from any known classical or first order theory solutions of axisymmetric circular plates for any set of boundary conditions and transverse loads the relationships may also be used to develop finite element models of axisymmetric bending of circular plates according to the first order and third order theories

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

decay rate of saint venant end effects for multilayered orthotropic strips saint venant decay rate of end effects is investigated for generally laminated orthotropic strips under self equilibrated end loads the problem is governed by a fourth order partial differential equation for the airy stress function with discontinuous coefficients at the layer interfaces where displacements and traction continuities are imposed the solution is found in the form of product of an exponentially decaying function along the strip and an unknown function defined over its total height external face and interface conditions are used to obtain the characteristic equation for the eigenvalues governing the decay rate of end effects along the strip for orthotropic and strongly orthotropic sandwich strips the transcendental eigenvalue equation is explicitly given the case of laminated strips with periodic layout is finally considered making use of the homogenization method both effective elastic constants and expressions for the local stress variation at the layer level are obtained numerical calculations confirm that the eigenvalues of the homogenized material are the asymptotic values of those of periodically laminated strips when the number of layers increases moreover homogenization method is shown to be very powerful also in the calculation of local stress distributions

consistency and correctness evaluation of shear deformable anisoparametric formulations the first two of the quartet completeness continuity consistency and correctness conditions required for the13 development of robust displacement based finite elements are well understood as far as the anisoparametric elements13 are concerned when such elements are formulated primarily for accuracy due to their improved consistent load and mass capabilities by incorporating transverse shear deformation effect it becomes necessary to study the next13 two of the quartet field consistency and variational correctness aspects and to explore further their susceptibility for spurious effects such as locking stress oscillations etc such a study on a typical anisoparametric13 element formulation is presented in this paper

deformation of short composite beam using refined theories high order flexural theories for short laminated composite beams subjected to mechanical and thermal loading are presented the formulation allows for warping of the cross section of the beam and eliminates the need for using arbitrary shear correction coefficients as in other theories based on higher order shear deformation theories the governing equations are obtained using the principle of virtual work pvw the justification for use of higher order shear deformation theories is established for short and composite beams where cross sectional warping is predominant

free vibrations of a simply supported rectangular plate an exact elasticity solution abstract an exact three dimensional solution for the free vibrations of simply supported rectangular plates of arbitrary thickness within the linear theory of elastodynamics is given in this paper the solution obtained in a semi inverse fashion as was the solution of the elastostatic problem for such plates satisfies all of the boundary conditions of the problem in a pointwise manner it is found that there are two types of modes of oscillation possible which are consistent with the kinematic assumptions made to find the semi inverse solution other modes of oscillation may exist in the three dimensional theory of elastodynamics for such plates but our kinematic assumptions would not be consistent with such modes the two types of modes found are analogous to the flexural modes of classical plate theory and the thickness twist modes here called breathing modes of mindlin plate theory some numerical results are given which indicate that the predictions of mindlin plates are uncannily good approximations to the flexural frequencies given by the present three dimensional analysis even for very thick plates however the predictions of mindlin plate theory for the thickness twist or breathing frequencies are not nearly so good these discrepancies are discussed briefly in an appendix

generalized variational principle on nonlinear theory of naturally curved and twisted beams the nonlinear behavior for naturally curved and twisted beams undergoing large displacements and rotations but small strains has been investigated in this paper an incomplete generalized variational functional for the beams with complete constrained boundaries at two ends is established by means of the lagrange multiplier method a set of equilibrium equations which takes into account the effects of transverse shear deformations as well as torsion related warping has been derived from the stationary conditions of the functional the various elastic couplings of bending twisting and or extension twisting are also included in this analysis the present theory can also be easily extended to other various incomplete constrained boundaries in case of small displacement the equilibrium equations obtained by the present theory reduce to the well known results in literature

a refined nonlinear theory of plates with transverse shear deformation abstract a higher order shear deformation theory of plates accounting for the von karman strains is presented the theory contains the same dependent unknowns as in the hencky mindlin type first order shear deformation theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the plate exact solutions of simply supported plates are obtained using the linear theory and the results are compared with the exact solutions of 3 d elasticity theory the first order shear deformation theory and the classical plate theory the present theory predicts the deflections stresses and frequencies more accurately when compared to the first order theory and the classical plate theory

prediction capabilities of classical and shear deformable beam models excited by a moving mass abstract in this paper a comprehensive assessment of design parameters for various beam theories subjected to a moving mass is investigated under different boundary conditions the design parameters are adopted as the maximum dynamic deflection and bending moment of the beam to this end discrete equations of motion for classical euler bernoulli timoshenko and higher order beams under a moving mass are derived based on hamilton s principle the reproducing kernel particle method rkpm and extended newmark method are utilized for spatial and time discretization of the problem correspondingly the design parameter spectra in terms of the beam slenderness mass weight and velocity of the moving mass are introduced for the mentioned beam theories as well as various boundary conditions the results indicate the existence of a critical beam slenderness mostly as a function of beam boundary condition in which for slenderness lower than this so called critical one the application of euler bernoulli or even timoshenko beam theories would underestimate the real dynamic response of the system moreover there would be a roughly linear relation between the weight of the moving mass and the design parameters for a certain value of the moving mass velocity in most cases of boundary conditions

beam theory for strongly orthotropic materials abstract this paper presents a displacement based model for orthotropic beams under plane linear elasticity based on the only kinematic assumption of transverse inextensibility any given axial and transverse loading as well as boundary conditions at the beam ends are considered the solution is decomposed into the principal and the residual part corresponding to the interior and the boundary problems which are obtained by series expansions of polynomial functions and eigenfunctions respectively it is proved that the proposed one dimensional theory gives both interior and boundary exact two dimensional elasticity solutions for strongly orthotropic materials i e for ratio between shear modulus and axial young modulus approaching zero for isotropic and orthotropic materials the accuracy of the beam model is also analysed and compared with that of other theories in particular the complementary energy error of the interior solution with respect to two dimensional elasticity is evaluated the asymptotic estimate of the characteristic decay length of end effects given in choi and horgan j appl mech asme 44 424 430 1977 by two dimensional analysis is reobtained and the range of validity of boundary solution is discussed the numerical results presented agree very well with exact and finite element solutions even in the neighbourhood of clamped cross sections where the solution is mainly governed by the boundary problem

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

classification of all cycles of the parabolic map abstract the problem of determining all cycles of the parabolic map p x for all parameters values 0 and all x 0 is addressed the method is one of decomposing the set of all fixed points of the nth iterate p n x for arbitrary n into cycles of p x toward this goal the inverse graph of p n x is constructed for all and its complete description given in terms of sequences 0 1 2 each i a nonnegative integer and their negatives and a total order relation on all such sequences it is shown how the properties of these sequences give a complete description of the evolution of the graph in the parameter with respect to changes in its shape and in its bifurcation structure these results are then used to infer properties of the decomposition of the set of fixed points of p n into cycles of p

a finite element model for a higher order shear deformable beam theory the theory for a higher order shear deformable beam model is first developed it is based on a higher order displacement model and incorporates linear and quadratic variation of transverse normal strain and transverse shearing strain respectively through the beam thickness the effects of the transverse normal and shear stresses are included in the definition of the material s constitutive law the warping of the transverse normal cross section of the beam is automatically incorporated in the mathematical model the question of selecting a shear correction coefficient as in a first order shear deformable timoshenko theory does not arise a linear two noded finite element model of this theory is introduced and developed next both static and free vibration results of this theory are presented and compared with those of euler and timoshenko theories for various boundary and loading conditions

asymptotic analysis of an end loaded transversely isotropic elastic semi infinite strip weak in shear abstract we use linear elasticity to study a transversely isotropic or specially orthotropic semiinfinite slab in plane strain free of traction on its faces and at infinity and subject to edge loads or displacements that produce stresses and displacements that decay in the axial direction the governing equations which are identical to those for a strip in plane stress free of traction on its long sides and at infinity and subject to tractions or displacements on its short side are reduced in the standard way to a fourth order partial differential equation with boundary conditions for a dimensionless airy stress function we study the asymptotic solutions to this equation for four sets of end conditions traction mixed two displacement as g3 the ratio of the shear modulus to the geometric mean of the axial and transverse extensional moduli approaches zero in all cases the solutions for consist of a wide boundary layer that decays slowly in the axial direction over a distance that is long compared to the width of the strip plus a narrow boundary layer that decays rapidly in the axial direction over a distance that is short compared to the width of the strip moreover we find that the narrow boundary layer has a sinuous part that varies rapidly in the transverse direction but which to lowest order does not enter the boundary conditions nor affect the transverse normal stress or the displacements because the exact biorthogonality condition for the cigenfunctions associated with can be replaced by simpler orthogonality conditions in the limit as e b 0 we are able to obtain to lowest order explicit formulae for the coeflicients in the eigenfunction expansions of for the four different end conditions