Showing papers on "Timoshenko beam theory published in 1988"

Mechanical deflection of cantilever microbeams: A new technique for testing the mechanical properties of thin films

Abstract: The mechanical deflection of cantilever microbeams is presented as a new technique for testing the mechanical properties of thin films. Single-layer microbeams of Au and SiO2 have been fabricated using conventional silicon micromachining techniques. Typical thickness, width, and length dimensions of the beams are 1.0,20, and 30 μm, respectively. The beams are mechanically deflected by a Nanoindenter, a submicron indentation instrument that continuously monitors load and deflection. Using simple beam theory and the load-deflection data, the Young’s moduli and the yield strengths of thin-film materials that comprise the beams are determined. The measured mechanical properties are compared to those obtained by indenting similar thin films supported by their substrate.

Nonlinear Composite Beam Theory

Abstract: Department of Mechanical Engineering, Aeronautical Engineering and Mechanics, Rensselaer Polytechnic Institute, Troy, New York, 12180-3590 The modeling of naturally curved and twisted beams undergoing arbitrarily large displacements and rotations, but small strains, is a common problem in numerous engineering applications. This paper has three goals: (J) present a new formulation of this problem which includes transverse shearing deformations, torsional warping effects, and elastic couplings resulting from the use of composite materials, (2) show that the small strain assumption must be applied in a consistent fashion for composite beams, and (3) present some numerical results based on this new formulation to assess its accuracy, and to point out some distinguishing feature of anisotropic beam behavior. First, the predictions of the formulation will be compared with experimental results for the large deflections and rotations of an aluminum beam. Then, the distinguishing features ofcomposite beams that are likely to impact the design ofrotating blades (such as helicopter blades) will be discussed. A first type of extension-twisting coupling introduced by the warping behavior of a pretwisted beam is discussed, then, a shearing strain squared term, usually neglected in small strain analyses, is shown to introduce a coupling between axial extension and twisting behavior, that can be significant when the ratio E/G is large (E and G are Young's and shearing moduli of the beam, respectively). Finally, the impact of inplane shearing modulus changes and torsional warping constraints on the behavior of beams exhibiting elastic couplings is investigated.

On a non‐linear formulation for curved Timoshenko beam elements considering large displacement/rotation increments

Abstract: SUMMARY An incremental Total Lagrangian Formulation for curved beam elements that includes the effect of large rotation increments is developed. A complete and symmetric tangent stiffness matrix is obtained and the numerical results show, in general, an improvement over the standard formulation where the assumption of infinitesimal rotation increments is made in the derivation of the tangent stiffness matrix.

Computed torque for the position control of open-chain flexible robots

Abstract: A technique is presented for the solution of the inverse dynamics of open-chain flexible robots. The proposed method finds the joint torques necessary to produce a specified end-effector motion. The formulation includes all the nonlinear terms due to the large rotation of the links, together with Timoshenko beam theory to model their elastic characteristics. The finite-element method is used to discretize the equations of motion. The performance and capabilities of this technique are tested through a simulation analysis. Results show the potential of the method not only for feedforward control, but also for incorporation in feedback control strategies. >

Reexamination of dynamic problems of elasticity for negative Poisson’s ratio

Abstract: Recently, isotropic elastic materials with a negative Poisson’s ratio have been manufactured. Since most of the theoretical results of linear elasticity focus on a positive Poisson’s ratio, the need arises for their extension and reexamination. The above materials may have a variety of technological applications so the motivation for this study is not purely academic. The article deals first with some of the limit cases arising when Poisson’s ratio takes on an extreme value. For models represented by these limit cases, the material and structure responses may not be treated independently from each other. Then such basic dynamic elasticity problems as reflection from a free surface, propagation of Rayleigh waves, and lateral vibrations of beams and plates are reconsidered for the case of a negative Poisson’s ratio. It is shown, in particular, that the static definition of the shear factor in Timoshenko beam theory may not be satisfactory in all cases. Extensive numerical results are also given.

Dynamic Timoshenko Beam‐Columns on Elastic Media

Abstract: Two approaches are used to derive differential equations, stiffness coefficients, and fixed‐end forces for the analysis of structural systems composed of Timoshenko beam‐columns that may be supported by an elastic foundation. The analysis is for the evaluation of the critical static axial load and buckling mode‐shapes of a structure with consideration of elastic media, bending, and shear deformations. The two approaches differ in terms of the assumed shear component of the static axial load on the cross section. The first approach is based on the assumption that the shear component of the axial load is calculated from the total slope, which consists of the bending and shear slope. In the second approach, the shear component of the axial load, however, is calculated only from the bending slope. Analytical expressions for a typical simple beam are derived to show the influence of a foundation parameter on the buckling modes. It is observed that the critical axial loads are significantly reduced when shear d...

Flexural stiffness and modulus of elasticity of flower stalks from allium sativum as measured by multiple resonance frequency spectra

Abstract: Multiple resonance frequency spectra (MRFS) provide a rapid and repeatable method for determining the flexural stiffness and modulus of elasticity, E, of segments of plant stems and leaves. Each resonance frequency in a spectrum can be used to compute E, and removal of the distal portion of an organ produces characteristic shifts in spectra dependent upon the geometry of an organ. Hence, MRFS can be used to quantitatively determine the extent to which a particular leaf or stem morphology can be modelled according to beam theory. MRFS of flower stalks of Allium sativum L. are presented to illustrate the technique. The fundamental, f, and higher resonance frequencies, f2 . .. fn, of stems and the ratios of f2/f,, f3/f, and f3/f2 increase as stalk length is reduced by clipping. The magnitudes of these shifts conform to those predicted from the MRFS of a linearly tapered beam. Morphometric data confirm this geometry in 21 flower stalks. Based on this model, the average modulus equals 3.71 x 108 ? 0.32 x 108 N/M2, which compares favorably with values of E determined by static loading (3.55 x 108 ? 0.22 x 108 N/M2) and is in general agreement with ultrasonic measurements (3.8 x 108 to 4.4 x 108 N/M2). Data indicate that determinations of E from a single resonance frequency are suspect, since each resonance frequency yields slightly different values for E. Statistical evaluations from all the frequencies within a MRFS are more reliable for determining E and testing the appropriateness of beam theory to evaluate the biomechanical properties of plants. THE EXTENT to which a stem can support a weight at its tip or continue to grow vertically before buckling under its own weight depends upon its flexural stiffness and the modulus of elasticity of constituent tissues (McMahon, 1975; King, 1981; Givnish, 1982). Provided it can be modelled according to some beamlike geometry, the critical buckling weight and critical buckling length of a stem can be calculated from empirically determined values of the elastic modulus, E, which measures the proportionality between stress and strain for a structure (Silk, Wang, and Cleland, 1982; Niklas and O'Rourke, 1982, 1987). Consequently, a number of workers have devised methods to determine E. Among the most common is the Instron method of testing specimens under a ' Received for publication 1 October 1987; revision accepted 25 January 1988. The authors wish to thank Mr. William Holmes and Scott Copeland (Department of Theoretical and Applied Mechanics, Cornell University) for their technical assistance, Ms. Barbara Bernstein (Section of Plant Biology, Cornell University) for rendering figures from computer hard-copy printouts, and Professor Wolfgang H. Sachse and Mr. Howard J. Susskind (Theoretical and Applied Mechanics, Cornell University) for providing preliminary data from a longitudinal ultrasonic examination of flower stalks. Support from a National Science Foundation grant BSR 8320272 (KJN) is gratefully acknowledged. uniaxial, constant strain rate (Cleland, 1967, 1971, 1984). Although this technique provides rapid and repeatable measurements of E, the appropriateness of a beam geometry to model a particular stem morphology often remains conjectural. In addition, plant organs capable of supporting their own static weight can undergo dynamic mechanical failure. We present a technique for measuring the flexural stiffness and modulus of elasticity of plant stems which can also be used to evaluate the extent to which a particular stem morphology conforms to each of a variety of beam models. The technique is based on the mathematical relationship between the elastic properties and the multiple resonance frequencies of vibration of tapered or untapered beams with various transverse geometries (Timoshenko and Gere, 1961; Gorman, 1975; Blevins, 1979). A large body of empirical and theoretical studies underpin this approach, and to a limited extent it has been applied to examining the turgor pressure and rigidity of tissues (Virgin, 1955; Falk, Hertz, and Virgin, 1958). However, to our knowledge, multiple resonance frequency patterns have not been used to study plant organs in the method presented here. We have selected the flower stalk of Allium sativum L., to illustrate this method. Garlic

Modeling of web conveyance systems for multivariable control

Abstract: A distributed parameter model of the lateral dynamic behavior of a moving web in an n-roller system is proposed A span of web moving longitudinally and under tension is modeled as a beam with shear flexibility (Timoshenko beam) A quasistatic simplification of the model is made on the basis of spectral separation Boundary conditions at the web-roller interface are considered Actuator dynamics are introduced through the formulation of the boundary conditions Closed-loop simulations using a state-space version of the simplified model are in agreement with experimental results >

The identification and elimination of artificial stiffening errors in finite elements

Abstract: The causes of shear locking and other discretization errors are analysed using a physically interpretable notation. This analysis provides insights that allow the errors due to shear locking to be removed either directly or indirectly. St. Venant's principle is incorporated into the stiffness matrix to directly eliminate shear locking from bending elements. Rules-of-thumb are suggested by the same analysis that will insure the absence of errors due to shear locking at the cost of additional degrees of freedom. It is also shown that aspect ratio stiffening in membrane elements is partially due to the same modelling error that produces shear locking. The source of parasitic shear is also identified and a direct procedure for eliminating it is given. A two node Timoshenko beam element and a four node membrane element are fully developed in symbolic form. The procedure is directly applicable to plate bending elements.

Nonlinear Shallow Curved‐Beam Finite Element

Abstract: A geometrically nonlinear curved-beam finite element is derived and is applied to the load-deflection and buckling analysis of shallow and deep arches. The element stiffness properties are derived relative to a moving reference frame having one axis along the element chord, and they are then transformed to a global reference frame. The initial shape of the element is a shallow cubic polynomial, and its deformed shape is the exact shape caused by end actions within the context of shallow beam theory with shear deformation included. The element stiffness properties are exact within the context of that theory, and are described by explicit formulas similar to those of an initially straight element. In a global reference frame the element applies to deep arches with large rotations. Numerical applications are implemented for the second-order theory. Good results are obtained for the snap-through analysis of a shallow arch and for buckling of deep and shallow arches.

Response of Infinite Railroad Track to Vibrating Mass

Abstract: The resonant frequency of the railroad track is first determined by Timoshenko by modeling the track as a beam on a massless Winkler Foundation. The mass of the foundation and the vibrating load is not included in the formulation. Timoshenko provides a simple expression for the resonant frequency ωR=k/m, where k=the Winkler constant; and m=the mass per unit length of the beam. In this paper, the effect of the mass of the vibrating load on the dynamic response of a railroad track is determined. A dynamic analysis, without assuming a priori a steady‐state solution, is carried out. This procedure demonstrates the significant effect of the load mass on the dynamic response of the beam resting on the Winkler foundation.

Timoshenko beam finite elements using trigonometric basis functions

Abstract: A Timoshenko beam finite element is developed using trigonometric basis functions. The properties and performance of these new elements are explored through a series of illustrative problems that treat both straight and curved geometries. Both linear and nonlinear strain-displacement relations are employed in the formulation and comparison is made to results obtained from full, completely reduced, and partially reduced integration of polynomial basis functions of degree one through three. The results obtained in this work indicate that the trigonometric basis functions are a competitive alternative to polynomial basis functions with regard to accuracy and convergence and most importantly they possess a freedom from shear and membrane locking. The trigonometric basis functions also allow the recovery of rigid-body motions in the case of straight and circular arc curved beams.

A simple and accurate beam theory

Abstract: A fourth-order theory for beam bending is presented along with based on it adequate displacement and stress distributions in the plane of the beam. Their relative mean square error is shown to be of the order of the beam depth cubed in comparison with exact plane stress elasticity solutions.

General dynamic shape functions and stiffness matrix of timoshenko beam and their application to structure-borne noise on ships

Abstract: The general dynamic shape functions and stiffness matrix of a Timoshenko beam are derived and used to establish the stiffness equations for an entire structural system. All the effects of rotary inertia of the mass, shear distortion, mass and structural dampings, axial force, elastic-spring and dashpot foundation are included in this formulation. These stiffness equations can then be easily modified to account for the influences from the concentrated dynamic properties. Both the low-and high-frequency dynamic responses are discussed generally, with special emphasis on the latter case. The problem of structure-borne noise on ships is also studied as an example for application.

Shape optimization of 3D continuum structures via force approximation techniques

Abstract: The existing need to develop methods whereby the shape design efficiency can be improved through the use of high quality approximation methods is addressed. An efficient approximation method for stress constraints in 3D shape design problems is proposed based on expanding the nodal forces in Taylor series with respect to shape variations. The significance of this new method is shown through elementary beam theory calculations and via numerical computations using 3D solid finite elements. Numerical examples including the classical cantilever beam structure and realistic automotive parts like the engine connecting rod are designed for optimum shape using the proposed method. The numerical results obtained from these methods are compared with other published results, to assess the efficiency and the convergence rate of the proposed method.