Showing papers on "Timoshenko beam theory published in 1987"

Boundary control of the Timoshenko beam

Abstract: It is shown that the Timoshenko beam can be uniformly stabilized by means of a boundary control. A numerical study on the spectrum is also presented.

Coupling of bending and torsional vibration of a cracked Timoshenko shaft

Abstract: A transverse surface crack is known to add to the shaft a local flexibility due to the stress-strain singularity in the vicinity of the crack tip. This flexibility can be represented by way of a 6 × 6 matrix describing the local flexibility in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling of motion along the directions which are indicated by the off-diagonal terms. Not all motions are coupled, however. To study the coupling of torsion and shear, a 3 × 3 flexibility matrix is used which includes the appropriate terms. Due to the shear terms of the Timoshenko beam equation of the shaft, bending vibration is finally coupled to torsional vibration. This effect is the subject of this investigation, which is of particular importance in turbomachinery operation. The equations of motion of a Timoshenko beam shaft with three degrees of freedom are derived. The free vibration of the shaft and the influence of the crack on the vibrational behaviour of the shaft is studied. The relation of the eigenvalues of the system, to the crack depth and the slenderness ratio of the shaft is derived. Moreover forced vibration analysis of the cracked shaft is performed. The significant influence of the bending vibration on the torsional vibration spectrum, and vice-versa, is demonstrated. It is believed that this effect can be very useful for rotor crack identification in service, which is of importance to turbomachinery.

Composite Box Beam Analysis: Theory and Experiments

Abstract: Beam theory is widely used as a first approximation in numerous structural applications. When applied to composite beams, the accuracy of beam theory becomes questionable because (1) the shearing and warping deformations become significant, as the shearing stiffness of composite laminates is often very low, and (2) several elastic couplings can occur that strongly influence the behavior of composite beams. The torsional behavior of thin-walled composite beams has important implications for aeronautical structures and is deeply modified by the above nonclassical effects. This paper presents two comprehensive analysis methodologies for composite beams and describes experimental results obtained from a thin-walled, rectangular cross-sectional beam. The theoretical predictions are found in good agreement with the observed twist and strain distributions. Out-of-plane torsional warping of the cross-section is found to be the key factor for an accurate modeling of the torsional behavior of such structures.

Errors Associated with Flexure Testing of Brittle Materials

Abstract: : Requirements for accurate bend-testing of four-point and three-point beams of rectangular cross-section are outlined. The so-called simple beam theory assumptions are examined to yield beam geometry ratios that will result in minimum error when utilizing elasticity theory. Factors that give rise to additional errors when determining bend strength are examined, such as: wedging stress, contact stress, load mislocation, beam twisting, friction at beam contact points, contact point tangency shift, and neglect of corner radii or chamfer in the stress determination. Also included are the appropriate Weibull strength relationships and an estimate of errors in the determination of the Weibull parameters based on sample size. Such analyses and results provide guidance for the accurate determination of flexure strength of brittle materials within the linear elastic regime. Error tables resulting from these analyses are presented.

Mixed Petrov-Galerkin methods for the Timoshenko beam problem

Abstract: A new mixed Petrov-Galerkin method is presented for the Timoshenko beam problem. The method has enhanced stability compared to the Galerkin formulation, allowing new combinations of interpolation, in particular, equal-order stress and displacement fields. The methodology is easily generalizable for multi-dimensional Hellinger-Reissner systems.

Finite rotation and nonlinear beam kinematics

Abstract: Standard means of representing finite rotation in rigid-body kinematics, including orientation angles, Euler parameters, and Rodrigues parameters, are reviewed and compared. General kinematical relations for a beam theory that treats arbitrarily large rotation are then presented. The standard methods of representing finite rotations are applied to these kinematical expressions, and comparison is made among the standard methods and additional methods found in the literature, such as quasi-coordinates and linear combinations of projection angles. The method of Rodrigues parameters is shown to stand out for both its simplicity and generality when applied to beam kinematics, a result that is really missing from the literature.

Warping effects in transverse bending of thin-walled beams

Abstract: A beam theory which accounts for cross-sectional warping caused by transverse shearing is presented. A stress resultant theory is formulated for elastic beams, and examples are provided to assess the effects of warping restraint. The ideas are extended to the case of elastoplastic warping. A numerical formulation for the analysis of the inelastic problem is presented and examples of elastoplastic warping are given.

Vibrations and transient responses of Timoshenko beams resting on elastic foundations

Abstract: A finite element technique is used to obtain the natural frequencies and transient responses of Timoshenko beams resting on elastic foundations The beam is discretized into beam elements, each with four degrees of freedom The equations of motion in terms of the nodal degrees of freedom are derived by applying Hamilton's principle The numerical results for a hinged-hinged beam are given to show the effects of rotatory inertia, shear deformation and foundation constants on the natural frequencies of the beam The transient responses of beams are subsequently presented and compared with the available solutions

Petrov-Galerkin formulations of the Timoshenko beam problem

Abstract: Petrov-Galerkin formulations of the Timoshenko beam problem are presented. They are shown to provide the best approximation property, optimal rate of convergence, and nodally exact solution for arbitrary loading, for all values of the thickness of the beam.

Boundary stabilization of an

Abstract: Boundary feedback schemes for stabilizing flexural vibrations in a linear viscoelastic beam are studied. It is shown that in the EulerBernoulli model an arbitrarily small feedback delay can cause unbounded vibrations with arbitrarily large exponential growth rates. For the Timoshenko beam: in the purely elastic case: a contrasting result is given, and formulas for decay rates of high frequency modes are developed for the case of no feedback delay. Numerical results for various no-delay cases are summarized.

Consistent and Inconsistent Higher Order Beam and Plate Theories: Some Surprising Comparisons

Abstract: In 1982 Bickford [1] used Hamilton’s principle to derive a consistent, higher order theory for the elastodynamics of beams based upon the kinematic and stress assumptions previously used by Levinson [2] for beams of rectangular cross-section. In [2] the usual beam equations of motion [3] were used to obtain a higher order beam theory. This latter theory provided a fourth order system of differential equations not too unlike the equations of Timoshenko beam theory with a difference being that in Levinson’s theory the shear stress boundary conditions on the lateral surfaces of the beam were satisfied. Bickford’s variationally consistent theory consists of a sixth order system of differential equations requiring the specification, provided by the variational formulation, of three boundary conditions at each end of the beam. A vectorial formulation of Bickford’s theory is achieved in [1] by defining a posteriori, a “...higher order moment resultant” but, as Bickford himself notes, “it remains to be seen whether or not there can be developed a rational method for constructing the vectorial equations without knowing, a priori, the variational equations.” This latter point is of no consequence since the validity of the variational equations is unquestioned.

A Simple Microcomputer Finite Strip Analysis

Abstract: The development of a finite strip analysis of thin-walled section buckling behaviour is outlined. The analysis uses plate theory for out of plane deflection behaviour and beam theory for in plane displacement behaviour, and is suitable for use on microcomputers. A number of typical buckling problems are examined to illustrate the general applicability of the method, and the results are discussed.

Boundary stabilization of an Euler-Bernoulli beam with viscoelastic damping

Abstract: Boundary feedback schemes for stabilizing flexural vibrations in a linear viscoelastic beam are studied. It is shown that in the Euler-Bernoulli model an arbitrarily small feedback delay can cause unbounded vibrations with arbitrarily large exponential growth rates. For the Timoshenko beam, in the purely elastic case, a contrasting result is given, and formulas for decay rates of high frequency modes are developed for the case of no feedback delay. Numerical results for various no-delay cases are summarized.

A mixed time integration method for Timoshenko and Mindlin type elements

Abstract: A brief summary of recent developments in numerical methods for doing transient analyses of Timoshenko and Mindlin types of structures is given. The pure shearing behaviour of these structures (with no transverse motion) is identified as the high-frequency mode which severely restricts the time step in standard explicit time integration methods. A mixed implicit–explicit scheme is proposed as a technique for circumventing this restriction, and it is shown that this scheme is equivalent to an approach suggested by Belytschko and Mindle9.

Refined Effects in Beam Theories and Their Influence on the Natural Frequencies of Beams

Abstract: The calculation of the natural frequencies, modes and dynamical responses of beam-like structures is an important subject in structural dynamics sciences. In a lot of practical cases, it is sufficient to base these calculations on the classical EULER-BERNOULLI beam theory. However, equally numerous cases exist where it is necessary to take into account refined effects that exceed the classical theory to ensure a right description of the behaviour of the beam. The most important refined effect is that of shear deformation, followed by the effects of rotary ineria, axial prestress etc.