second order linear plate theories partial differential equations stress resultants and displacements abstract by combining the uniform approximation technique with the pseudo reduction approach a consistent second order plate theory is developed with recourse to neither kinematical assumptions nor to shear correction factors by truncation of the elastic energy the governing partial differential equations and the expressions for the stress resultants are compared with those of other authors free coefficients of the resulting displacement ansatz are determined a posteriori in order to satisfy three dimensional boundary conditions and local equilibrium equations

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

a new simple third order shear deformation theory of plates an improved simple third order shear deformation theory for the analysis of shear flexible plates is presented in this paper this new plate theory is composed of three parts the simple third order kinematics of displacements reduced from the higher order displacement field derived previously by the author a system of 10th order differential equilibrium equations in terms of the three generalized displacements of bending plates five boundary conditions at each edge of plate boundaries although the resulting displacement field is the same as that proposed by murthy the variational consistent governing equations and the associated proper boundary conditions are derived and identified in this work for the first time in the literature the applications and accuracy of the present shear deformation theory of plates are demonstrated by analytically solving the differential governing equations of a twisting plate a bending beam and two bending plates to which the 3 d elasticity solutions are available and excellent agreements are achieved even for the torsion of a plate with square cross section as well the local effects of stresses at plate boundaries can be characterized accurately these analytical solutions clearly show that the simple third order shear deformation theory developed in this work indeed gives better results than the first order shear deformation theories and other simple higher order shear deformation theories since the present third order shear flexible theory is based on a more rigorous kinematics of displacements and consists of not only a system of variational consistent differential equations but also a group of consistent boundary conditions associated with the differential equations the present simple third order shear deformation theory can easily be applied to the static and dynamic finite element analysis of laminated plates just like the applications of other popular shear flexible plate theories and improved results could be obtained from the present simple third order shear deformable theories of plates

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

a quadrilateral element based on refined global local higher order theory for coupling bending and extension thermo elastic multilayered plates in this paper a refined higher order global local theory is presented to analyze the laminated plates coupled bending and extension under thermo mechanical loading the in plane displacement fields are composed of a third order polynomial of global coordinate z in the thickness direction and 1 2 3 order power series of local coordinate fk in the thickness direction of each layer which is identical to the 1 2 3 global local higher order theory by li and liu li x y liu d 1997 generalized laminate theories based on double superposition hypothesis int j numer methods eng 40 1197 1212 moreover a second order polynomial of global coordinate z in the thickness direction is chosen as transverse displacement field the transverse shear stresses can satisfy continuity at interfaces and the number of unknowns does not depend on the layer numbers of the laminate based on this theory a quadrilateral laminated plate element satisfying the requirement of c 1 continuity is presented by solving both bending and thermal expansion problems of laminates it can be found that the present refined theory is very accurate and obviously superior to the existing 1 2 3 global local higher order theory the most attractive feature of this theory is that the transverse shear stresses can be accurately predicted from direct use of constitutive equations without any post processing method it is also shown that the present quadrilateral element possesses higher accuracy 2006 elsevier ltd all rights reserved

free vibrations of plate using two variable refined plate theory the novelty of this paper is the use of two variable refined plate theory for free vibration analysis of plate unlike any other theory the theory is variationally consistent and gives two governing equations which are only inertially coupled and there is no elastic coupling at all number of unknown functions involved is only two as against three in case of mindlin s theory the theory has strong similarity with classical plate theory in many aspects the theory does not require shear correction factor and transverse shear stress variation is parabolic across the thickness simple variant of the theory is also presented

exact closed form frequency equations for thick circular plates using a third order shear deformation theory abstract this paper presents for the first time exact closed form frequency equations and transverse displacement for thick circular plates with free soft simply supported hard simply supported and clamped boundary conditions based on reddy s third order shear deformation theory hamiltonian and minimum potential energy principles are used to extract the equations of dynamic equilibrium and natural boundary conditions of the plate the new formulation is verified by comparing the results with their counterparts reported in open literature natural frequencies of circular plates with different boundary conditions are tabulated in dimensionless form for various values of thickness radius ratios the results presented on the basis of exact closed form frequency equations are expected to serve as reliable benchmarks

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 solvability of second kind integral equations on the real line abstract this paper considers general second kind integral equations of the form s r k s t t dt s in operator form k k where the functions k and are assumed known with y the space of bounded continuous functions on r and k such that the mapping s k s from r to l1 r is bounded and continuous the function y is the solution to be determined conditions on a set w bc r l1 r are obtained such that a generalised fredholm alternative holds if w satisfies these conditions and i k k is injective for all k w then i k k is also surjective for all k w and moreover the inverse operators i k k 1 on y are uniformly bounded for k w the approximation of the kernel in the integral equation by a sequence kn converging in a weak sense to k is also considered and results on stability and convergence are obtained these general theorems are used to establish results for two special classes of kernels k s t s t z t and k s t s t s t t where l1 r z l r and bc r 0 r kernels of both classes arise in problems of time harmonic wave scattering by unbounded surfaces the general integral equation results are here applied to prove the existence of a solution for a boundary integral equation formulation of scattering by an infinite rough surface and to consider the stability and convergence of approximation of the rough surface problem by a sequence of diffraction grating problems of increasingly large period

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

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

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

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

scattering by infinite one dimensional rough surfaces we consider the dirichlet boundary value problem for the helmholtz equation in a non locally perturbed half plane this problem models time harmonic electromagnetic scattering by a one dimensional infinite rough perfectly conducting surface the same problem arises in acoustic scattering by a sound soft surface chandler wilde and zhang have suggested a radiation condition for this problem a generalization of the rayleigh expansion condition for diffraction gratings and uniqueness of solution has been established recently an integral equation formulation of the problem has also been proposed and in the special case when the whole boundary is both lyapunov and a small perturbation of a flat boundary the unique solvability of this integral equation has been shown by chandler wilde and ross by operator perturbation arguments in this paper we study the general case with no limit on surface amplitudes or slopes and show that the same integral equation has exactly one solution in the space of bounded and continuous functions for all wavenumbers as an important corollary we prove that for a variety of incident fields including the incident plane wave the dirichlet boundary value problem for the scattered field has a unique solution

differential quadrature method for nonlinear vibration of orthotropic plates with finite deformation and transverse shear effect based on the reddy s theory of plates with the effect of higher order shear deformations the governing equations for nonlinear vibration of orthotropic plates with finite deformations are presented the nonlinear free vibration is analyzed by the differential quadrature method the differential quadrature approach suggested by wang and bert is extended to handle the multiple boundary conditions of the plate a new technique is also further extended to simplify nonlinear computations and the harmonic balance method is used in deriving the equation of motion the numerical convergence and comparison studies are carried out to validate the present solutions the results show that the presented differential quadrature method is fairly reliable and valid influences of geometric and material parameters transverse shear deformations and rotation inertia as well as vibration amplitudes on the nonlinear free vibration characteristics of orthotropic plates are studied

a uniqueness result for scattering by infinite rough surfaces consider the dirichlet boundary value problem for the helmholtz equation in a non locally perturbed half plane with an unbounded piecewise lyapunov boundary this problem models time harmonic electromagnetic scattering in transverse magnetic polarization by one dimensional rough perfectly conducting surfaces a radiation condition is introduced for the problem which is a generalization of the usual one used in the study of diffraction by gratings when the solution is quasi periodic and allows a variety of incident fields including an incident plane wave to be included in the results obtained we show in this paper that the boundary value problem for the scattered field has at most one solution for the case when the whole boundary is lyapunov and is a small perturbation of a flat boundary we also prove existence of solution and show a limiting absorption principle

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

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

a two variable refined plate theory for orthotropic plate analysis a new theory which involves only two unknown functions and yet takes into account shear deformations is presented for orthotropic plate analysis unlike any other theory the theory presented gives rise to only two governing equations which are completely uncoupled for static analysis and are only inertially coupled i e no elastic coupling at all for dynamic analysis number of unknown functions involved is only two as against three in case of simple shear deformation theories of mindlin and reissner the theory presented is variationally consistent has strong similarity with classical plate theory in many aspects does not require shear correction factor gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions well studied examples available in literature are solved to validate the theory the results obtained for plate with various thickness ratios using the theory are not only substantially more accurate than those obtained using the classical plate theory but are almost comparable to those obtained using higher order theories having more number of unknown functions

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

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

nonlinear free vibration analysis of composite plates with material uncertainties a monte carlo simulation approach abstract in this paper the effects of dispersion in material properties on free vibration response of composite plates with geometric nonlinearity in von karman sense are investigated the higher order shear deformation theory is employed for the study reported here an efficient c0 finite element formulation is developed for the analysis using monte carlo simulation the second order statistics i e mean and standard deviation of the nonlinear free vibration response of the composite laminates are obtained for different thickness and amplitude ratios the input statistics mean and variance of the material property are assumed to be known as a priori the computed results demonstrate the influence of the variations in the material properties and amplitudes on the nonlinear free vibration response of the composite plates the results are compared with those available in the literature

stability and natural vibration analysis of laminated plates by using a mixed element based on a refined plate theory abstract a mixed shear flexible finite element with relaxed continuity is developed for the geometrically linear and non linear analysis of layered anisotropic plates the element formulation is based on a refined higher order theory which satisfies the zero transverse shear stress boundary conditions on the top and bottom faces of the plate and requires no shear correction coefficients the mixed finite element developed herein consists of eleven degrees of freedom per node which include three displacements two rotations and six moment resultants the element is evaluated for its accuracy in the analysis of the stability and vibration of anisotropic rectangular plates with different lamination schemes and boudary conditions the mixed finite element described here for the higher order theory gives very accurate results for buckling loads and natural frequencies

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 note on generalized g matrices abstract in this paper we slightly generalize the notion of g matrices which has been recently introduced a real nonsingular matrix a is called a g matrix if there exist nonsingular diagonal matrices d 1 and d 2 such that d 1 a t d 2 a 1 we generalize this definition to the case where a can be singular we say that a real matrix a which is not necessarily square is a generalized g matrix gg matrix if there exist nonsingular diagonal matrices d 1 and d 2 such that d 1 a t d 2 is a g inverse of a the main purpose of this paper is to show that any generalized cauchy matrix is a gg matrix

generalized power law exponent based shear deformation theory for free vibration of functionally graded beams free vibration of functionally graded fg beams is presented based on a newly proposed shear deformation theory pesdbt the material properties of the beam vary along the thickness direction according to the power law form rayleigh ritz method is used in the modeling to obtain the eigenfrequency equation relationships of different shear deformation theories with pesdbt have also been noticed and incorporated here results in special cases are compared and new results are also reported present study deals with free vibration of functionally graded fg beams subject to different sets of boundary conditions the analysis is carried out on the basis of a newly proposed generalized power law exponent based shear deformation beam theory pesdbt the proposed theory exactly satisfies the transverse stress boundary conditions on the bottom and top surfaces of the beam material properties of the beam vary continuously in the thickness direction according to the power law exponent form the displacement components of the cross sections of the beam are expressed in simple algebraic polynomials rayleigh ritz method is used to estimate frequency parameters in order to handle various sets of boundary conditions at the edges the objective is to study the effects of constituent volume fractions slenderness ratios and the beam theories on the natural frequencies new results for frequency parameters are also incorporated after checking the convergence pattern and validation of results computed for previously published shear deformation beam theories

on timoshenko reissner type theories of plates and shells abstract linear tr theories timoshenko reissner theories of isotropic plates and shells are discussed these theories take into account the transversal shear deformation and rotation inertia the main subject under consideration is the construction of these theories by the asymptotic method and the related error estimates for static and dynamic problems in the dynamic case a method is suggested for the extension of the range of applicability of the tr theory

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

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

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

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

combined matrices in special classes of matrices abstract the combined matrix of a nonsingular matrix a is the matrix a a 1 t where means the hadamard entrywise product it has simple properties its row as well as column sums are always one and it is not changed if a is multiplied from either side by a nonsingular diagonal matrix although it is usually difficult to compute its further properties deserve attention in the paper we concentrate on the sequence of the diagonal entries of combined matrices in various classes of matrices

refined global local higher order theory and finite element for laminated plates based on completely three dimensional elasticity theory a refined global local higher order theory is presented as enhanced version of the classical global local theory proposed by li and liu int j numer meth engng 1997 40 1197 1212 in which the effect of transverse normal deformation is enhanced compared with the previous higher order theory the refined theory offers some valuable improvements these are able to predict accurately response of laminated plates subjected to thermal loading of uniform temperature however the previous higher order theory will encounter difficulty for this problem a refined three noded triangular element satisfied the requirement of c1 weak continuity conditions in the inter element is also presented the results of numerical examples of moderately thick laminated plates and even thick plates with span thickness ratios l h 2 are given to show that in plane stresses and transverse shear stresses can be reasonably predicted by the direct constitutive equation approach without smooth technique in order to accurately obtain transverse normal stresses the local equilibrium equation approach in one element is employed herein copyright 2006 john wiley sons ltd

a new approach to the analysis of rough surface scattering a novel approach to the problem of scattering by a randomly rough surface is developed based on combining normalization and the method of smoothing previous uses of smoothing have been forced to assume small surface heights in order to guarantee the dominance of the scattered field by its average or specular part the term that causes the average scattered field to be so sensitive to the height is identified and is normalized out of the basic integral equation smoothing is then applied to the normalized integral equation with a subsequent reintroduction of the normalizing factor after smoothing the results are shown to extend beyond existing approximations and they contain none of the divergent integral behavior that has recently been observed with a pure second order iterative approximation an examination of the conditions under which the technique provides very accurate results shows that it is essentially an extension of the so called composite surface scattering model

further results on cauchy tensors and hankel tensors in this article we present various new results on cauchy tensors and hankel tensors we first introduce the concept of generalized cauchy tensors which extends cauchy tensors in the current literature and provide several conditions characterizing positive semi definiteness of generalized cauchy tensors with nonzero entries furthermore we prove that all even order generalized cauchy tensors with positive entries are completely positive tensors which means every such that generalized cauchy tensor can be decomposed as the sum of nonnegative rank 1 tensors we also establish that all the h eigenvalues of nonnegative cauchy tensors are nonnegative secondly we present new mathematical properties of hankel tensors we prove that an even order hankel tensor is vandermonde positive semi definite if and only if its associated plane tensor is positive semi definite we also show that if the vandermonde rank of a hankel tensor a is less than the dimension of the underlying space then positive semi definiteness of a is equivalent to the fact that a is a complete hankel tensor and so is further equivalent to the sos property of a thirdly we introduce a new class of structured tensors called cauchy hankel tensors which is a special case of cauchy tensors and hankel tensors simultaneously sufficient and necessary conditions are established for an even order cauchy hankel tensor to be positive definite

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

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

a new approach to plate theory based on through thickness homogenization a new approach to linear plate theory for layered structures is introduced the new theory distances itself from the classical strategy where through the thickness displacement functions are assumed a priori instead the homogenization is based on a single assumption that the plate response is a superposition of fundamental states an equivalent single layer esl formulation is presented which for the first time allows accurate three dimensional stress and strain fields to be recovered through a postprocessing strategy that is consistent with the theory s assumptions copyright 2010 john wiley sons ltd

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

a hierarchy of dynamic equations for micropolar plates this work considers homogeneous isotropic micropolar plates adopting a power series expansion method in the thickness coordinate variationally consistent equations of motion and end boundary conditions are derived in a systematic fashion up to arbitrary order for extensional and flexural displacement cases the plate equations are asymptotically correct to all studied orders numerical results are presented for various orders of the present method other approximate theories as well as the exact three dimensional theory the results illustrate that the present approach may render benchmark solutions provided higher order truncations are used and act as engineering plate equations using low order truncation

uniqueness of a scattering problem with l2 boundary values in this paper we consider the scattering problem for a locally perturbed half plane with l 2 boundary values and we obtain a uniqueness result to the solution especially we prove our result for the two dimensional dirichlet boundary value problem in l 2 sense for the helmholtz equation in a locally perturbed half plane

scattering of scalar waves by two dimensional gratings of arbitrary shape application to rough surfaces at near grazing incidence we present the numerical study of scattering of scalar waves from impenetra ble two dimensional periodic surfaces of arbitrary shape nearly all numerical simulations of scattering of waves from rough surfaces in the past have been limited to one dimensional surfaces and moderate angles of incidence by making the surface infinite and bi periodic it becomes possible to simulate numerically scattering from two dimensional surfaces even down to grazing angle only impenetrable surfaces are considered some calculations are presented and are used to compare with the small perturbation or rayleigh rice theory it is found that for near grazing incidence neumann boundary condition the small perturbation theory gives inaccurate values especially near the backscatter direction

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

analysis of the nonlinear vibrations of unsymmetrically laminated composite beams large amplitude free vibrations of unsymmetrically laminated beams using von karman large deflection theory are investigated herein one dimensional finite elements based on classical lamination theory first order shear deformation theory and higher order shear deformation theory having 8 10 and 12 degrees of freedom per node respectively are developed

a generalized multilength scale nonlinear composite plate theory with delamination abstract a new type of plate theory for the nonlinear analysis of laminated plates in the presence of delaminations and other history dependent effects is presented the formulation is based on a generalized two length scale displacement field obtained from a superposition of global and local displacement effects the functional forms of global and local displacement fields are arbitrary the theoretical framework introduces a unique coupling between the length scales and represents a novel two length scale or local global approach to plate analysis appropriate specialization of the displacement field can be used to reduce the theory to any currently available variationally derived displacement based discrete layer smeared or zig zag plate theory the theory incorporates delamination and or nonlinear elastic or inelastic interfacial behavior in a unified fashion through the use of interfacial constitutive cohesive relations arbitrary interfacial constitutive relations can be incorporated into the theory without the need for reformulation of the governing equations the theory is sufficiently general that any material constitutive model can be implemented within the theoretical framework the theory accounts for geometric nonlinearities to allow for the analysis of buckling behavior the theory represents a comprehensive framework for developing any order and type of displacement based plate theory in the presence of delamination buckling and or nonlinear material behavior as well as the interactions between these effects the linear form of the theory is validated by comparison with exact solutions for the behavior of perfectly bonded and delaminated laminates in cylindrical bending the theory shows excellent correlation with the exact solutions for both the inplane and out of plane effects and the displacement jumps due to delamination the theory can accurately predict the through the thickness distributions of the transverse stresses without the need to integrate the pointwise equilibrium equations the use of a low order of the general theory i e use of both global and local displacement fields reduces the computational expense compared to a purely discrete layer approach to the analysis of laminated plates without loss of accuracy the increased efficiency compared to a solely discrete layer theory is due to the coupling introduced in the theory between the global and local displacement fields

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

effective models of composite periodic plates iii two dimensional approaches abstract this paper is concerned with two dimensional approximations of the three dimensional local problems of caillerie 1984 math meth appl sci 6 159 191 and kohn and vogelius 1984 int j solids structures 20 333 350 the solutions to these problems make it possible to evaluate the effective stiffnesses of periodic plates the kirchhoff type approximation results in the formulae shown by duvaut 1976 in theoretical and applied mechanics edited by w t koiter pp 119 132 north holland amsterdam by imposing hencky reissner type constraints one is led to new formulae which have a wider range of applicability the paper also discusses the formulae which result from homogeneizing two dimensional kirchhoff s reissner hencky s and reddy s equations of plates with periodic structure

shear deformable versus classical theories for nonlinear vibrations of rectangular isotropic and laminated composite plates in the present study i the classical von karman theory ii the first order shear deformation theory and iii the higher order third order shear deformation theory are compared for studying the nonlinear forced vibrations of isotropic and laminate composite rectangular plates in particular the harmonic response in the frequency neighborhood of the fundamental mode of rectangular plates is investigated and the response curves computed by using the three different theories are compared the boundary conditions of the plates are simply supported with immovable edges geometric imperfections are taken into account calculations for isotropic and laminated composite plates are presented and results are discussed for isotropic plates the frequency response curves for large amplitude vibrations obtained by using the three theories are almost coincident for laminated composite plates differences arise for relatively thick plates ratio between the thickness and the edge equal to 0 1 while for thin plates ratio between the thickness and the edge equal to 0 01 no difference is obtained for all cases the first order shear deformation with shear correction factor 3 2 and the higher order shear deformation theories give practically coincident results and differences are observed with respect to the classical von karman theory

combined matrices of sign regular matrices abstract the combined matrix of a nonsingular matrix a is the hadamard entry wise product c a a a 1 t since each row and column sum of c a is equal to one the combined matrix is doubly stochastic when it is nonnegative in this work we study the nonnegativity of the combined matrix of sign regular matrices based upon their signature in particular a few coordinates of the signature e of a play a crucial role in determining whether or not c a is nonnegative

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

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

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

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

more on g matrices abstract the first author and hall defined recently a g matrix as a real nonsingular matrix a such that there exist diagonal matrices d 1 and d 2 for which a t 1 d 1 ad 2 the class of g matrices was shown to possess interesting properties in this paper some new characterizations are found and extensions to rectangular and complex matrices are discussed

notes on hilbert and cauchy matrices inspired by examples of small hilbert matrices the author proves a property of symmetric totally positive cauchy matrices called at property and consequences for the hilbert matrix