Showing papers on "Timoshenko beam theory published in 1974"

On the theory of rods I. Derivations from the three-dimensional equations

Abstract: Starting with the three-dimensional theory of classical continuum mechanics, some aspects of both the nonlinear and the linear theories of elastic rods are discussed. Detailed attention is given to the derivation of constitutive equations for the linear isothermal theory of elastic rods of an isotropic material and of variable cross-section, deduced by an approximation procedure from the three-dimensional equations. Explicit linear constitutive relations are obtained for straight isotropic circular rods of non-uniform cross-section; the explicit calculation is carried out (in terms of an approximate specific Gibbs free energy function) in four distinct parts, since the complete system of equations involved separate into those appropriate for extensional, torsional and two flexural modes of deformation. A system of displacement differential equations is derived for flexure of a beam of variable circular cross-section; they reduce to those of the Timoshenko beam theory when the radius of the cross-section is constant.

Analysis of the Flexure Test for Laminated Composite Materials

Abstract: Equations applicable to a general class of symmetrically laminated beams are derived by considering a beam as a special case of a laminated plate. The beam bending stiffness thus becomes a function of all the bending stiffness coefficients of a laminated plate. The validity of this approach is verified by comparing theoretical results to flexure data on graphite/epoxy angle-ply and quasi-isotropic laminates. In addition, it is shown that flex strength on general composite laminates is extremely difficult to interpret, even though the stresses can be calculated from the modified beam theory. Discontinuities in the in-plane stresses at layer interfaces lead to a state of stress which is difficult to compare to standard laminate tensile coupons.

Layered Beam Analysis

Abstract: Governing equations for layered Timoshenko beam theory for the general case of unsymmetrically stacked laminated beams are presented. A computer program is developed to use the theory to investigate layered beam problems with any kind of end conditions and any transverse loading. Capability is also demonstrated for varying properties such as modulus of elasticity along the length of the beam. The method is applied to the study of laminated wooden beams. Variations of deflection, shear stress, and normal stress with percentage of strong wood along the outer fibers is shown. The program is not meant to be a design tool. The main purpose of the program is to furnish an analytical tool to use in correlating experimental data with theoretical results so that assumptions regarding physical properties can be altered if need be. A study of the effect of changing the properties of a wooden laminated beam is presented.

Computer analysis of a rotating axial-turbomachine blade in coupled bending-bending-torsion vibrations

Abstract: Basic equations are derived for a comprehensive computer analysis for an equivalent lumped parameter system which simulates a pre-twisted, rotating or non-rotating. Timoshenko beam in coupled bending-bending-torsion vibrations. These equations enable one to develop straightforwardly digital computer programs for studying vibration problems related to compressor or turbine blades in turbomachinery as well as in other structural dynamics applications. The validity of the lumped parameter approach is established through a free vibration study. Good agreement between the present computer results and those obtained theoretically or experimentally by other investigators is demonstrated. In particular, it is found that the natural frequencies obtained in the free vibration study for all beams examined converge from below the exact values at a convergence rate of N−4, where N is the total number of the lumped parameters used in the computer simulation. The rapid rate of convergence shows that the mathematical model used in deriving the basic equations can adequately represent the type of beams under consideration. The known convergence rate also provides an estimate on the accuracy of the computer results, and a means to improve these results by extrapolation.

Flexural Vibrations and Timoshenko's Beam Theory

Abstract: This paper is a study of flexural elastic vibrations of Timoshenko beams with due allowance for the effects of rotary inertia and shear. Two independent formulations are developed, one based on the concepts proposed by Timoshenko and the other on the extended Rayleigh-Ritz energy method. The results obtained from the two formulations are matched in order to determine the correcting coefficients in the simplified formulae which are proposed for the frequency of the first two flexural branches. The proposed formulae are shown to achieve greater accuracy in describing the flexural motions. New accurate solutions are offered for several cross-sectional geometries which enable the accuracy of the available methods to be assessed. Correction coefficients are evaluated for several sections and are compared with previous work.

A Critical Assessment of the Cantilever Beam Method for the Determination of Dynamic Young's Modulus

Abstract: The application of a noncontacting optical transducer for the measurement of flexural resonant frequencies and normal mode shapes of a cantilever beam is described. Results on the resonant frequencies of cantilever beams, gripped at the fixed end by a vise, are compared with the Bernoulli-Euler and Timoshenko beam theories. Although the mode shapes agree well with the theories, there are large discrepancies between the experimental variation of resonant frequency with inverse slenderness ratio and that predicted by the theories. These discrepancies are attributed to imperfect clamping. It is concluded that for accurate Young's modulus calculations from the measured resonant frequencies of cantilever beams, tests must be restricted to long slender beams. For beams which cannot be fabricated to meet this criterion, an empirical method is presented which yields Young's modulus to within ±3 percent on the specimens tested. For increased accuracy the free-free reasonant beam method should be used.

Stability analysis of a damped polygenic systems with relocatable mass along its length

Abstract: The stability of a clamped-free rod subjected to a compressive follower force at its tip, and bearing one or two relocatable mass points, is investigated. The outcomes of the calculations either using Euler's beam theory and neglecting damping and rotary inertia or using Rayleigh's beam theory including damping and rotary inertia are compared with each other: An optimal location exists for the mass points in both cases, however, the optimum is less pronounced with Rayleigh's theory including damping and rotary inertia than with Euler's theory neglecting damping and rotary inertia. Hence, one may conclude that for a safe design damping and rotary inertia, of the beam's mass elements as well as of the masspoints, has to be taken into account. The results obtained are of interest in connection with aircraft wings carrying jet engines, with turbine blades having varying cross-sections, etc.

An equation for fracture energy determination by double-cantilever cleavage

Abstract: With the aid of elementary beam theory a Lagrangian equation of motion was developed to relate the crack velocity to cleavage energy in a double‐cantilevered single‐crystal specimen when shear contributes significantly to the cleavage energy. For a constant cleavage energy the equation predicts that the square of crack length is proportional to time when the crack is long with respect to transverse specimen dimensions. For shorter cracks, crack length is proportional to time.