exact closed form solution for the vibration modes of the euler bernoulli beam with multiple open cracks abstract in this study exact closed form expressions for the vibration modes of the euler bernoulli beam in the presence of multiple concentrated cracks are presented the proposed expressions are provided explicitly as functions of four integration constants only to be determined by the standard boundary conditions the enforcement of the boundary conditions leads to explicit expressions of the natural frequency equations besides the evaluation of the natural frequencies neither computational work nor recurrence expressions for the vibration modes are required the cracks that are not subjected to the closing phenomenon are modelled as a sequence of dirac s delta generalised functions in the flexural stiffness the eigen mode governing equation is formulated over the entire domain of the beam without enforcement of any continuity conditions which are already accounted for in the adopted flexural stiffness model the vibration modes of beams with different numbers of cracks under different boundary conditions have been analysed by means of the proposed closed form expressions in order to show their efficiency

the hu washizu variational principle for the identification of imperfections in beams this paper presents a procedure for the identification of imperfections of structural parameters based on displacement measurements by static tests the proposed procedure is based on the well known hu washizu variational principle suitably modified to account for the response measurements which is able to provide closed form solutions to some inverse problems for the identification of structural parameter imperfections in beams copyright 2008 john wiley sons ltd

a family of distributions on the circle with links to and applications arising from mobius transformation we propose a family of four parameter distributions on the circle that contains the von mises and wrapped cauchy distributions as special cases the family is derived by transforming the von mises distribution via a mobius transformation which maps the unit circle onto itself the densities in the family have a symmetric or asymmetric unimodal or bimodal shape depending on the values of the parameters conditions for unimodality are explored further properties of the proposed model are obtained many by applying the theory of mobius transformation properties of a three parameter symmetric submodel are investigated as well these include maximum likelihood estimation its asymptotics and a reparameterization that proves useful quite generally a three parameter asymmetric subfamily which often proves to be an adequate model is also discussed with emphasis on its mean direction and circular skewness the proposed family and subfamilies are used to model an asymmetrically distributed data set and ar

optimal selection of the forgetting matrix into an iterative learning control algorithm a recursive optimal algorithm based on minimizing the input error covariance matrix is derived to generate the optimal forgetting matrix and the learning gain matrix of a p type iterative learning control ilc for linear discrete time varying systems with arbitrary relative degree this note shows that a forgetting matrix is neither needed for boundedness of trajectories nor for output tracking in particular it is shown that in the presence of random disturbances the optimal forgetting matrix is zero for all learning iterations in addition the resultant optimal learning gain guarantees boundedness of trajectories as well as uniform output tracking in presence of measurement noise for arbitrary relative degree

physically based dirac s delta functions in the static analysis of multi cracked euler bernoulli and timoshenko beams abstract dirac s delta functions enable simple and effective representations of point loads and singularities in a variety of structural problems leading very often to elegant and otherwise unworkable closed form solutions this is the case of cracked beams under static loads whose theoretical and practical significance has attracted in recent years the interest of many researchers nevertheless analytical formulations currently available for this problem are not completely satisfactory either in terms of computational efficiency when the continuity conditions must be enforced with auxiliary equations or in terms of physical consistency when the singularities in the beam s flexural rigidity are represented with dirac s delta functions having a questionable negative sign these considerations motivate the present study which offers a novel and physically based modelling of slender euler bernoulli beams and short timoshenko beams with any number and severity of cracks conducing in both cases to exact closed form solutions for validation purposes a standard finite element code is used along with two nascent deltas uniform and gaussian density functions to describe a smeared increase in the bending flexibility around the abscissa of the crack

maximum response statistics of mdof linear structures excited by non stationary random processes the paper deals with the problem of predicting the maximum response statistics of multi degree of freedom mdof linear structures subjected to non stationary non white noises the extension of two different censored closures of gumbel type originally proposed by the authors for the response of single degree of freedom oscillators it is presented the improvement associated with the introduction in the closure of a consistent censorship factor accounting for the response bandwidth it is pointed out simple and effective step by step procedures are formulated and described in details numerical applications on a realistic 25 storey moment resisting frame along with comparisons with classical approximations and monte carlo simulations are also included

analysis of an interfacial crack in a piezoelectric bi material via the extended green s functions and displacement discontinuity method abstract based on the extended stroh formalism we first derive the extended green s functions for an extended dislocation and displacement discontinuity located at the interface of a piezoelectric bi material these include green s functions of the extended dislocation displacement discontinuities within a finite interval and the concentrated displacement discontinuities all on the interface the green s functions are then applied to obtain the integro differential equation governing the interfacial crack to eliminate the oscillating singularities associated with the delta function in the green s functions we represent the delta function in terms of the gaussian distribution function in so doing the integro differential equation is reduced to a standard integral equation for the interfacial crack problem in piezoelectric bi material with the extended displacement discontinuities being the unknowns a simple numerical approach is also proposed to solve the integral equation for the displacement discontinuities along with the asymptotic expressions of the extended intensity factors and j integral in terms of the discontinuities near the crack tip in numerical examples the effect of the gaussian parameter on the numerical results is discussed and the influence of different extended loadings on the interfacial crack behaviors is further investigated

a tractable and interpretable four parameter family of unimodal distributions on the circle this article presents a class of four parameter distributions for circular data that are unimodal possess simple characteristic and density functions and a tractable distribution function can be interpretably parameterized directly in terms of their trigonometric moments afford a very wide range of skewness and kurtosis envelop numerous interesting submodels including the wrapped cauchy and cardioid distributions allow straightforward parameter estimation by both method of moments and maximum likelihood and are closed under convolution this class of distributions exhibits the widest range of attractive properties yet available while retaining unimodality

on applications of generalized functions to beam bending problems using a mathematical approach this paper seeks an eacient solution to the problem of beams bending under singular loading conditions and having various jump discontinuities for two instances the boundary value problem that describes beam bending cannot be written in the space of classical functions in the first instance the beam is under singular loading conditions such as point forces and moments and in the second instance the dependent variable s and its derivatives have jump discontinuities in the most general case we consider both instances first we study singular loading conditions and present a theorem by which the equivalent distributed force of a general class of singular loading conditions can be found as a consequence of obtaining the equivalent distributed force of a distributed moment we find a mathematical explanation for the corner condition in classical plate theory while plate theory is not the focus of this paper this explanation is interesting then beams with various jump discontinuities are considered when beams have jump discontinuities the form of the governing diaerential equations changes we find the governing diaerential equations in the space of generalized functions it is shown that for euler bernoulli beams with jump discontinuities the operator of the diaerential equation remains unchanged only the force term changes so that delta function and its distributional derivatives appear within it but for timoshenko beams with jump discontinuities in addition to changes in the force terms the operator of one of the governing diaerential equations changes we then propose a new method for solving these equations this method which we term the auxiliary beam method is to solve the governing diaerential equations not in the space of generalized functions but rather to solve them by means of solving equivalent boundary value problems in the space of classical functions the auxiliary beam method reduces the number of diaerential equations and at the same time obviates the need to solve these diaerential equations in the space of generalized functions which can be more diacult 7 2000 elsevier science ltd all rights reserved

brief paper noise tolerant iterative learning control for a class of continuous time systems the paper proposes a noise tolerant iterative learning control ilc for a class of linear continuous time systems which achieves high precision tracking for uncertain plants by iteration of trials in the presence of heavy measurement noise the robustness against measurement noise is achieved through i projection of continuous time i o signals onto a finite dimensional parameter space ii using error data of all past iterations via an integral operation in the learning law and iii noise reduction by h 2 optimization subject to a specified convergence speed of the ilc

closed form solutions for frequency equation and mode shapes of elastically supported euler bernoulli beams abstract this paper is concerned with the problem of free vibration of an euler bernoulli beam supported with an arbitrary number of translational springs of different constant stiffnesses the natural frequencies and mode shapes are calculated by means of the green s function method seven different boundary conditions are considered at both ends of the beam fixed pinned sliding free translational spring rotational spring and combined translational rotational spring each boundary condition on one end is combined with all other boundary conditions on the other yielding a total of 49 possible cases it has been shown in this paper that a green s function for any set of boundary conditions can be represented as a linear combination of function coefficients calculated for homogeneous boundary conditions and also a new form of the general solution is formulated using this principle green s functions are calculated and tabulated for each of the 49 possible cases which in essence closes the question of analytical solutions for this particular kind of problem using green s functions the tabulated solutions were used as the starting point in the derivation of the general formulae and to provide a new insight into the underlying mathematical structure of the solutions formulated with green s functions this can be of practical interest to the end user who wishes to implement them in computer code the validity of the obtained solutions is demonstrated with examples whose solutions are compared to the results in the published literature and to the results obtained with fem

multi cracked euler bernoulli beams mathematical modeling and exact solutions localized flexibility models of cracks enable one for simple and effective representation of the behavior of damaged beams and frames important applications such as the determination of closed form solutions and the development of diagnostic methods of analysis have attracted the interest of many researchers in recent years nevertheless certain fundamental questions have not been completely clarified yet one of these issues concerns with the mechanical justification of the macroscopic model of rotational elastic spring commonly used to describe the presence of an open crack in a beam under bending deformation two main analytical formulations have been recently proposed to take into account the singularity generated by the crack the crack is represented by suitable dirac s delta functions either in the beam s flexural rigidity or in the beam s flexural flexibility both approaches require some caution due to mathematical subtleties of the analysis motivated by these considerations in this paper we propose a justification of the rotational elastic spring model of an open crack in a beam in bending deformation we show that this localized flexibility model of a crack is the variational limit of a family of one dimensional beams when the flexural stiffness of these beams tends to zero in an interval centered at the cracked cross section and simultaneously the length of the interval vanishes in a suitable way we also show that the static and dynamic problem for the flexibility model of cracked beam can be formulated within the classical context of the theory of distributions avoiding the hindrances encountered in previous approaches to the problem in addition the proposed treatment leads to a simple and efficient determination of exact closed form solutions of both static and dynamic problems for beams with multiple cracks

on a discrete time stochastic learning control algorithm the learning gain for a selected learning algorithm is derived based on minimizing the trace of the input error covariance matrix for linear time varying systems it is shown that if the product of the input output coupling matrices is a full column rank then the input error covariance matrix converges uniformly to zero in the presence of uncorrelated random disturbances however the state error covariance matrix converges uniformly to zero in presence of measurement noise moreover it is shown that if a certain condition is met then the knowledge of the state coupling matrix is not needed to apply the proposed stochastic algorithm the proposed algorithm is shown to suppress a class of nonlinear and repetitive state disturbance the application of this algorithm to a class of nonlinear systems is also considered a numerical example is included to illustrate the performance of the algorithm

multiplication of distributions in one dimension possible approaches and applications to δ function and its derivatives we introduce a new class of multiplications of distributions in one dimension merging two different regularizations of distributions some of the features of these multiplications are discussed in detail we use our theory to study a number of examples involving products between dirac delta functions and its successive derivatives

closed form solutions of a multi cracked circular arch under static loads abstract generalised functions have been widely adopted in structural mechanics to treat singularities of beam like structures however due to the curved geometry that couples axial and transversal displacements their use has never been explored for curved beams in this paper the capability of distributions of leading to closed form exact solutions for multi cracked circular arch is shown the exact closed form solution of a circular euler arch in presence of any number of discontinuities due to concentrated damage and subjected to an arbitrary distribution of static loads is obtained damage under the form of concentrated cracks has been modeled through the widely adopted and validated equivalent elastic hinge concept and has been introduced in the governing differential equations by making use of dirac s delta functions the resulting nontrivial generalised six order differential equations have been derived and solved in closed form independently of the number of along arch concentrated cracks the solution is expressed as a function of six integration constants only in which the damage positions and intensities are given data appearing explicitly in the solution expression this latter aspect constitutes a fundamental aid towards the resolution of the static damage inverse identification problem the results have been validated through some comparisons with finite element numerical simulations examples referred to multi cracked euler arches with different boundary conditions damage and load scenarios are presented

inverse batschelet distributions for circular data we provide four parameter families of distributions on the circle which are unimodal and display the widest ranges of both skewness and peakedness yet available our approach is to transform the scale of a generating distribution such as the von mises using various nontrivial extensions of an approach first used in batschelet s 1981 circular statistics in biology book the key is to employ inverses of batschelet type transformations in certain ways these exhibit considerable advantages over direct batschelet transformations the skewness transformation is especially appealing as it has no effect on the normalizing constant as well as a variety of interesting theoretical properties when likelihood inference is explored these distributions display orthogonality between elements of a pairing of parameters into location skewness and concentration peakedness further the location parameter can sometimes be made approximately orthogonal to all the other parameters profile likelihoods come to the fore in practice two illustrative applications one concerning the locomotion of a drosophila fly larva the other analyzing a large set of sudden infant death syndrome data are investigated

an exact generalised function approach to frequency response analysis of beams and plane frames with the inclusion of viscoelastic damping abstract this paper concerns the frequency response analysis of beams and plane frames with an arbitrary number of kelvin voigt viscoelastic dampers typical external and internal dampers are considered as grounded translational tuned mass rotational and axial dampers for bending and axial vibrations respectively using the theory of generalised functions within a 1d formulation of equations of motion exact closed form expressions are derived for beam dynamic green s functions and frequency response functions under arbitrary polynomial load for any number of dampers for a plane frame exact global frequency response matrix and load vector are built with size depending only on the number of beam to column nodes for any number of dampers and point polynomial loads along the frame members from the nodal displacement solution the exact frequency response in all frame members is also obtained in closed analytical form numerical applications show many of the advantages of the proposed method

sinh arcsinh distributions we introduce the sinh arcsinh transformation and hence by applying it to a generating distribution with no parameters other than location and scale usually the normal a new family of sinh arcsinh distributions this four parameter family has symmetric and skewed members and allows for tailweights that are both heavier and lighter than those of the generating distribution the central place of the normal distribution in this family affords likelihood ratio tests of normality that are superior to the state of the art in normality testing because of the range of alternatives against which they are very powerful likelihood ratio tests of symmetry are also available and are very successful three parameter symmetric and asymmetric subfamilies of the full family are also of interest heavy tailed symmetric sinh arcsinh distributions behave like johnson su distributions while their light tailed counterparts behave like sinh normal distributions the sinh arcsinh family allowing a seamless transition between the two via the normal controlled by a single parameter the sinh arcsinh family is very tractable and many properties are explored likelihood inference is pursued including an attractive reparameterization illustrative examples are given a multivariate version is considered options and extensions are discussed

closed form solutions of euler bernoulli beams with singularities abstract the problem of the integration of the static governing equations of the uniform euler bernoulli beam with discontinuities is studied in particular two types of discontinuities have been considered flexural stiffness and slope discontinuities both the above mentioned discontinuities have been modeled as singularities of the flexural stiffness by means of superimposition of suitable distributions generalized functions to a uniform one dimensional field closed form solutions of governing differential equation requiring the knowledge of the boundary conditions only are proposed and no continuity conditions are enforced at intermediate cross sections where discontinuities are shown the continuity conditions are in fact embedded in the flexural stiffness model and are automatically accounted for by the proposed integration procedure finally the proposed closed form solution for the cases of slope discontinuity is compared with the solution of a beam having an internal hinge with rotational spring reproducing the slope discontinuity

fixed point theorem based iterative learning control for ltv systems with input singularity in this paper we address a challenging and open problem how to design a suitable iterative learning control ilc system in the presence of input singularity which is incurred by the singularities of the system direct feed through term considering two typical types of input singularities we first revise the ilc operators accordingly by adding a forgetting factor and incorporating a time varying learning gain in the sequel guarantee ilc operators to be contractible next using the banach fixed point theorem we demonstrate that the output sequence can either enter and remains ultimately in a designated neighborhood of the target trajectory or is bounded by a class k function finally an illustrative example is presented

generalized solutions to a linear discontinuous differential equation in this paper we construct generalized solutions weak solutions to a linear discontinuous differential equation which appears as mathematical model in mechanics and for which usual methods are not quite adequate the result is applied to the equation of an elastic beam on a winkler type of foundation

exact solution of the multi cracked euler bernoulli column abstract the use of distributions generalized functions is a powerful tool to treat singularities in structural mechanics and besides providing a mathematical modelling their capability of leading to closed form exact solutions is shown in this paper in particular the problem of stability of the uniform euler bernoulli column in presence of multiple concentrated cracks subjected to an axial compression load under general boundary conditions is tackled concentrated cracks are modelled by means of dirac s delta distributions an integration procedure of the fourth order differential governing equation which is not allowed by the classical distribution theory is proposed the exact buckling mode solution of the column as functions of four integration constants and the corresponding exact buckling load equation for any number position and intensity of the cracks are presented as an example a parametric study of the multi cracked simply supported and clamped clamped euler bernoulli columns is presented

dynamics of multiple cracked prismatic beams with uncertain but bounded depths under deterministic and stochastic loads abstract dynamic response of cracked beams changes with respect to undamaged ones due to the change in mechanical properties moreover usually the crack characteristics crack size crack position applied stress etc are assumed to be known however in practice they possess considerable scatter of uncertainty the aim of this paper is to evaluate the time varying upper and lower bounds of the response of damaged beams with multiple cracks subjected to deterministic or stochastic excitations assuming uncertain crack depths as interval variables in this study the analysis is performed adopting a finite element model for the damaged beam extended to the case where multiple non propagating cracks are presented in order to provide the bounds of the response the authors propose an approach developed in the time domain starting from the derivation of the bounds of the interval eigenvalues evaluated as solution of two appropriate deterministic eigenvalue problems to assess the efficiency and the accuracy of the presented method a damaged cantilever prismatic aluminum beam with multiple uncertain but bounded cracks is examined the bounds of the free end displacement and variance are calculated and compared with the exact solution via combinatorial procedure the proposed results show a high level of accuracy for both increasing amplitude value of interval deviation and increasing number of uncertain parameters

on euler bernoulli discontinuous beam solutions via uniform beam green s functions abstract the bending problem of euler bernoulli discontinuous beams is dealt with the purpose is to show that uniform beam green s functions can be used to build efficient solutions for beams with internal discontinuities due to along axis constraints and flexural stiffness jumps specifically upon deriving the equilibrium equation in the space of generalized functions first it is seen that the original bending problem may be recast as linear superposition of a principal and an auxiliary bending problem both involving a uniform reference beam and homogeneous boundary conditions then based on the green s functions of the reference beam closed form solutions are developed for the principal beam response while the auxiliary beam response is obtained by solving in general r 2 s algebraic equations written at the discontinuity locations being r the number of discontinuities due to along axis constraints and s the number of flexural stiffness jumps in this manner an appreciable reduction of computational effort is achieved as compared to alternative analytical solutions in the literature

computational analysis of multi stepped beams and beams with linearly varying heights implementing closed form finite element formulation for multi cracked beam elements abstract the model where the cracks are represented by means of internal hinges endowed with rotational springs has been shown to enable simple and effective representation of transversely cracked slender euler bernoulli beams subjected to small deflections it namely provides reliable results when compared to detailed 2d and 3d models even if the basic linear moment rotation constitutive law is adopted this paper extends the utilisation of this model as it presents the derivation of a closed form stiffness matrix and a load vector for slender multi stepped beams and beams with linearly varying heights the principle of virtual work allows for the simple inclusion of an arbitrary number of transverse cracks the derived at matrix and vector define an exact finite element for the utilised simplified computational model the presented element can be implemented for analysing multi cracked beams by using just one finite element per structural beam member the presented expressions for a stepped beam are not exclusively limited to this kind of height variation as by proper discretisation an arbitrary variation of a cross section s height can be adequately modelled the accurate displacement functions presented for both types of considered beams complete the derivations all the presented expressions can be easily utilised for achieving computationally efficient and truthful analyses

vibration of a beam with a breathing crack a continuous cracked beam vibration theory is used for the prediction of changes in transverse vibration of a simply supported beam with a breathing crack the equation of motion and the boundary conditions of the cracked beam considered as a one dimensional continuum were used the eigenfrequency changes due to a breathing edge crack are shown to depend on the bi linear character of the system the associated linear problems are solved over their respective domains of definition and the solutions are matched through the initial conditions the changes in vibration frequencies for a fatigue breathing crack are smaller than the ones caused by open cracks the method has been tested for the evaluation of the lowest natural frequency of lateral vibration for beams with a single edge breathing crack experimental results from aluminium beams with fatigue cracks are used for comparison with the analytical results

stochastic p type d type iterative learning control algorithms this paper presents stochastic algorithms that compute optimal and sub optimal learning gains for a p type iterative learning control algorithm ilc for a class of discrete time varying linear systems the optimal algorithm is based on minimizing the trace of the input error covariance matrix the state disturbance reinitialization errors and measurement errors are considered to be zero mean white processes it is shown that if the product of the input output coupling matrices c t 1 b t is full column rank then the input error covariance matrix converges to zero in presence of uncorrelated disturbances another sub optimal p type algorithm which does not require the knowledge of the state matrix is also presented it is shown that the convergence of the input error covariance matrices corresponding to the optimal and sub optimal p type and d type algorithms are equivalent and all converge to zero at a rate inversely proportional to the number of learning iterations a transient response pe

meshfree co rotational formulation for two dimensional continua in this paper a meshfree co rotational formulation for two dimensional continua is proposed in a co rotational formulation the motion of a body is separated into rigid motion and strain producing deformation traditionally this has been done in the setting of finite elements for beams and shell type elements in the present work every node in a meshfree discretized domain has its own co rotating coordinate system three key ingredients are established in order to apply the co rotational formulation i the relationship between global and local variables ii the angle of rotation of a typical co rotating coordinate system and iii a variationally consistent tangent stiffness matrix an algorithm for the co rotational formulation based on load control is provided maximum entropy basis functions are used to discretize the domain and stabilized nodal integration is implemented to construct the global system of equations numerical examples are presented to demonstrate the validity of the meshfree co rotational formulation copyright 2009 john wiley sons ltd

detection of damage in beams subjected to diffused cracking this paper addresses the problem of identifying structural damage affecting one zone of a beam using measured frequencies the beam model has a zone in which the stiffness is lower than the undamaged value damage is defined by three parameters position extension and degree which need to be identified in the inverse problem the solution of the direct problem is first obtained and the peculiarities of damage detection are examined two different procedures for damage identification are proposed which use frequency measurements and take advantage of the peculiarities of the problem the first procedure is based on the characteristic equation error and the second on the comparison between analytical and experimental frequency values the identifiability and ill conditioning properties are discussed by referring to cases with pseudo experimental data

absolute versus relative formulations of the moving oscillator problem in the general framework of the bridge vehicle dynamic interaction the so called moving oscillator problem is revisited in order to provide a deeper insight into some analytical and physical aspects not specifically analyzed in previous investigations without lack of generality the case of a stream of moving oscillators crossing a simply supported beam with arbitrary time law is considered the formulations in terms of both absolute and relative displacements of the moving oscillators are critically reviewed and compared and alternative sets of differential equations with time dependent coefficients are derived the study enlightens both theoretically and numerically that impulsive contributions to the dynamic response appear in the relative displacement formulation at the time instants in which each vehicle enters or exits the bridge it is demonstrated that such contributions somehow hidden in the absolute displacement formulation may have a significant influence on the vibration of the moving oscillators and thus cannot be a priori neglected in the analysis it is also shown that the analytical and computational difficulties associated with these additional impulses make preferable the use of the absolute displacement formulation far from being restricted to the case of simply supported beams these findings are valid for any type of bridge structure which induces a discontinuity in the slope of the road profile experienced by the vehicles

a family of symmetric distributions on the circle we propose a new family of symmetric unimodal distributions on the circle that contains the uniform von mises cardioid and wrapped cauchy distributions among others as special cases the basic form of the densities of this family is very simple although its normalization constant involves an associated legendre function the family of distributions can also be derived by conditioning and projecting certain bivariate spherically and elliptically symmetric distributions on to the circle trigonometric moments are available and a measure of variation is discussed aspects of maximum likelihood estimation are considered and likelihood is used to fit the family of distributions to an example set of data finally extension to a family of rotationally symmetric distributions on the sphere is briefly made

a stochastic iterative learning control algorithm with application to an induction motor a recursive optimal algorithm based on minimizing the input error covariance matrix is derived to generate the learning gain matrix of a p type ilc for linear discrete time varying systems with arbitrary relative degree it is shown that in the case where the number of inputs is not greater than the number of outputs the input error covariance matrix converges to zero at a rate inversely proportional to the number of iterations in the presence of uncorrelated random state disturbance reinitialization errors and measurement noise the state error covariance matrix converges to zero at a rate inversely proportional to the number of iterations in the presence of measurement noise in the case where the number of inputs is greater than the number of outputs then the system output error converges to zero at a rate inversely proportional to the number of iterations in presence of measurement noise another suboptimal recursive algorithm is also proposed based on unknown system dynamics and unknown disturb

on nonuniform euler bernoulli and timoshenko beams with jump discontinuities application of distribution theory in this article bending of nonuniform euler bernoulli and timoshenko beams with jump discontinuities in the slope deflection and mechanical properties are studied the governing equations are obtained in the space of generalized functions and the expression of its governing differential equations in terms of a single displacement function and a single rotation function is shown always to be possible in contrast for a nonuniform euler bernoulli beam with jump discontinuities in slope and deflection and abrupt changes in flexural stiffness the governing equation can be written in terms of a single displacement function only under certain conditions it is observed that for most discontinuous nonuniform euler bernoulli beams we cannot write the governing differential equation in terms of a single displacement function usually if there are n discontinuity points on a nonuniform euler bernoulli beam n 1 displacement functions appear in the governing equilibrium equation