Showing papers on "Timoshenko beam theory published in 1993"

On the Mechanics of Thin-Walled Laminated Composite Beams:

Abstract: A formal engineering approach of the mechanics of thin-walled laminated beams based on kinematic assumptions consistent with Timoshenko beam theory is pre sented. Thin-walled composite beams with open or closed cross section subjected to bend ing and axial load are considered. A variational formulation is employed to obtain a com prehensive description of the structural response. Beam stiffness coefficients, which account for the cross section geometry and for the material anisotropy, are obtained. An explicit expression for the static shear correction factor of thin-walled composite beams is derived from energy equivalence. A numerical example involving a laminated I-beam is used to demonstrate the capability of the model for predicting displacements and ply stresses.

Stresses in Narrow Regions

Abstract: A method for calculating the stress distribution in a narrow region of an elastic plate is presented. It consists in treating the narrow region by beam theory, treating the rest of the plate by any computational or analytical method, and matching the results of these two calculations. It is illustrated by finding the stress distribution in the narrow region of a plate, between a straight edge and a nearby hole, when the plate is under tension. The same method can be applied to three-dimensional bodies with thin plate-like or shell-like regions.

Elastica of cantilevered beams with variable cross sections

Abstract: Numerical and experimental methods are developed for solving the elastica of cantilevered beams of variable cross sections subjected to combined loading. The beam model is based on Bernoulli-Euler beam theory. The Runge-Kutta and Regula — Falsi methods, respectively, are used to solve the governing differential equations and to compute the beam's horizontal deflection at the free end. Extensive numerical results, including deflected shapes and free-end rotations, are presented in non-dimensional form for elastic beams whose area moment of inertia (bending stiffness) varies linearly with beam length. In these studies, such beams are subjected to combinations of tip vertical loads, tip bending moments, and vertical distributed loads that vary linearly with beam length. Experimental studies are presented that complement the theoretical results.

Adaptively shifted integration technique for finite element collapse analysis of framed structures

Abstract: The present study is concerned with the improvement of the previously proposed ‘shifted integration technique’ for the plastic collapse analysis of framed structures using the linear Timoshenko beam element or the cubic beam element based on the Bernoulli-Euler hypothesis. In the newly proposed ‘adaptively shifted integration technique’, the numerical integration points for the evaluation of the stiffness matrices are automatically shifted immediately after. the occurrence of plastic hinges according to the previously established relations between the locations of numerical integration points and those of plastic hinges. By using the adaptively shifted integration technique, sufficiently accurate solutions can be obtained in the non-linear frame analysis by two-linear-element or only one-cubic-element idealization for each structural member. The present technique can easily be implemented in the existing finite element codes utilizing the linear or the cubic beam element.

A three‐dimensional non‐linear Timoshenko beam based on the core‐congruential formulation

Abstract: A three-dimensional, geometrically nonlinear two-node Timoshenkoo beam element based on the total Larangrian description is derived. The element behavior is assumed to be linear elastic, but no restrictions are placed on magnitude of finite rotations. The resulting element has twelve degrees of freedom: six translational components and six rotational-vector components. The formulation uses the Green-Lagrange strains and second Piola-Kirchhoff stresses as energy-conjugate variables and accounts for the bending-stretching and bending-torsional coupling effects without special provisions. The core-congruential formulation (CCF) is used to derived the discrete equations in a staged manner. Core equations involving the internal force vector and tangent stiffness matrix are developed at the particle level. A sequence of matrix transformations carries these equations to beam cross-sections and finally to the element nodal degrees of freedom. The choice of finite rotation measure is made in the next-to-last transformation stage, and the choice of over-the-element interpolation in the last one. The tangent stiffness matrix is found to retain symmetry if the rotational vector is chosen to measure finite rotations. An extensive set of numerical examples is presented to test and validate the present element.

Control of planar networks of Timoshenko beams

Abstract: The present study is concerned with the questions of controllability and stabilizability of planar networks of vibrating beams consisting of several Timoshenko beams connected to each other by rigid joints at all interior nodes of the system. Some of the exterior nodes are either clamped or free; controls may be applied at the remaining exterior nodes and/or at interior joints in the form of forces and/or bending moments. For a given configuration, is it at all possible to drive all vibrations to the rest configuration in a given finite time interval by means of controls acting at some or all of the available (nonclamped) nodes of the network and, if so, where should such controls be placed? Alternatively, a control objective is to construct energy absorbing boundary-feedback controls that will guarantee uniform energy decay. It is demonstrated that if such a network does not contain closed loops and if at most one of the exterior nodes is clamped, exact controllability and uniform stabilizability of the ...

A finite element for the vibration analysis of a fluid-conveying Timoshenko beam

Abstract: A finite element was developed for use in the vibration analysis of fluid-conveying pipes. The pipe was represented as a Timoshenko beam possessing stiffness and mass while the fluid was iciealized as incompressible and inviscid. With these simplifications the equations of motion were derived by the use of Hamilton's principle. Coriolis and centripetal terms in the equation of motion were the result of the fluid flowing in a moving frame of reference (Le. the vibrating pipe). Formulation of a two-node, Co continuous, fluid-conveying beam element followed from the weak form of the equation of motion. Inclusion of the Coriolis term is what made this element unique with respect to previous work. Verification of the element was accomplished by modeling Coriolis mass flowmeters and then predicting their frequency and relative phase delay for the mode of operation. Results compared favorably to experimental data for commercially available Coriolis mass flowmeters.

Observations on Higher‐Order Beam Theory

Abstract: A parabolic sheardeformation beam theory assuming a higherorder variation for axial displacement has been recently presented. In this theory, the axial displacement variation can be selected so tha...

Axially-loaded damped timoshenko beam on viscoelastic foundation

Abstract: The study of an axially-loaded damped Timoshenko beam on a viscoelastic foundation is presented. The effects of the axial force, viscoelastic foundation and all the various damping components to a Timoshenko beam are considered in the formulation. The dynamic stiffness matrix is established and intensively discussed for applications. Free vibrations and the overall dampings are solved accordingly. Examples and discussions are also included.

Transverse vibrations of a rotating twisted Timoshenko beam under axial loading

Abstract: Transverse bending vibrations of a rotating twisted beam subjected to an axial load and spinning about its axial axis are establihed by using the Timoshenko beam theory and applying Hamilton's principle. The equation of motion of the twisted beam are derived in the twist nonorthogonal coordinate system. The finite element method is employed to discretize the equations of motion into time dependent ordinary differential equations that have gyroscopic terms

Mixed finite element methods for elastic rods of arbitrary geometry

Abstract: The boundary-value problem for rods having arbitrary geometry, and subjected to arbitrary loading, is studied within the context of the small-strain theory. The basic assumptions underlying the rod kinematics are those corresponding to the Timoshenko hypotheses in the plane rectilinear case: that is, plane sections normal to the line of centroids in the undeformed state remain plane, but not necessarily normal. The problem is formulated in both the standard and mixed variational forms, and after establishing the existence and uniqueness of solutions to these equivalent problems, the corresponding discrete problems are studied. Finite element approximations of the mixed problem are shown to be stable and convergent. It is shown that the equivalence between the mixed problem and the standard problem with selective reduced integration holds only for the case of rods having constant curvature and torsion, though. The results of numerical experiments are presented; these confirm the convergent behaviour of the mixed problem.

Formulation and solution of inverse spaghetti problem - Application to beam deployment dynamics

Abstract: A finite element model of the dynamics of axially moving, highly flexible beams is presented. The model is based on a geometrically nonlinear beam formulation that allows for large overall motions. To account for the varying length of a deploying beam, a moving finite element reference grid is incorporated within the nonlinear beam formulation such that the number of finite element nodes remains fixed and the finite element length is allowed to vary. Hamilton's law is used to formulate the equations of motion, and a transient integration solution procedure is derived from a space-time finite element discretization of the Hamiltonian variational principle. Computational results of the methodology are presented for a planar inverse-spaghetti problem. HE dynamics of axially moving beamlike structures are becoming increasingly important for the analysis of spacecraft and large space structures that deploy flexible appendages such as antenna, stabilizing booms, solar arrays, and long trusslike structures as well as for other applications such as magnetic tape drives, printing machines, traveling cables, and band saws. An extensive amount of research has focused on the modeling of axially moving continua to analyze the dynamic behavior of such structures. One of the most common models of axially moving continua that has received the most attention is the traveling string problem. In what he termed the "spaghetti problem," Carrier1 analyzed the motion of a string being accelerated upward into a fixed orifice. In other works since then, analytical expressions representing both linear and nonlinear vibrations of axially moving strings subject to various support conditions have been derived.2'9 The dynamics of beamlike structures traveling between or over fixed supports have also been studied to address industrial applications such as high-speed tape drives and band saws.10'13 To address spacecraft applications in which antenna or other long flexible structures are deployed from a host satellite, the dynamics of a cantilever beam that is ejected from a single fixed guide are modeled using Bernoulli-Euler beam theory in Ref. 14. In a quite different approach, the extrusion of a beam from a rotating base is modeled using a series of elastically connected rigid links in which the number of rigid links are continuously increased/decreased to account for deployment/ retrieval of the beam.15 General formulations of axially moving Bernoulli-Euler beam models cantilevered to a rotating rigid host body have also been developed to study the effect of appendage deployment on the attitude dynamics of orbiting spacecraft.16"20 All of these given analyses are based on an assumption of small elastic deformations. With the exception of the discrete spring-mass model of Ref. 15, all of the other models that have been referenced thus far use linear combinations of space-depende nt admissible functions of a time-varying length weighted by time-dependen t generalized coordi

Static and dynamic analysis of space frames using simple Timoshenko type elements

Abstract: In this paper a finite element method for geometrically and materially non-linear analyses of space frames is described. Beams with both solid and thin-walled open cross-sections are considered. The equations of equilibrium are formulated using an updated incremental Lagrangian description. The elements developed can undergo large displacements and rotations, but the incremental rotations are assumed to be small. The material behaviour is described by elastoplastic, temperature-dependent elastoplastic and viscoplastic models with special reference to metals. Computationally, more economical formulations based on the relationship between stress resultants and generalized strain quantities are also presented. In the case of thin-walled beams the torsional behaviour is modelled using a two-parameter warping model, where the angle of twist and the axial variation of warping have independent approximations. This approach yields average warping shear strains directly from the displacement assumptions and no discrepancy between stress and strain fields exists.

Wave splitting of the Timoshenko beam equation in the time domain

Abstract: In recent years, wave splitting in conjunction with invariant imbedding and Green’s function techniques has been applied with great success to a number of interesting inverse and direct scattering problems. The aim of the present paper is to derive a wave splitting for the Timoshenko equation, a fourth order PDE of importance in beam theory. An analysis of the hyperbolicity of the Timoshenko equation and its, in a sense, less physical relatives—the Euler-Bernoulli and the Rayleigh equations—is also provided.

Flexure-torsion behavior of prismatic beams. I - Section properties via power series

Abstract: The behavior of a tip-loaded cantilever beam with an arbitrary cross section is studied using Saint-Venant's semi-inverse method along with a power series solution for the out-of-plane flexure and torsion warping functions. The power series coefficients are determined by solving a set of variationally derived linear algebraic equations. For complex cross sections, the calculated coefficients represented a 'best-fit approximation' to the exact warping function. The resulting warping functions are used to determine the cross-sectional properties (torsion constant, shear correction factors, shear deformation coefficients, and shear center location). A new linear relation is developed for locating the shear center, where the twist rate is zero about the line of shear centers. Moreover, the kinematic relations for a new fully compatible one-dimensional beam theory are developed. Numerical results are presented first to verify the approach and second to provide section data on NACA four-series airfoils not currently found in the literature.

Random vibrations of externally damped viscoelastic Timoshenko beams with general boundary conditions

Abstract: A complex mode approach is presented. The eigenfunction of the original and adjoint equations are used to decouple the equations. Any conventional or nonstandard boundary conditions at beam ends can be handled with equal ease

Mixed-mode crack-tip deformations studied using a modified flexural specimen and coherent gradient sensing

Abstract: An optical mapping of deformation fields and evaluation of fracture parameters near mixed-mode cracks in homogeneous specimens under elastostatic conditions is undertaken. A modified edge notched flexural geometry is used in the study and its ability in providing a relatively wide range of mode mixities is demonstrated. A full-field, optical shearing interferometry called ‘coherent gradient sensing’ (CGS) is used in the study. Crack-tip parameters such as stress-intensity factors, mode mixity and energy-release rate are measured from the interference patterns. The patterns are analyzed using Williams' mixed-mode, asymptotic expansion field. An expression for energy-release rate for the specimen is also derived using beam theory. The theoretical stress-intensity factors are then obtained using a mode-partitioning method based on moment decomposition. Experimental measurements and theoretical predictions are found to be in good agreement. Limitations of the mode-partitioning method used in the investigation are also pointed out.