brief paper identification and data driven model reduction of state space representations of lossless and dissipative systems from noise free data we illustrate procedures to identify a state space representation of a lossless or dissipative system from a given noise free trajectory important special cases are passive systems and bounded real systems computing a rank revealing factorization of a gramian like matrix constructed from the data a state sequence can be obtained the state space equations are then computed by solving a system of linear equations this idea is also applied to perform model reduction by obtaining a balanced realization directly from data and truncating it to obtain a reduced order model

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

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

the mechanics of running how does stiffness couple with speed a mathematical model for terrestrial running is presented based on a leg with the properties of a simple spring experimental force platform evidence is reviewed justifying the formulation of the model the governing differential equations are given in dimensionless form to make the results representative of animals of all body sizes the dimensionless input parameters are u a horizontal froude number based on forward speed and leg length v a vertical froude number based on vertical landing velocity and leg length and kleg a dimensionless stiffness for the leg spring results show that at high forward speed kleg is a nearly linear function of both u and v while the effective vertical stiffness is a quadratic function of u for each u v pair the simulation shows that the vertical force at mid step may be minimized by the choice of a particular step length a particularly useful specification of the theory occurs when both kleg and v are assumed fixed when kleg 15 and v 0 18 the model makes predictions of relative stride length s and initial leg angle theta o that are in good agreement with experimental data obtained from the literature

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

leg stiffness in human running comparison of estimates derived from previously published models to direct kinematic kinetic measures abstract it is not presently clear whether mathematical models used to estimate leg stiffness during human running are valid therefore leg stiffness during the braking phase of ground contact of running was calculated directly using synchronous kinematic high speed motion analysis and kinetic force platform analysis and compared to stiffness calculated using four previously published kinetic models nineteen well trained male middle distance runners age 21 1 4 1 yr vo2max 69 5 7 5 mlo2 kg 1 min 1 completed a series of runs of increasing speed from 2 5 to 6 5 m s 1 leg stiffness was calculated directly from kinetic kinematic analysis using both vertical and horizontal forces to obtain the resultant force in the line of leg compression model 1 values were also estimated using four previously published mathematical models where only force platform derived and anthropometric measures were required models 2 5 morin et al 2005 morin et al 2011 blum et al 2009 farley et al 1993 respectively the greatest statistical similarity between leg stiffness values occurred with models 1 and 2 the poorest similarity occurred when values from model 4 were compared with model 1 analyses suggest that the poor correlation between model 1 other models may have resulted from errors in the estimation in change in leg length during the braking phase previously published mathematical models did not provide accurate leg stiffness estimates although model 2 used by morin et al 2005 provided reasonable estimates that could be further improved by the removal of systematic error using a correction factor k 1 0496 kmodel2

leg stiffness measures depend on computational method leg stiffness is often computed from ground reaction force grf registrations of vertical hops to estimate the force resisting capacity of the lower extremity during ground contact with leg stiffness values incorporated in a spring mass model to describe human motion individual biomechanical characteristics including leg stiffness were investigated in 40 healthy males our aim is to report and discuss the use of 13 different computational methods for evaluating leg stiffness from a double legged repetitive hopping task using only grf registrations four approximations for the velocity integration constant were combined with three mathematical expressions giving 12 methods for computing stiffness using double integrations one frequency based method that considered ground contact times was also trialled the 13 methods thus defined were used to compute stiffness in four extreme cases which were the stiffest and most compliant consistent and variable subjects all methods provided different stiffness measures for a given individual but the between method variations in stiffness were consistent across the four atypical subjects the frequency based method apparently overestimated the actual stiffness values whereas double integrations measures were more consistent in double integrations the choice of the integration constant and mathematical expression considerably affected stiffness values as variations during hopping were more or less emphasized stating a zero centre of mass position at take off gave more consistent results and taking a weighted average of the force or displacement curve was more forgiving to variations in performance in any case stiffness values should always be accompanied by a detailed description of their evaluation methods as our results demonstrated that computational methods affect calculated stiffness

state maps from integration by parts we develop a new approach to the construction of state vectors for linear time invariant systems described by higher order differential equations the basic observation is that the concatenation of two solutions of higher order differential equations results in another weak solution once their remainder terms resulting from repeated integration by parts match these remainder terms can be computed in a simple and efficient manner by making use of the calculus of bilinear differential forms and two variable polynomial matrices factorization of the resulting two variable polynomial matrix defines a state map as well as a state map for the adjoint system minimality of these state maps is characterized the theory is applied to three classes of systems with additional structure namely self adjoint hamiltonian conservative port hamiltonian and time reversible systems for the first two classes it is shown how the factorization leading to a minimal state map is equivalent to the factorization of another two variable polynomial matrix which is immediately derived from the external system characterization and defines a symplectic respectively symmetric bilinear form on the minimal state space

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

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

on two variable rational interpolation abstract the goal of this contribution is to investigate interpolation of two variable rational functions the tool is the two variable loewner matrix which is an extension of its single variable counterpart the main property of the loewner matrix is that its rank encodes the information concerning minimal complexity interpolants both polynomial and generalized state space descriptor realizations of interpolants are presented examples illustrate how two variable rational functions can be recovered from appropriate measurements

from time series to linear system part ii exact modelling abstract in the second part of this paper the problem of finding an exact model for a q dimensional infinite time series is considered first a mathematical vocabulary for discussing exact modelling is developed it is then shown how the results of part i guarantee the existence of a most powerful ar model for an observed time series two algorithms for obtaining such an ar model are subsequently derived one of these algorithms gives a shortest lag input output model the problem of obtaining a minimal state space realization of the observed time series is also considered in order to do that realization theory based on the truncated behaviour is developed as an extensive example the classical situation with impulse response measurements is discussed

comparison of whole body vertical stiffness and leg stiffness during single leg hopping in place in children and adults in the hopping literature whole body vertical stiffness and leg stiffness are used interchangeably due to most of the movement occurring in the vertical direction however there is some anterior posterior movement of the center of mass and displacements of the foot during hopping in place in both children and adults further it is not understood if leg stiffness show a similar pattern as whole body vertical stiffness when increasing hopping frequency the purpose of this study was to test if whole body vertical stiffness and leg stiffness are different during single leg hopping in place in children and adults across a range of frequencies seventeen children aged 5 11years and 16 young adults participated in this study the subjects hopped at their preferred frequency as well as 20 below 20 above and 40 above preferred frequency our results demonstrate that both whole body vertical stiffness and leg stiffness increase when increasing hopping frequency for children and adults however whole body vertical stiffness consistently overestimates leg stiffness due to a similar peak force but a greater leg length change compared to vertical com displacement this suggests a considerable horizontal com movement from landing to mid stance during hopping children aged 5 11years old showed lower absolute values but higher normalized values of two stiffness measures than adults this suggests somewhat adult like stiffness control in children but a reduced ability to manipulate the horizontal movement during single leg hopping in place when compared to adults

the behavioral approach to linear parameter varying systems linear parameter varying lpv systems are usually described in either state space or input output form when analyzing system equivalence between different representations it appears that the time shifted versions of the scheduling signal dynamic dependence need to be taken into account therefore representations used previously to define and specify lpv systems are not equal in terms of dynamics in order to construct a parametrization free description of lpv systems that overcomes these difficulties a behavioral approach is introduced that serves as a basis for specifying system theoretic properties lpv systems are defined as the collection of trajectories of system variables like inputs and outputs and scheduling variables lpv kernel input output and state space system representations are introduced with appropriate equivalence transformations

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

on quadratic differential forms this paper develops a theory around the notion of quadratic differential forms in the context of linear differential systems in many applications we need to not only understand the behavior of the system variables but also the behavior of certain functionals of these variables the obvious cases where such functionals are important are in lyapunov theory and in lq and h infty optimal control with some exceptions these theories have almost invariably concentrated on first order models and state representations in this paper we develop a theory for linear time invariant differential systems and quadratic functionals we argue that in the context of systems described by one variable polynomial matrices the appropriate tool to express quadratic functionals of the system variables are two variable polynomial matrices the main achievement of this paper is a description of the interaction of one and two variable polynomial matrices for the analysis of functionals and for the application of higher order lyapunov functionals

the subspace nevanlinna interpolation problem and the most powerful unfalsified model a generalization of the tangential nevanlinna interpolation problem will be studied from a behavioral point of view necessary and sufficient conditions for its solvability and a characterization of all its solutions are derived these results are obtained by associating to the interpolation data a behavior that enjoys a special structure

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

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

non uniform warping including the effects of torsion and shear forces part ii analytical and numerical applications abstract this two part contribution presents a beam theory bt with a non uniform warping nuw including the effects of torsion and shear forces and valid for any homogeneous cross section made of isotropic elastic material in part i the governing equations of the nuw bt has been established and simplified nuw bt versions has been deduced wherein the number of degrees of freedom is reduced in this part ii these theories are used to analyze for a representative set of cross sections cs solid cs and thin walled open closed cs bi symmetric or not the elastic behavior of cantilever beams subjected to torsion or shear bending for bi symmetrical cs torsion and shear bending are analyzed separately analytical and numerical results are given for the distributions along the beam axis of the cross sectional displacements and stresses for the nuw bt and its simplified versions numerical results are also given for the three dimensional stress distributions close to the embedded section the stress predictions of the nuw bt are compared to those obtained by three dimensional finite elements computations it can be drawn from all these results indications that can help to decide when the simplified theories may be applied and hence when the warping parameters may be reduced as specified in nuw bt torsion and bending are coupled for non symmetrical cs even if the bending moments refer to the centroid while the torsional moment refers to the shear center to illustrate this coupling effect the particular example of the channel cs presented in kim and kim kim n i kim m y 2005 exact dynamic static stiffness matrices of non symmetric thin walled beams considering coupled shear deformation effects thin walled structures 43 701 734 is analyzed and the results are compared

a comparison and update of direct kinematic kinetic models of leg stiffness in human running abstract direct kinematic kinetic modelling currently represents the gold standard in leg stiffness quantification during three dimensional 3d motion capture experiments however the medial lateral components of ground reaction force and leg length have been neglected in current leg stiffness formulations it is unknown if accounting for all 3d would alter healthy biologic estimates of leg stiffness compared to present direct modelling methods this study compared running leg stiffness derived from a new method multiplanar method which includes all three cartesian axes against current methods which either only include the vertical axis line method or only the plane of progression uniplanar method twenty healthy female runners performed shod overground running at 5 0 m s three dimensional motion capture and synchronised in ground force plates were used to track the change in length of the leg vector hip joint centre to centre of pressure and resultant projected ground reaction force leg stiffness was expressed as dimensionless units as a percentage of an individual s bodyweight divided by standing leg length bw ll leg stiffness using the line method was larger than the uniplanar method by 15 6 bw ll p

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

leg stiffness and stride frequency in human running when humans and other mammals run the body s complex system of muscle tendon and ligament springs behaves like a single linear spring leg spring a simple spring mass model consisting of a single linear leg spring and a mass equivalent to the animal s mass has been shown to describe the mechanics of running remarkably well force platform measurements from running animals including humans have shown that the stiffness of the leg spring remains nearly the same at all speeds and that the spring mass system is adjusted for higher speeds by increasing the angle swept by the leg spring the goal of the present study is to determine the relative importance of changes to the leg spring stiffness and the angle swept by the leg spring when humans alter their stride frequency at a given running speed human subjects ran on treadmill mounted force platform at 2 5ms 1 while using a range of stride frequencies from 26 below to 36 above the preferred stride frequency force platform measurements revealed that the stiffness of the leg spring increased by 2 3 fold from 7 0 to 16 3 knm 1 between the lowest and highest stride frequencies the angle swept by the leg spring decreased at higher stride frequencies partially offsetting the effect of the increased leg spring stiffness on the mechanical behavior of the spring mass system we conclude that the most important adjustment to the body s spring system to accommodate higher stride frequencies is that leg spring becomes stiffer

reprint of a bilinear differential forms approach to parametric structured state space modelling abstract we use one variable loewner techniques to compute polynomial parametric models for mimo systems from vector exponential data gathered at various points in the parameter space instrumental in our approach are the connections between vector exponential modelling via bilinear differential forms and the loewner framework

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

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

effects of altered stride frequency and contact time on leg spring behavior in human running abstract many studies have demonstrated that contact time is a key factor affecting both the energetics and mechanics of running the purpose of the present study was to further explore the relationships between contact time t c step frequency f and leg stiffness k leg in human running since f is a compound parameter depending on both contact and aerial time the specific goal of this study was to independently vary f and t c and to investigate their respective effects on spring mass characteristics during running seeking to determine if the changes in k leg observed when running at different f are mainly due to inherent changes in t c we compared three types of constant 3 33 m s 1 running conditions in 10 male subjects normal running at the subject s freely chosen f running with decreased and increased f and decreased and increased t c at the imposed freely chosen f the data from the varied f trials showed that the variation of t c was strongly correlated to that of k leg r 2 0 90 and the variation of f was also significantly correlated to that of k leg r 2 0 47 further changes in t c obtained in various t c conditions were significantly correlated to changes in k leg r 2 0 96 these results confirm that leg stiffness was significantly influenced by step frequency variations during constant speed running as earlier demonstrated but our more novel finding is that compared to step frequency the effect of contact time variations appears to be a stronger and more direct determinant of k leg indeed 90 96 of the variance in k leg can be explained by contact time whether this latter parameter is directly controlled or indirectly controlled through its close relationship with step frequency in conclusion from the comparison of two experimental procedures i e imposing various step frequency conditions vs asking subjects to intentionally vary contact time at their freely chosen step frequency it appears that changes in leg stiffness are mainly related to changes in contact time rather than to those in step frequency step frequency appears to be an indirect factor influencing leg stiffness through its effect on contact time which could be considered a major determinant of this spring mass characteristic of human running

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

non uniform warping including the effects of torsion and shear forces part i a general beam theory this two part contribution presents a beam theory with a non uniform warping including the effects of torsion and shear forces and valid for any homogeneous cross section made of isotropic elastic material part i is devoted to the theoretical developments and part ii discusses analytical and numerical results obtained for torsion and shear bending of cantilever beams made of different kinds of cross section the theory is based on a kinematics assuming that the cross section maintains its shape and including three independent warping parameters associated to the three warping functions corresponding to torsion and shear forces starting from this displacement model and using the principle of virtual work the corresponding beam theory is derived for this theory closed form results are obtained for the cross sectional constants and the three dimensional expressions of the normal and shear stresses comparison with classical beam theories is carried out and additional effects due to the non uniformity of the warping are highlighted in particular the contributions of primary and secondary internal forces and the effect of the non symmetry of the cross section on the structural behavior of the beam are specified simplified versions of this theory wherein the number of degrees of freedom is reduced are also presented the analytical and numerical analyzes presented in part ii give responses on the quality of this non uniform beam theory and indicate also when its simplified versions could be applied

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

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

scattered data interpolation subject to piecewise quadratic range restrictions abstract the constriction of range restricted univariate and bivariate c1 interpolants to scattered data is considered in particular we deal with quadratic spline interpolation on a refined univariate grid respectively on a powell sabin refinement of a triangulation of the data sites subject to piecewise quadratic lower and upper obstacles to the values of the interpolant the derived sufficient conditions for the fulfillment of the range restrictions result in a system of linear inequalities for the slopes respectively gradients as parameters which is separated with respect to the data sites this system is shown to be always solvable for important special forms of the obstacles if at all in general there exist an infinite number of spline interpolants meeting the constraints the selection of a visually pleasant one is based on the minimization of a suitable choice functional

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

on the takagi interpolation problem in this paper we approach the interpolation problem of takagi from the perspective of time series modeling the framework of exact identification and the notion of mpum are used in order to generalize some results on the nevanlinna interpolation problem these results are of independent interest and concern the relation between the roots of the determinant of the denominator of an unfalsified model and the signature of the pick matrix of the data

a note on the form of the shear coefficient abstract in an earlier paper it was noted that the expressions derived for the shear stiffness coefficients for various cross sectional shapes all had a common general form it was found that poisson s ratio always appeared to be encapsulated within a simple rational function this led to the conjecture that this would prove to be true in general new results obtained by different means have supported this conjecture and have lead to a re examination of the theory yielding a proof which is valid for simply connected cross sections

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