Showing papers on "Timoshenko beam theory published in 1984"

Toward a consistent beam theory

Abstract: It is well known that the Euler-Bernoulli theory of the bending of beams makes use of a contradicting assumption of zero shear strains and nonzero shear stresses. Sometimes, this type oJ assumption is also carried over to more refined shear deformation theories. This paper outlines a theory thai avoids this assumption. With the aid of the specific example of a tip loaded cantilever beam, it is shown that the present theory gives Euler Bernoulli solutions in that part of the beam where shear deformation is unimportant and a shear deformation type of solution in the pari of the beam where shear deformation is important, with transition stress patterns between the two. Numerical studies, with a shear modulus representative of sandwich beams, bring out the usefulness of the present theory for the analysis of such soft-cored beams.

R/C Structures Under Impulsive Loading

Abstract: A finite element method is presented to analyze the effects of airblast‐induced ground shock on shallow‐buried, flat‐roofed, reinforced concrete structures. A finite element based on Timoshenko beam theory is adopted. Material properties are defined in terms of nonlinear stress‐strain relations in each of several layers through the thickness of the element. Elastic, ideally‐plastic constitutive properties for plain concrete are cast in terms of shear‐stress/normal‐stress variables. Elastic, strain‐hardening constitutive properties are assumed for steel. Dynamic explicit and implicit and static solution algorithms are available. General two‐dimensional structures, such as rings and arches as well as boxes, can be represented. This analysis method is applied to simulation of static beam‐column tests reported by ACI Committee 318‐77. It is then applied to simulation of structural response in FOAM HEST Events 2, 3 and 4 in which shear, flexure and combined shear‐flexure damage was observed. In these post‐test...

A refined finite element model for the analysis of elastic–plastic frames

Abstract: A finite element model for the elastic–plastic analysis of plane frames is proposed. The formulation is based on the independent modelling of the displacement and plastic strain fields; the latter is modelled both over the cross-section and along the element length as function of a finite number of parameters, which are considered as an extra set of independent variables, in addition to nodal displacements. Stress redistribution is allowed for over the cross-section, but not over the element length, where the distribution of stress resultants (axial forces and bending moments) is imposed consistently with the assumed displacement model; stress redistribution in terms of stress resultants becomes possible only because of the finite number of redundancies introduced when assembling. It is shown that the model can be formulated in such a way that not only compatibility and elasticity, but also equilibrium (in the sense of beam theory), are fully complied with and only the plastic portion of the constitutive relationship is approximately fulfilled, even if, in principle, to any desired level of accuracy. The model produces accurate results, including a detailed representation of the spreading of plastic zones, with a fairly limited number of elements.

Bounds for Stiffness of Prismatic Beams

Abstract: A linearly elastic beam finite element, dual to the Timoshenko beam model, is developed. Shear effects are included, but unlike the Timoshenko element, which is essentially an equilibrium model (stress compatibility enforced), the proposed element is a displacement model (strain and displacement compatibility enforced). Shear distorsions are accounted for by defining a set of internal degrees of freedom, which are subsequently eliminated through use of a condensation procedure to yield the element stiffness matrix in terms of nodal degrees of freedom only. The stiffnesses thus obtained are shown to be upper bound estimates of the corresponding exact values, whereas the theory of the Timoshenko beam provides the associated lower bound estimates. A comparison of the predictions of the two models shows a good agreement over a wide range of aspect ratios.

Simple formulas for dynamic fracture mechanics parameters of elastic and viscoelastic three-point bend specimens based on timoshenko's beam theory

Abstract: Simple formulas for the dynamic fracture mechanics parameters of a three-point elastic or viscoelastic bend specimen are derived by using the Timoshenko's beam theory. A numerical analysis is performed for a steel or PMMA specimen impacted by a falling steel cylinder or sphere. The results are compared with those obtained by using the formulas which were obtained previously using the Euler-Bernoulli's beam theory, and the effects of rotatory inertia and transverse-shear deformation of specimen are discussed.

Vertical Vibration in Timoshenko Beam Suspension Bridges

Abstract: A method of analysis is presented for free vertical vibration of suspension bridges. The method takes into account the effects of shear deformation and rotary inertia, and uses a linearized theory which maintains small amplitudes of vibration. The formulation of the problem is based on the Timoshenko beam theory, and the differential equations of motion and the associated boundary conditions are derived by applying Hamilton's principle. The analysis is conducted by using general solutions for the fourth order differential equation of motion. The objective of the study is to determine a sufficient number of natural frequencies and mode shapes, and to enable an accurate vibration analysis for higher modes. A detailed numerical example, which includes the various boundary conditions of the stiffening girders and the elasticity of the towers, is shown to illustrate the applicability of the analysis and to investigate the dynamic characteristics of vertical vibrating suspension bridges.

Dynamic Analysis of Impact Test Specimens

Abstract: Two methods are represented by means of which the dynamic behaviour of impact test specimens can be analysed. First such problems can be treated by use of a finite difference scheme. On the basis of the results of this technique, a second method can be developed, which founds upon a beam model of the impact test specimen. In both cases, the dynamic stress intensity factor can be calculated from the measured hammer force.

Versatile and Improved Higher‐Order Beam Element

Abstract: The degeneration of two classes of deep beam elements is conducted, one (DB6) based on the conventional Timoshenko beam assumptions and the other (DB7) based on the assumed cubic order longitudinal displacement profile. While an adjustable shear correction factor is required for the DB6 element to compensate for the unrealistic distribution of a shear strain across the beam depth, the DB7 element assumes the more realistic quadratic profile of shear strain at the outset. With the plane-stress continuum solution serving as reference in static and free-vibration analyses, solutions obtained by these two element models are compared with the analytical Timoshenko solution, the analytical thin beam solution and several available solutions of existing beam elements. The result indicates that the performance of the higher order beam element DB7 is seen to be more versatile than other models previously developed by various investigators. Also, superior accuracy of the results is evident in both analyses over a wide range of the beam aspect ratios.

Penalty Finite Element Formulation for Curved Elastica

Abstract: A simple penalty finite element formulation is presented for the large‐rotation and postbuckling analyses of curved beams. The analytical formulation is based on a form of Reissner's large‐deformation theory with the transverse shear deformation and the extensibility of the center line constrained through the use of the penalty method. Reduced integration is used in evaluating the elemental arrays, and a procedure is outlined for determining the critical points in the solution path. Numerical results are presented to demonstrate the effectiveness of the finite elements developed.

On the natural frequencies of a stepped thickness Timoshenko shaft

Abstract: In this note the free vibrations of stepped thickness shafts are investigated taking into account the effects of rotatory inertia and transverse shear force. The solution of the system of indefinite equations is written first, together with boundary and continuity conditions. Then, a practical application is presented and the values of the first six natural frequencies are given, using both the Timoshenko and Euler-Bernoulli theories. Finally, the dependence of the eigenvalues on the extension of the central, thicker portion of the shaft, are studied.

Vibration analysis of frame and cable-stayed footbridges by vafcaf

Abstract: The computer program "VAFCAF" (Vibration Analysis of Frame and Cable-Stayed Footbridges) has been developed; it is based on the finite element method and is intended for use in analysing the behaviours of cable-stayed footbridges, continuous beams and plane frames. The Timoshenko beam element has been incorporated into the program. Jennings' solution for eigenvalue problems is adopted to minimise the size of computer storage thus gaining advantage over storing the large sparseness in matrix. Natural frequencies, mode shapes and maximum vertical acceleration are given in the output. Comparisons of results with computer program FREMOD, model tests and full size cable-stayed footbridges show that the program is accurate enough for practical use. The theoretical background is given. (Author/TRRL)

Formulation of a beam theory due to higher-order approximation and accuracy of typical beam theories

Abstract: In this paper, the most general higher-order equations of one-dimensional static and dynamic theories for generally anisotropic elastic beams are proposed by expanding the displacements of infinite Power series with respect to the transverse coordinate of the beam.Several beam theories which were proposed previously, can be deduced by the particular cases with employment of the lower-order terms of our theory and with specialization of the constitutive equations.On the other hand, we apply the previously proposed plate theories including the effect of transverse deformations to the beam analysis, and examine the special theoretical and numerical charachteristics of the various beam theories through the application to some static and dynamic problems.

Transient flexural vibrations of a timoshenko column supported by an elastic half‐space

Abstract: Transient flexural vibrations of an elastic column supported by an elastic half-space are investigated analytically under the condition that an arbitrarily shaped free-field lateral acceleration is given as an input. Applying the Timoshenko theory to the column and making use of Laplace transformations with respect to time and numerical inverse Laplace transformations, the time histories of the column free end acceleration are presented. Numerical results obtained from the Timoshenko theory are compared with those of a previous paper1 (applying the Bernoulli-Euler theory to the column), and the effects of column slenderness and foundation stiffness on the transient flexural vibrations of the column are clarified.