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

semiparametric bayesian inference for phage display data we discuss inference for a human phage display experiment with three stages the data are tripeptide counts by tissue and stage the primary aim of the experiment is to identify ligands that bind with high affinity to a given tissue we formalize the research question as inference about the monotonicity of mean counts over stages the inference goal is then to identify a list of peptide tissue pairs with significant increase over stages we use a semi parametric dirichlet process mixture of poisson model the posterior distribution under this model allows the desired inference about the monotonicity of mean counts however the desired inference summary as a list of peptide tissue pairs with significant increase involves a massive multiplicity problem we consider two alternative approaches to address this multiplicity issue first we propose an approach based on the control of the posterior expected false discovery rate we notice that the implied solution ignores the relative size of the increase this motivates a second approach based on a utility function that includes explicit weights for the size of the increase

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

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

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

sample function behavior of increasing processes of class l we consider increasing processes x t t 0 of classl that is increasing self similar processes with inswpendent increments leth t be an increasing positive function on 0 withh 0 0 andh by virtue of the zero one laws there existsc resp c 0 such that lim inf resp lim sup x t h t c resp c a s both ast tends to 0 and ast tends to we decide a necessary and sufficient condition for the existence ofh t withc orc 1 and explicitly constructh t in caseh t exists withc orc 1 moreover we give a criterion to classify functionsh t withc orc 0 andh t withc orc in caseh t does not exist withc orc 1

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

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

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

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

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

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

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

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

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

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 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

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 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

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

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

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

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