Timoshenko Beam Structures Research Articles

An improved isogeometric collocation formulation for spatial multi-patch shear-deformable beams with arbitrary initial curvature

The present paper presents a robust multi-patch formulation based on the isogeometric collocation (IGA-C) method for the solution of linear, spatial Timoshenko beam structures with complex geometry. The proposed approach is based on the combination of the (local) Bishop frame with the exponential map for SO(3) (Rodrigues formula) to compute the beam curvature and its derivative. This choice permits bypassing known issues related to the Serret–Frenet (SF) frame (in cases where the local beam curvature vanishes) and does not require the Darboux vector and its derivative, which are both affected by limitations of the SF frame. Moreover, in contrast to existing formulations mostly derived in the local SF frame, here the formulation is consistently derived by linearizing the governing equations in the material setting, where the multi-patch coupling can be enforced in a straightforward way. Numerical tests are performed on complex spatial curved beams, including a demanding biomechanical problem of a braided stent, proving the accuracy and the robustness of the formulation.

Flexibility-based large increment method for nonlinear analysis of Timoshenko beam structures controlled by a bilinear material model

The displacement-based finite element method is currently used for solving nonlinear structural problems. This method utilizes the step-by-step solution approach to perform the nonlinear analysis of structures. This solution approach requires linearization of the constitutive equations. Therefore, there are several shortcomings of the displacement-based method that can affect the computational efficiency of this method such as errors accumulation and propagation in the step-by-step solution approach and the consumption of the computational time due to the high computational efforts during the calculations. The large increment method (LIM) can overcome these shortcomings by separating the linear global equilibrium and compatibility equations from the local constitutive equations which are considered the source of the nonlinearity.This paper discusses the development and application of the large increment method for the nonlinear analysis of Timoshenko beam structures. Timoshenko finite beam element is developed and formulated for solving beam structures with rectangular and wide flange sections. A nonlinear strain-hardening plasticity (bilinear model) material model is used for the newly developed beam elements in LIM algorithm. The proposed elements were verified by two numerical examples covering all possible cases and the results of these examples are compared with the displacement-based finite element software ABAQUS. This work proves that the LIM can be used successfully for solving the static nonlinear analysis of structures and can produce accurate results with less computational efforts and time consumption compared with the common displacement method.

Medium-Frequency Vibration Analysis of Timoshenko Beam Structures

Medium-frequency (mid-frequency) vibration analysis of complex structures plays an important role in automotive, aerospace, mechanical, and civil engineering. Flexible beam structures modeled by the classical Euler–Bernoulli beam theory have been widely used in various engineering problems. A kinematic hypothesis made in the Euler–Bernoulli beam theory is that the plane sections of a beam normal to its neutral axis remain planes after the beam experiences bending deformation, which neglects shear deformation. However, previous investigations found out that the shear deformation of a beam (even with a large slenderness ratio) becomes noticeable in high-frequency vibrations. The Timoshenko beam theory, which describes both bending deformation and shear deformation, would naturally be more suitable for medium-frequency vibration analysis. Nevertheless, vibrations of Timoshenko beam structures in a medium frequency region have not been well studied in the literature. This paper presents a new method for mid-frequency vibration analysis of two-dimensional Timoshenko beam structures. The proposed method, which is called the augmented Distributed Transfer Function Method (DTFM), models a Timoshenko beam structure by a spatial state-space formulation in the [Formula: see text]-domain. The augmented DTFM determines the frequency response of a beam structure in an exact and analytical form, in any frequency region covering low, middle, or high frequencies. Meanwhile, the proposed method provides the local information of a beam structure, such as displacement, shear deformation, bending moment and shear force at any location, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated in numerical examples, where the efficiency and accuracy of the proposed method is demonstrated. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are examined through comparison of the Timoshenko beam and Euler–Bernoulli beam theories.

Substitution method: A technique to study dynamics of both non-gyroscopic and gyroscopic systems

Although the substitution idea has been used occasionally in some engineering structures to study vibration characteristics, it is still short of systematic study in the vibrational engineering field. We summarize the techniques of the substitution method in this paper and apply such skill further to continuous and nonlinear systems, by which the dimension of the system can be reduced. By illustrated examples of frequency analysis on Timoshenko beam and composite sandwich structure, the substitution method has verified to be valid and efficient. The gyroscopic systems, represented by both rotating structures and orbits around libration points have also been presented to show the power of substitution method. Illustrated by the current examples, it is concluded that the substitution method has wide potential applications via studying the functional relations among all the degree-of-freedoms (DOFs) of non-gyroscopic or gyroscopic systems.

Stress Wave Attenuation in Noncollinear Structures Subjected to Impulsive Transient Loadings

AbstractThis paper presents an exploration of noncollinear Timoshenko beam structures for attenuating the stress waves generated from impulsive transient loadings. Due to the existence of the noncollinear segments, flexural waves are generated in addition to the longitudinal waves. Furthermore, the resulting waves are dispersive and thus provide higher potential for the attenuation of the stress waves. Symmetric patterns are found to be more appropriate for increasing the stress-wave attenuation capacity, as these patterns can eliminate bending stress effects at the boundary of the structures. More generally, an adaptive optimization methodology is developed to find the most efficient layout of the noncollinear stress-wave attenuators (NSWAs). This methodology uses genetic algorithms (GA), coupled with an explicit finite-element (FE) method for analyzing the wave-propagation behavior of the structures. The developed methodology is capable of updating the FE model of each trial solution, as the geometric c...

Analytical analysis of the thermal effect on vibrations of a damped Timoshenko beam

We analyze theoretically the thermal effect on the dynamics of a damped-cantilever Timoshenko beam structure, which is featured with cross-sectional temperature gradients. The thermal effect is incorporated in the dynamic model via the temperature-dependent elastic modulus and the temperature-gradient-induced thermal force. The inhomogeneous boundaries introduced by the thermal moment are handled by decomposing the total displacement into static and dynamic displacements. In the end, numerical results based on the analytical solutions are given for a better understanding of the thermal effect on the dynamic behaviors. The solutions reveal the circumstances in which thermal eigenvalues should be considered together with the cross-sectional temperature gradients in predicting thermal vibrations. Moreover, the results also highlight the role of axial temperature gradients on the dynamic response induced by equivalent shearing forces.

Analytical and experimental analysis of Lamb wave generation in piezoelectrically driven Timoshenko beam

Piezoelectric-generated Lamb wave in a Timoshenko beam is studied, analytically, experimentally, and using finite element simulation in this article. The governing equations of the beam with the piezoelectric patches are derived based on integration of the piezoelectric transducer vibration equations into the vibration equations of the Timoshenko beam structure. Finite element model of the beam and the piezoelectric patches is then developed in which the Rayleigh damping factors have been estimated using modal analysis. The results obtained by proposed analytical and finite element models are compared with experimental ones. A significant agreement has been observed and three principal sets of signals have been distinguished by experimental, analytical, and finite element method results: the direct emitted waves and reflected waves from the clamped and tip boundaries. The velocity of wave propagation in a brass beam is determined about 4800 m/s with the actuation frequency of 200 kHz using Lamb wave propagation signals. The results show that exciting the structure in high frequency prevents to overlap the emitted waves from the actuator and reflected waves from the boundaries in short structures.

General exact harmonic analysis of in-plane timoshenko beam structures

The exact solution for the problem of damped, steady state response, of in-plane Timoshenko frames subjected to harmonically time varying external forces is here described. The solution is obtained by using the classical dynamic stiffness matrix (DSM), which is non-linear and transcendental in respect to the excitation frequency, and by performing the harmonic analysis using the Laplace transform. As an original contribution, the partial differential coupled governing equations, combining displacements and forces, are directly subjected to Laplace transforms, leading to the member DSM and to the equivalent load vector formulations. Additionally, the members may have rigid bodies attached at any of their ends where, optionally, internal forces can be released. The member matrices are then used to establish the global matrices that represent the dynamic equilibrium of the overall framed structure, preserving close similarity to the finite element method. Several application examples prove the certainty of the proposed method by comparing the model results with the ones available in the literature or with finite element analyses.

Damage detection in Timoshenko beam structures by multilayer perceptron and radial basis function networks

This study investigates the efficiency of artificial neural networks (ANNs) in health monitoring of pristine and damaged beam-like structures. Beam modeling is based on Timoshenko theory. Two commonly used network models, multilayer perceptron (MLP) and radial basis neural network (RBNN), are used. Beam material and geometrical properties, beam end conditions and dynamically obtained data are used as input to the neural networks. The combinations of these parameters yield umpteenth input data. Therefore, to examine the effectiveness of ANNs, the frequency of intact beams is first tried to be determined by the network models, given the material and geometrical characteristics of beam elements and support conditions. The methodology to compute the vibrational data utilized in training the networks is provided. Showing the robustness of network models, the second stage of the study is carried out. At this stage, the crack parameters (e.g. the location and severity of crack) are estimated by the ANNs using the beam properties, beam end conditions and vibrational data, which consist of natural frequencies and mode shape rotation values. Despite the multiplexed input data, no data reduction schemes or multistage computations are executed in training and validation of neural network models. As a result of analysis runs, the optimal MLP and RBNN models are determined. Comparison of these models shows that the optimal RBNN algorithm performs better. The effectiveness of optimal ANN models in the presence of noise is also presented. As a conclusion, the trained network can be used as a diagnosis method in structural health monitoring of beam-like structures.

Power secant method applied to natural frequency extraction of Timoshenko beam structures

This work deals with an improved plane frame formulation whose exact dynamic stiffness matrix (DSM) presents, uniquely, null determinant for the natural frequencies. In comparison with the classical DSM, the formulation herein presented has some major advantages: local mode shapes are preserved in the formulation so that, for any positive frequency, the DSM will never be ill-conditioned; in the absence of poles, it is possible to employ the secant method in order to have a more computationally efficient eigenvalue extraction procedure. Applying the procedure to the more general case of Timoshenko beams, we introduce a new technique, named "power deflation", that makes the secant method suitable for the transcendental nonlinear eigenvalue problems based on the improved DSM. In order to avoid overflow occurrences that can hinder the secant method iterations, limiting frequencies are formulated, with scaling also applied to the eigenvalue problem. Comparisons with results available in the literature demonstrate the strength of the proposed method. Computational efficiency is compared with solutions obtained both by FEM and by the Wittrick-Williams algorithm.

Free vibration analysis of axially loaded cracked Timoshenko beam structures using the dynamic stiffness method

In this article, the purpose is to investigate the changes in the magnitude of natural frequencies and modal response introduced by the presence of a crack on an axially loaded uniform Timoshenko beam using a particular member theory. A new and convenient procedure based on the coupling of dynamic stiffness matrix and line-spring element is introduced to model the cracked beam. The application of the theory is demonstrated by two illustrative examples of bending–torsion coupled beams with different end conditions, for which the influence of axial force, shear deformation and rotatory inertia on the natural frequencies is studied. Moreover, a parametric study to investigate the effect of the crack on the modal characteristics of the beam is conducted.

Structural power flow analysis of Timoshenko beam with an open crack

In this paper, the structural power flow characteristics of cracked Timoshenko beams are presented. The case of an infinite Timoshenko beam with an open crack is considered. The crack in the Timoshenko beam structure is modeled as a rotational spring with a constant flexibility which is deduced from the relationship between the strain energy and stress intensity factor in fracture mechanics. The flexural wave motions of perfect Timoshenko beam and cracked Timoshenko beam are investigated, then the input power flow and transmitted power flow of perfect and cracked beams are calculated. The results show that the vibrational power flow of cracked beam is highly related to the location and depth of the crack. Contours of input power flow with different frequencies are constructed to identify the location and depth of the crack. This method is very promising for the detection and location of structural damage in the future work.

Differential Transformation Approach for Free Vibration Analysis of a Centrifugally Stiffened Timoshenko Beam

In this paper, the differential transformation approach is applied to analyze the free vibration of centrifugally stiffened Timoshenko beam structures. Such structures involve variable coefficients in the governing equations, which in general cannot be solved analytically in closed form. Both the natural frequencies and the mode shapes are obtained using the differential transformation technique. Numerical examples are presented and results are compared with available results in the literature.

Free and Forced Wave Vibration Analysis of Axially Loaded Materially Coupled Composite Timoshenko Beam Structures

In this paper, wave vibration analysis of axially loaded bending-torsion coupled composite beam structures is presented. It includes the effects of axial force, shear deformation, and rotary inertia; namely, it is for an axially loaded composite Timoshenko beam. The study also includes the material coupling between the bending and torsional modes of deformations that is usually present in laminated composite beam due to ply orientation. From a wave standpoint, vibrations propagate, reflect, and transmit in a structure. The transmission and reflection matrices for various discontinuities on an axially loaded materially coupled composite Timoshenko beam are derived. Such discontinuities include general point supports, boundaries, and changes in section. The matrix relations between the injected waves and externally applied forces and moments are also derived. These matrices can be combined to provide a concise and systematic approach to vibration analysis of axially loaded materially coupled composite Timoshenko beams or complex structures consisting of such beam components. The systematic approach is illustrated through numerical examples for which comparative results are available in the literature.

Vibrational Energy Flow Analysis of Corrected Flexural Waves in Timoshenko Beam – Part II: Application to Coupled Timoshenko Beams

This paper presents the methodology for the energy flow analysis of coupled Timoshenko beam structures and various numerical applications to verify the developed methodology. To extend the application of the energy flow model for corrected flexural waves in the Timoshenko beam, which is developed in the other companion paper, to coupled structures, the wave transmission analyses of general coupled Timoshenko beam systems are performed. First, power transmission and reflection coefficients for all kinds of propagating waves in the general, coupled Timoshenko beam structures are derived by the wave transmission approach. In numerical applications, the energy flow solutions using the derived coefficients agree well with the classical solutions for various exciting frequencies, damping loss factors, and coupled Timoshenko beam structures. Additionally, the numerical results for the Timoshenko beam are compared with those for the Euler‐Bernoulli beam.

Wave Reflection and Transmission in Timoshenko Beams and Wave Analysis of Timoshenko Beam Structures

This paper concerns wave reflection, transmission, and propagation in Timoshenko beams together with wave analysis of vibrations in Timoshenko beam structures. The transmission and reflection matrices for various discontinuities on a Timoshenko beam are derived. Such discontinuities include general point supports, boundaries, and changes in section. The matrix relations between the injected waves and externally applied forces and moments are also derived. These matrices can be combined to provide a concise and systematic approach to vibration analysis of Timoshenko beams or complex structures consisting of Timoshenko beam components. The approach is illustrated with several numerical examples.

Dynamic Response of Timoshenko Beam Structures Solved by DQEM Using EDQ

The dynamic response of Timoshenko beam structures is solved by using the DQEM to the space discretization and EDQ to the time discretization. In the DQEM discretization, DQ is used to define the discrete element model. Discrete dynamic equilibrium equations defined at interior nodes in all elements, transition conditions defined on the inter-element boundary of two adjacent elements and boundary conditions at the structural boundary form a dynamic equation system at a specified time stage. The dynamic equilibrium equation system is solved by the direct time integration schemes of time-element by time-element method and stages by stages method which are developed by using EDQ and DQ. Numerical results obtained by the developed numerical algorithms are presented. They demonstrate the developed numerical solution procedure.Copyright © 2002 by ASME

Exact Solutions for Bending-Torsion Coupled Vibration of Composite Timoshenko Beam

This paper proposes a dynamic analysis method for obtaining exact solutions of composite Timoshenko beams, which are inherently subjected to both the bending , and torsional vibrations. In this paper, the bending-torsion coupled vibration of composite Timoshenko beam is rigorously modelled and analyzed. Two numerical examples are provided to validate and illustrate the bending-torsion coupled vibration of composite Timoshenko beam structure. The numerical examples prove that the proposed method is of great use for the dynamic analysis of dynamic structures composed of multiply connected composite Timoshenko beams.

Dynamics of fractured Timoshenko beams moving in wavy fluids

The fail-safe criterion is applied to floating ship-like structures moving forward in waves. The proposed model utilizes thick-walled Timoshenko beam structures excited by waves. The response to wave excitation is evaluated numerically, including both hydrodynamic and structural damping. Added mass and damping terms appear on both sides of the global equations describing vibrations excited by waves. A coupled direct method has been adopted for the evaluation of the corresponding terms. The nominal stress near the crack, needed as input to the Paris equation for the evaluation of the cycles required to failure, is computed assuming known values for the stress intensity factor. The resulting fail-safe diagrams indicate the effect of various parameters of the problem.

The Timoshenko beam model of the differential quadrature element method

A new numerical approach for solving Timoshenko beam problems is proposed. The approach uses the differential quadrature method (DQM) to discretize the Timoshenko beam equations defined on all elements, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of Timoshenko beam structures. The resulting overall discrete equation can be solved by using a solver of the linear algebra. Numerical results of the DQEM Timoshenko beam model are presented. They demonstrate the DQEM numerical method.