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Showing papers on "Timoshenko beam theory published in 1970"


Journal ArticleDOI
TL;DR: In this paper, the associated periodicity extension of Fourier analysis is used to obtain an exact solution of the classical, three-dimensional elasticity problem of free vibration of the rectangular parallelepiped.
Abstract: The “associated‐periodicity” extension of Fourier analysis is used to obtain an exact solution of the classical, three‐dimensional elasticity problem of free vibration of the rectangular parallelepiped. This problem has been completely stated for more than a century and has been solved for only a very few special cases. The characteristic determinant yielding the eigenvalues is formulated for the completely free, rectangular parallelepiped, although the method of associated periodicity can be straightforwardly applied to arbitrary boundary conditions. Modes are classified into eight mutually exclusive and collectively exhaustive symmetry classes. Numerical results are presented for the frequency spectrum of plane‐strain vibrations of completely free rectangles according to two‐dimensional elasticity and are compared with classical Bernoulli‐Euler beam theory and Timoshenko beam theory (including the effects of shear deformation and rotary inertia).

54 citations


Journal ArticleDOI
TL;DR: In this paper, Timoshenko beam theory was used to account quantitatively for the dependence of ex perimental resonant frequencies on mode of vibration, length/thickness ratio, and the ratio of Young's modulus/shear modulus of the com posite beam.
Abstract: The resonant frequencies of unidirectional graphite epoxy com posite beams were found to deviate markedly from classical beam theory predictions at higher modes of vibration. Timoshenko beam theory was used to account quantitatively for the dependence of ex perimental resonant frequencies on mode of vibration, length/thickness ratio, and the ratio of Young's modulus/shear modulus of the com posite beam. By using reasonable values for the longitudinal-trans verse shear moduli, the longitudinal Young's modulus of anisotropic composite beams under vibration agreed well with values determined by static tests and became independent of mode of vibration and L/t. The vibrating beam test method was employed for the deter mination of E11, E22, and G12.

51 citations


Journal ArticleDOI
TL;DR: In this paper, a general matrix formulation suitable for a use of the digital computer is presented for dynamic analysis of frameworks composed of prismatic members, which can be applied to a problem with various considerations of Timoshenko theory, Rayleigh theory, bending and shear, and of Bernoulli-Euler theory.
Abstract: A general matrix formulation suitable for a use of the digital computer is presented for dynamic analysis of frameworks composed of prismatic members. The dynamic stiffness coefficients are derived in the form of nondimensional parameters corresponding to the effects of rotatory inertia, shear deformation, and of bending deformation. The individual parameter may be dropped when the appropriate effect is not considered; hence, the stiffness coefficients can be applied to a problem with various considerations of Timoshenko theory, Rayleigh theory, bending and shear, and of Bernoulli-Euler theory. Input data for the computer include the configurations of the framework and the elastic properties of constituent members. The method may be applied to irregular frameworks composed of sloping members with and without sidesway. Numerical examples presented indicate that the effect of rotatory inertia and of shear deformation on the frequencies of frameworks without sidesway is more significant than that of swayed structures.

49 citations


Journal ArticleDOI
TL;DR: In this paper, a general dynamic three-moment equation including the effects of rotary inertia and shear deformation has been derived for the determination of natural frequencies of continuous beams.

21 citations


Journal ArticleDOI
TL;DR: In this paper, a partial differential equation is derived that describes the motion of a transversely isotropic Timoshenko beam under initial stress, initial displacement, and transverse loading.
Abstract: A partial differential equation is derived that describes the motion of a transversely isotropic Timoshenko beam under initial stress, initial displacement, and transverse loading. Suitable specializations of this equation permits one to investigate buckling behavior, vibrational be havior as affected by initial stress, and wave propagation behavior in initially stressed infinite beams. Each separate topic provides important results for designing structures with high values of the ratio of longi tudinal modulus to longitudinal-transverse shear modulus which are common among advanced composites. The buckling investigation pre dicts moderate decreases in the buckling coefficient for simply sup ported beams but predicts very large decreases for clamped beams. The vibration investigation shows that initial tension and compression have practically no effect on the thickness shear frequencies for all modes. The wave propagation investigation shows (a) the extreme sensitivity of the thickness shear cutoff freque...

19 citations


Journal ArticleDOI
TL;DR: Time-average holographic interferometry applied to HF transverse vibrations of uniform cantilever beam, noting correlation with Timoshenko beam theory as mentioned in this paper, was applied to the transverse vibration of the cantilevered beam.
Abstract: Time-average holographic interferometry applied to HF transverse vibrations of uniform cantilever beam, noting correlation with Timoshenko beam theory

17 citations


Journal ArticleDOI
TL;DR: In this article, the bending of a cantilever brought gradually into contact with a cylindrical supporting surface is studied, and a comparison is made with a solution obtained by means of the theory of elasticity.

15 citations


Journal ArticleDOI
TL;DR: In this paper, the Timoshenko theory was extended to account for the internal damping of the beams in addition to the effects of rotary inertia and shear displacement, and the driving point impedances and natural frequencies of small, rigidly terminated, metal cantilever beams in bending vibration have been measured at low strain amplitudes.
Abstract: The driving‐point impedances and natural frequencies of small, rigidly terminated, metal cantilever beams in bending vibration have been measured at low strain amplitudes. The results show excellent agreement with the predictions of the Timoshenko theory, which has been extended to account for the internal damping of the beams in addition to the effects of rotary inertia and shear displacement. Values of impedance measured at the free end of a magnesium alloy beam have a maximum dynamic range of 6×106; the lowest measured value of impedance was 4.1×10−5 lb‐sec/in. Attempts to avoid the excitation of the third beam mode by driving the beam at a nodal point near its midpoint have proved both successful and straight‐forward. Driving‐point impedance has also been measured at the ends of a manganese‐copper alloy beam and an aluminum beam coated with an unconstrained layer of damping compound. The driving‐point impedance of this damped beam has been closely matched theoretically by the impedance of an equivalen...

11 citations


Journal ArticleDOI
TL;DR: In this article, the exact equations of generalized plane elasticity are reduced to coupled sets of ordinary differential equations by using series representations in Legendre polynomials for all dependent variables.

10 citations


Journal ArticleDOI
TL;DR: In this article, the behavior and response of structural reinforced concrete elements under severe blast loads are investigated numerically using the Timoshenko beam theory for the analysis of reinforced concrete beams, and a formally proposed rate-sensitive nonlinear material models are used to predict the dynamic property of structural steel and concrete under the blast loads.
Abstract: The behavior and response of structural reinforced concrete elements under severe blast loads are investigated numerically. The analytical approach utilizes the Timoshenko beam theory for the analysis of reinforced concrete beams. A formally proposed rate-sensitive nonlinear material models are used to predict the dynamic property of structural steel and concrete under the blast loads. Comparison between the rate-sensitive analysis and rate-in sensitive analysis in which the static strength of material is enhanced by a factor to roughly reflect the effect of strain-rate is given. The numerical results are also compared with the limited experimental data. It is demonstrated that the present consideration of the influence of strain-rate on the structural responses in design manuals leads to over conservative.

6 citations


Journal ArticleDOI
TL;DR: Euler-Bernoulli and Timoshenko beam impact models were compared for case of finite beam resting on spring supports as discussed by the authors, where the beam was assumed to rest on a spring support.
Abstract: Euler-Bernoulli and Timoshenko beam impact models compared for case of finite beam resting on spring supports

Journal ArticleDOI
TL;DR: In this paper, the nonlinear response of an elastic beam to a moving transverse load is studied, using a special perturbation method, and solutions are obtained that remain valid throughout some neighborhood of the critical speed of the linear beam theory.

Journal ArticleDOI
TL;DR: In this paper, the resonance frequencies of unidirectional carbon fiber reinforced/epoxy composite beams were studied over the temperature range 24-225°C and the effects of transverse shear deformation were found to increase in importance with increasing temperature.
Abstract: The resonance frequencies of unidirectional carbon fiber reinforced/epoxy composite beams were studied over the temperature range 24–225°C. Longitudinal Young's moduli E11 and longitudinal-transverse shear moduli G12 were computed from the experimental data by the use of Timoshenko beam theory. The effects of transverse shear deformation (a function of E11/G12) were found to increase in importance with increasing temperature. Values of G12 were found to be approximately proportional to the shear modulus Gm of matrix material but were about 30% lower than predicted by the theory of Hashin and Rosen. The anisotropy of the carbon filaments and voids in the composite samples were proposed to account for the discrepancy between theory and experiment.

Journal ArticleDOI
TL;DR: In this paper, the resonant frequencies of unidirectional graphite fiber-reinforced polyimide (Skybond 703) and polyquinoxaline resin composite beams were determined.
Abstract: The resonant frequencies of unidirectional graphite fiber-reinforced polyimide (Skybond 703) and polyquinoxaline resin composite beams were determined. The Timoshenko beam theory was employed to compute both the longitudinal Young's modulus (E11) and the effective transverse-longitudinal shear modulus (G12) from the set of resonant frequencies of the beams. E11, E22, and G12 were determined for a 64% by volume Modmor II-reinforced polyimide (Skybound 703) composite, and E11 and G12 were determined for cured and postcured Modmor II-reinforced polyquinoxaline (PQ) composites. Dynamic E11 and E22 results were found to agree with experimentally determined static flexural moduli. Voids present in these high-temperature resin composites to an extent of 5–13% by volume appeared to lower the effective shear and longitudinal moduli of the composites.

Journal ArticleDOI
TL;DR: In this article, a two dimensional finite element (FE) analysis was carried out to determine the strain energy release rates during the delamination of two different cross-ply laminated three-point bending specimens.
Abstract: An investigation was performed to study the delamination growth in two different cross-ply laminated three-point bending specimens. A two dimensional (2D) finite element (FE) analysis was firstly carried out to determine the strain energy release rates during the delamination of the beams. Contact elements were used to prevent the material interpenetration on the crack surfaces. To study three dimensional (3D) effects on the crack growth in the composite beams, 3D FE analysis was developed. The 3D results showed that the distribution of the energy release rates along the delamination front are not constant. At the free surface a coupling between mode II and mode III energy release rates was also observed, and it was further noticed that the dominating deformation mode is different for distinct laminate lay-ups even under a pure mode I loading condition. Comparison of the 2D and 3D FE analyses suggested that the critical delamination toughness obtained from 2D computations be conservative. INTRODUCTION Laminated composites have demonstrated their usefulness and potential increases in many structural applications. These materials must be designed to meet various engineering requirements and to withstand the service loads. Delamination is the major failure mode of laminated composites and is now widely investigated by static and/or low velocity impact loading tests. Delamination is initiated by two types of cracks: shear cracks in matrix and normal tension cracks perpendicular to fibres. These interlaminar cracks immediately propagate into ply interface and the associated stress concentration may then initiate delamination cracks. The initiation and growth of delamination is governed by the interlaminar fracture toughness of the material. For most of the specimen geometries used in laminate testings it is possible to obtain analytical solutions using beam theory [1,2,3,4], A general method for calculating the total energy release rates Gj from the local values of forces and * Dept. of Matr. Sci and Eng., Oregon Grad. Inst, P.O. Box 91000, Portland, OR 97291-1000, USA. * Dept. of Strength of Materials, Riga Tech. University, Kalku St. 1, LV 1047, Riga, LATVIA. Transactions on Engineering Sciences vol 6, © 1994 WIT Press, www.witpress.com, ISSN 1743-3533 252 Localized Damage bending moments in a cracked laminates has been suggested by Williams [5]. These formulas [5] are the same as well known formulas for Double Cantilever Beam (DCB), End Notched Flexure (ENF) and other specimens developed previously by other methods. Formulas are simple only for the case of isotropic layers. Fig. la shows a laminated cross ply beam with a central edge notch or edge crack in the bottom layer. In this case delamination crack with length a runs into the ply interface normal to the notch. This case of delamination growth due to bending in graphite/epoxy laminates was investigated by Sun [6]. A formula to calculate the total energy release rate was also given as Here D = /4,, •/),, -B^, where AH, EH and D%% are beam stiffnesses, and subscripts 1 and 2 denote values in section 1 and 2 respectively (see Fig. Ib). |2P |2P

Journal ArticleDOI
01 Mar 1970
TL;DR: In this article, the authors present an analysis of the effect of different types of beams with varying degrees of bendability on the tension side of the beam and the location of the central axis.
Abstract: PRESENTS AN ELASTIC STRESS ANALYSIS FOR RECTANGULAR REINFORCED CONCRETE CURVED BEAMS WITH CIRCULAR AXES SUBJECT TO PURE BENDING. THE ANALYSIS DEALS WITH RECTANGULAR SECTIONS REINFORCED ON THE TENSION SIDE ONLY. THE LOCATION OF THE NEUTRAL AXIS AND THE CALCULATION OF MAXIMUM STRESSES IN CONCRETE AND STEEL IS GIVEN IN CHART FORM. THE LIMITING CASE FOR BEAMS WITH STRAIGHT AXES IS ALSO PRESENTED FOR COMPARISON. IF THE RADIUS OF CURVATURE OF THE MEMBER IS SMALL COMPARED TO THE BEAM DEPTH, THE STRESSES DIFFER SIGNIFICANTLY FROM CALCULATED USING LINEAR BEAM THEORY. /AUTHOR/

Journal ArticleDOI
TL;DR: In this article, a flexible beam with a tip mass and attached to a rotating base is modelled to include the tensile forces due to centripetal acceleration, and an approximate solution is approached via the Rayleigh-Ritz method.
Abstract: A flexible beam with a tip mass and attached to a rotating base is modelled to include the tensile forces due to centripetal acceleration. The equations of motion are derived using the extended Hamilton's Principle and an approximate solution is approached via the Rayleigh-Ritz method. The system is simulated for both a prescribed torque profile and a prescribed velocity profile. The results indicate that the beam stiffens when these tensile forces are included. This is evidenced by an increased frequency and reduced amplitude of the flexural vibration of the beam.

Journal ArticleDOI
TL;DR: In this article, the problem of determining the natural frequencies and modes of a statically indeterminant,Timoshenko beam is considered by lumping the beam properties of linear and rotary inertia at discrete points along the length of the beam and by employing the complementary, variational principle.
Abstract: The Problem of determining the natural frequencies and modes of a statically indeterminant,Timoshenko beam is considered By lumping the beam properties of linear and rotary inertia at discrete points along the length of the beam and by employing the complementary, variational principle, an approximate solution is obtained by simple matrix iteration