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Showing papers on "Timoshenko beam theory published in 1982"


Journal ArticleDOI
TL;DR: In this paper, a penalty type argument is used to degenerate thick elements to thin elements, and various approximations of the shear related energy terms act as different types of constraints.
Abstract: 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.

231 citations


01 Jan 1982

174 citations


Journal ArticleDOI
W.L. Hallauer1, R.Y.L. Liu1
TL;DR: In this paper, the exact dynamic stiffness matrix for a straight and uniform beam element whose elastic and inertial axes are not coincident is derived for planar assemblages of connected bending-torsion beams.

98 citations


Journal ArticleDOI
TL;DR: In this paper, it has been shown that there is only a single frequency spectrum for the transverse vibrations of the Timoshenko beam and not two distinct spectra of frequencies, as has been claimed by a number of prior authors for the case of the simply supported beam.

71 citations


Journal ArticleDOI
W. Soedel1
TL;DR: In this paper, the authors derived the Timoshenko and Mindlin equations from the derived shell equations by geometrical reduction and proved the consistency of these equations with the free vibration behavior of a cylindrical shell with the behavior of the timoshenko beam and the Mindlin plate.

36 citations


Proceedings ArticleDOI
TL;DR: In this article, it was shown that beam theory is generally inadequate to determine the free vibration frequencies and mode shapes of turbomachinery blades and that shallow shell theory is capable of representing all the vibration modes accurately.
Abstract: Vibration analysis of turbomachinery blades has traditionally been carried out by means of beam theory. In recent years two-dimensional methods of blade vibration analysis have been developed, most of which utilize finite elements and tend to require considerable computation time. More recently a two-dimensional method of blade analysis has evolved which does not require finite elements and is based upon shell equations. The present investigation has the primary objective to demonstrate the accuracy and limitations of blade vibration analyses which utilize one-dimensional, beam theories. It is found that beam theory is generally inadequate to determine the free vibration frequencies and mode shapes of moderate to low aspect ratio turbomachinery blades. The shallow shell theory, by contrast, is capable of representing all the vibration modes accurately. However, the one-dimensional beam theory has an important advantage over the two-dimensional shell theory for blades and vibration modes. It uses fewer degrees of freedom, thus requiring less computer time.

36 citations


Journal ArticleDOI
TL;DR: In this article, a previously developed analysis of the flexural vibration of isotropic rectangular plates is extended to include the presence of a membrane stress system, which consists of biaxial direct stress plus inplane shearing stress and is uniform throughout the plate.

34 citations



Journal ArticleDOI
TL;DR: In this paper, a Timoshenko beam finite element is constructed with an arbitrary number of degrees of freedom, including the displacement and cross-section rotation at each of two end nodes, together with coefficients of polynomial expansions of the transverse displacement and the shear deformation.

18 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the vibration and stability of a two-layered beam with imperfect bonding subjected to axial and tangential loads and found that the modes of vibration are divided into three classes, that is, the bending mode, the thickness stretch mode, and the longitudinal mode.
Abstract: This paper is a study of the vibration and stability of a two‐layered beam with imperfect bonding subjected to axial and tangential loads. The beam consists of two identical layers joined together by a bonding agent. The normal and shear bond stresses at the interface of the layers are taken to depend on the respective relative vertical and horizontal displacements of the layers. Each layer is assumed to bend according to the Timoshenko beam theory. It is found that the modes of vibration are divided into three classes, that is, the bending mode, the thickness‐stretch mode, and the longitudinal mode. The flutter and divergence instability loads are obtained for clamped‐free beams and are shown graphically as functions of the shear and normal stiffnesses of the bond. The numerical results are also compared with those from the classical Euler beam theory.

17 citations


Journal ArticleDOI
TL;DR: In this paper, a transfer matrix technique for Timoshenko beams of varying cross-sections was used to calculate the frequency and mode shape of the non-uniform beam represented by a series of uniform segments.

01 Jun 1982
TL;DR: In this paper, the authors present cubic and linear spline-based approximation schemes for models of beams based on the Timoshenko theory, which are used in parameter estimation algorithms; convergence results and numerical findings are reported.
Abstract: : The authors present cubic and linear spline-based approximation schemes for models of beams based on the Timoshenko theory. The schemes are used in parameter estimation algorithms; convergence results and numerical findings are reported. (Author)

Journal ArticleDOI
TL;DR: In this article, Fourier techniques have been used to predict transmitted and reflected waves at a T-joint in rods of square cross-section for an arbitrary longitudinal impulse approaching the joint in the terminating rod.

01 Dec 1982
TL;DR: In this paper, simple beam theory has been extended to treat torsional, distortional and shear-lag effects in straight, thin-walled box beams of uniform section, and all the calculations have been performed on a non-programmable hand calculator.
Abstract: Following the generalized co-ordinate method of Vlasov, simple beam theory has been extended to treat torsional, distortional and shear-lag effects in straight, thin-walled box beams of uniform section. Single-cell or multi-cell sections with side cantilevers can be analysed, and the structure can be single-span or continuous. Examples suitable for use in design are given, and all the calculations have been performed on a non-programmable hand calculator. Comparisons are made with the results of finite strip and finite element computations. It is expected that, when the method of hand analysis has been programmed, computing times will be so short that interactive design procedures will become feasible, making it possible to develop relatively simple design rules which take adequately into consideration the types of structural action specific to box beams. A physical understanding of box-beam behaviour is facilitated by the use of the theory given here. As the elastic solution developed satisfies equilibrium, it can be factored to give a lower bound to the collapse load, for use in limit state design. (Author/TRRL)

Journal ArticleDOI
TL;DR: In this paper, the authors considered a group of problems associated with rotating Timoshenko beams and obtained analytical results for both the clamped-free and clampedclamped boundary conditions, augmented by results obtained from numerical solution of the corresponding boundary value problems.
Abstract: This work considers a group of problems associated with rotating Timoshenko beams. The beam is not assumed to be hubclamped, i.e. the axis of rotation does not necessarily pass through the beam's clamped end. Cases of physical interest involving off-clamped beams include wobbling rotors, impellor blades, and turbine blades. For clamped-free boundary conditions, we seek solutions of the governing equations which correspond to transverse buckling. For the rotor, it is known that Euler-Bernoulli beams do not have buckled modes. By contrast, the Timoshenko beam will have an infinite number of buckled modes. In the impellor blade case, both Euler-Bernoulli and Timoshenko beams will have an infinite number of buckled modes. However, the Timoshenko beam will buckle at a lower eigenrotation speed. This is also true for the case of a rotating Timoshenko beam with clamped-clamped boundary conditions, e.g. a turbine blade clamped at both the rim and hub of a rotating platform. Analytic results for both the clamped-free and clamped-clamped cases are augmented by results obtained from numerical solution of the corresponding boundary value problems.

Journal ArticleDOI
TL;DR: In this paper, a variational theorem is presented which may be used as a basis for developing the equations of motion and the boundary conditions appropriate for studying the vibrational behavior of flexible bodied systems and the surrounding acoustic medium.

01 Jun 1982
TL;DR: In this article, a semi-discrete approximation of the Euler-Bernoulli equation with structural and viscous damping and the Timoshenko equation for transverse vibration of a beam is presented.
Abstract: : Numerical methods for approximate identification or estimation of constant parameters in certain fourth-order partial differential equations (distributed parameter systems) from data are proposed based upon a reformulation of the problem as an abstract equation in a Hilbert space. Projections onto suitable subspaces of splines are used to obtain a semi-discrete approximation which is used to estimate the unknown parameters. Covergence of the approximations is proved using linear semigroup theory and the Trotter-Kato theorem. The proposed methods are applied to estimation of parameters in both the Euler-Bernoulli equation with structural and viscous damping and the Timoshenko equation for transverse vibration of a beam. Numerical results are presented.

Dissertation
01 Jan 1982
TL;DR: In this article, an analysis was carried out to determine the transient response of finite beams with discontinuities of cross section subiected to eccentric longitudinal impact, and the analysis was based on the Timoshenko beam theory which takes into account the effects of shear deformation and rotatory inertia.
Abstract: An investigation was carried out to determine the transient response of finite beams with discontinuities of cross section subiected to eccentric longitudinal impact. Experiments were performed on several stepped beams with increased and reduced cross section and with various end conditions. The analysis was based on the Timoshenko beam theory which takes into account the effects of shear deformation and rotatory inertia. The governing equations were solved as a system of two second order hyperbolic partial differential equations. The numerical solution was obtained by the method of characteristics and theoretical predictions were in excellent agreement with experimental observations at several monitoring positions along the various test beams. The agreement between theoretical and experimental results verified the adequacy of the Timoshenko theory and its numerical solution for describing the flexural wave propagation in beams with discontinuities of cross section. The effect of the discontinuity of beam cross section on the bending moment time distribution showed the importance of reflections in estimating the level of stresses and strains in structural elements when subjected to transient dynamic loading. The computer program developed in this work can be used to obtain numerical solutions for. a wide range of flexural wave propagation problems in beams with discontinuities of cross section with various end conditions and loading configurations.

Journal ArticleDOI
01 Jan 1982
TL;DR: In this paper, the optimal shape of the beam is defined as the shape which minimizes the maximum root-mean-square value of the bending stresses in the beam, and their optimal combination is found by sequential linear programming with move limits.
Abstract: Nonuniform Timoshenko beams subjected to a given stationary random excitation are considered. The general equations relating the spectral density function of the response to the cross spectral density of the load are derived. The optimal shape of the beam is defined as the shape which, for given constant volume of the beam, minimizes the maximum root-mean-square value of the bending stresses in the beam. The shape of the beam is described by a limited number of orthogonal design functions, and their optimal combination is found by sequential linear programming with move limits. From numerical results it is seen that slight modifications of the beam shape give a considerable reduction of maximum r.m.s. stress for most loading cases.

Journal Article
TL;DR: In this article, the results of analytical methods designed to predict the crashworthiness of automobile structures were discussed, under torsional moments on square and rectangular section tubes (closed section), a hat section beam closed by spot welding (semi closed section) and channel beams (open section).
Abstract: The results are discussed of analytical methods designed to predict the crashworthiness of automobile structures. Collapse tests were carried out under torsional moments on square and rectangular section tubes (closed section), a hat section beam closed by spot welding (semi-closed section) and channel beams (open section). Yield and maximum moments in the collapsing process are detailed in respect of cross-section geometry and material properties of the beams. In the case of channels, the effect of constraints against warning deformation of yielding moments was considered using thin-walled beam theory. (Author/TRRL)

Journal ArticleDOI
S. Chonan1
TL;DR: In this paper, the impulse response of a two-layered prestressed beam with flexible bonding resting on an elastic foundation is studied, where two dissimilar layers of the beam are assumed to bend according to the Timoshenko beam theory.

Journal ArticleDOI
TL;DR: In this paper, free flexural vibrations of vertical rods having an attached mass at an intermediate point and different end constraints are investigated, with consideration of the axial force owing to the weight of a mass.


01 Dec 1982
TL;DR: In this article, three large-scale prestressed concrete box beams of deformable cross-section were tested in a service load range under concentric and eccentric loading, and results of bending tests i.e. deflections, longitudinal and transverse normal stresses, and shear stresses were compared with finite element predictions and also hand calculations based on simple beam theory.
Abstract: Three large-scale prestressed concrete box beams of deformable cross-section were tested in a service load range under concentric and eccentric loading. Results of bending tests i. e. deflections, longitudinal and transverse normal stresses, and shear stresses are compared with finite element predictions and also hand calculations based on simple beam theory. Similarly, results of torsion tests are compared with finite element predictions and calculations made by the appropriate hand method. The limitations of these calculation methods are discussed and recommendations are made concerning the limits of their applicability. (Author/TRRL)

Journal ArticleDOI
TL;DR: In this paper, the steady state out-of-plane response of an internally damped ring supported by springs in some bays to a sinusoidally varying point force or moment is determined by use of the transfer matrix technique.

Journal ArticleDOI
TL;DR: In this article, the stability of a slender fixed-free elastic strip in a uniform air stream is studied analytically using slender body aerodynamic theory and beam theory, and it is seen that the aerodynamic instability occurs as a result of coupling between first and second bending modes.
Abstract: The stability of a slender fixed-free elastic strip in a uniform air stream is studied analytically. The analysis is performed using slender body aerodynamic theory and beam theory. It is seen that the aerodynamic instability occurs as a result of coupling between first and second bending modes.

01 Aug 1982
TL;DR: In this paper, the static vibration and flutter analysis of a straight elastic axis blade was performed based on a finite element method solution, and the total potential energy functional was formulated according to linear beam theory.
Abstract: Static vibration and flutter analysis of a straight elastic axis blade was performed based on a finite element method solution. The total potential energy functional was formulated according to linear beam theory. The inertia and aerodynamic loads were formulated according to the blade absolute acceleration and absolute velocity vectors. In vibration analysis, the direction of motion of the blade during the first out-of-lane and first in-plane modes was examined; numerical results involve NASA/DOE Mod-0, McCauley propeller, north wind turbine and flat plate behavior. In flutter analysis, comparison cases were examined involving several references. Vibration analysis of a nonstraight elastic axis blade based on a finite element method solution was performed in a similar manner with the straight elastic axis blade, since it was recognized that a curved blade can be approximated by an assembly of a sufficient number of straight blade elements at different inclinations with respect to common system of axes. Numerical results involve comparison between the behavior of a straight and a curved cantilever beam during the lowest two in-plane and out-of-plane modes.

Journal ArticleDOI
TL;DR: In this article, an exact solution to the title problem using classical beam theory was obtained using beam theory, and the frequency and mode shapes were determined as a function of the end flexibility coefficient and of the ratio concerned, end mass/beam mass.

01 Jan 1982
TL;DR: In this paper, Bernoulli-Euler beam theory and modal analysis are used to obtain analytical solutions for the motion of simply supported and fixed ended beams after impact with a spring support at midspan.
Abstract: The effect of gaps present in the seismic supports of nuclear piping systems has been studied with the use of such large general purpose analysis codes as ANSYS. Exact analytical solutions to two simple beam impact problems are obtained to serve as benchmarks for the evaluation of the ability of such codes to model impact between beam elements and their supports. Bernoulli-Euler beam theory and modal analysis are used to obtain analytical solutions for the motion of simply supported and fixed ended beams after impact with a spring support at midspan. The solutions are valid up to the time the beam loses contact with the spring support. Numerical results are obtained which show that convergence for both contact force and bending moment at the point of impact is slower as spring stiffness is increased. Finite element solutions obtained with ANSYS are compared to analytical results and good agreement is obtained.