Analysis Of Timoshenko Beams Research Articles

Analysis of Timoshenko beam with Koch snowflake cross-section and variable properties in different boundary conditions using finite element method

This study analyzed a Timoshenko beam with Koch snowflake cross-section in different boundary conditions and for variable properties. The equation of motion was solved by the finite element method and verified by Solidworks simulation in a way that the maximum error was about 2.9% for natural frequencies. Displacement and natural frequency for each case presented and compared to other cases. Significant research achievements illustrate that if we change the Koch snowflake cross-section of the beam from the first iteration to the second, the area and moment of inertia will increase, and we have a 5.2% rise in the first natural frequency. Similarly, by changing the cross-section from the second iteration to the third, a 10.2% growth is observed. Also, the hollow cross-section is considered, which can enlarge the natural frequency by about 26.37% compared to a solid one. Moreover, all the clamped-clamped, hinged-hinged, clamped-free, and free-free boundary conditions have the highest natural frequency for the Timoshenko beam with the third iteration of the Koch snowflake cross-section in solid mode. Finally, examining important physical parameters demonstrates that variable density from a minimum value to the standard value along the beam increases the natural frequencies, while variable elastic modulus decreases it.

Exact Solutions for Buckling and Second-Order Effect of Shear Deformable Timoshenko Beam–Columns Based on Matrix Structural Analysis

Shear deformable beams have been widely used in engineering applications. Based on the matrix structural analysis (MSA), this paper presents a method for the buckling and second-order solutions of shear deformable beams, which allows the use of one element per member for the exact solution. To develop the second-order MSA method, this paper develops the element stability stiffness matrix of axial-loaded Timoshenko beam–columns, which relates the element-end deformations (translation and rotation angle) and corresponding forces (shear force and bending moment). First, an equilibrium analysis of an axial-loaded Timoshenko beam–column is conducted, and the element flexural deformations and forces are solved exactly from the governing differential equation. The element stability stiffness matrix is derived with a focus on the element-end deformations and the corresponding forces. Then, a matrix structural analysis approach for the elastic buckling analysis of Timoshenko beam–columns is established and demonstrated using classical application examples. Discussions on the errors of a previous simplified expression of the stability stiffness matrix is presented by comparing with the derived exact expression. In addition, the asymptotic behavior of the stability stiffness matrix to the first-order stiffness matrix is noted.

Dynamic Analysis of Timoshenko Beam with Arbitrary Constraints and a Further Optimization Based on Least Energy Principle

Based on Timoshenko’s beam theory, this paper adopts segmented strategy in establishing the governing equations of a multibeam system subjected to various boundary conditions, in which free, clamped, hinged, and elastic constraints are considered. Meanwhile, Galerkin method is incorporated as a competitive alternative, in which a new set of unified, efficient, and reliable trial functions are proposed. A further optimization in regard to boundary distributions under forces is implemented and established on the least absorbed energy principle. High agreement is observed between the analytical results and the FEM results, verifying the correctness of the derivations. Complete comparisons between the analytical and the numerical results indicate the Galerkin method is beneficial when slender ratio is larger than 30, in which the continuity of the deformation is proved to be a crucial influencing factor. A modified numerical strategy about optimal boundary is employed and the remarks imply the algorithm can be availably used to reduce the energy absorption of the whole system.

Dynamic analysis of Timoshenko beam subjected to support motions

In present study, the analytical solution procedure for the dynamic responses of Timoshenko beam excited by support motions is presented by using the eigenfunction expansion method with the aid of the quasi-static decomposition method. The quasi-static decomposition method decomposes the total displacements into static ones and dynamic ones. And it transforms the time-dependent boundary conditions due to support motions into stationary ones. The static components are obtained by solving the static equilibrium equation of Timoshenko beam and the dynamic components are obtained by employing the eigenfunction expansion method. Hinged-hinged Timoshenko beams excited by support motions are illustrated to check the correctness of the solutions and to get insights into structural behaviors.

General Analysis of Timoshenko Beams on Elastic Foundation

General analytical solutions for stability, free and forced vibration of an axially loaded Timoshenko beam resting on a two-parameter foundation subjected to nonuniform lateral excitation are obtained using recursive differentiation method (RDM). Elastic restraints for rotation and translation are assumed at the beam ends to investigate the effect of support weakening on the beam behavior. However, the effects of rotational inertia and shear stress induced from the axial load are considered. The obtained solutions are verified first and then used to investigate the significance of different parameters on the beam behavior. In addition, solutions of forced vibration are analyzed to highlight the effects of excitation nonhomogeneity on the beam behavior.

Finite Element Formulation for Stability and Free Vibration Analysis of Timoshenko Beam

A two-node element is suggested for analyzing the stability and free vibration of Timoshenko beam. Cubic displacement polynomial and quadratic rotational fields are selected for this element. Moreover, it is assumed that shear strain of the element has the constant value. Interpolation functions for displacement field and beam rotation are exactly calculated by employing total beam energy and its stationing to shear strain. By exploiting these interpolation functions, beam elements' stiffness matrix is also examined. Furthermore, geometric stiffness matrix and mass matrix of the proposed element are calculated by writing governing equation on stability and beam free vibration. At last, accuracy and efficiency of proposed element are evaluated through numerical tests. These tests show high accuracy of the element in analyzing beam stability and finding its critical load and free vibration analysis.

FREE VIBRATION AND BUCKLING ANALYSIS OF BEAMS WITH A MODIFIED COUPLE-STRESS THEORY

A model based on a modified couple stress theory for the free vibration and buckling analyses of beams is presented. The model also incorporates the Poisson's effect and allows the analysis of Timoshenko beams with any arbitrary end boundary condition. The natural frequencies and buckling loads are computed using the Ritz method. Parametric studies show that, while the natural frequencies and the buckling loads increase monotonically with the increase of the material length scale, they present a minimum in certain values of the Poisson's ratio. A study relating the classical elasticity and the couple stress strain energies is also presented. By establishing this relation, explicit formulas to obtain the natural frequencies and buckling loads, in which the couple stress and Poisson's effects are accounted for, in terms of the buckling loads of the classical elasticity are found. These formulas, which are valid when the shear strain and stress are zero, allow an expedite computation of natural frequencies and buckling loads of beams with couple stress and Poisson's effect.

Biomechanics of hair cell kinocilia: experimental measurement of kinocilium shaft stiffness and base rotational stiffness with Euler–Bernoulli and Timoshenko beam analysis

Vestibular hair cell bundles in the inner ear contain a single kinocilium composed of a 9+2 microtubule structure. Kinocilia play a crucial role in transmitting movement of the overlying mass, otoconial membrane or cupula to the mechanotransducing portion of the hair cell bundle. Little is known regarding the mechanical deformation properties of the kinocilium. Using a force-deflection technique, we measured two important mechanical properties of kinocilia in the utricle of a turtle, Trachemys (Pseudemys) scripta elegans. First, we measured the stiffness of kinocilia with different heights. These kinocilia were assumed to be homogenous cylindrical rods and were modeled as both isotropic Euler-Bernoulli beams and transversely isotropic Timoshenko beams. Two mechanical properties of the kinocilia were derived from the beam analysis: flexural rigidity (EI) and shear rigidity (kGA). The Timoshenko model produced a better fit to the experimental data, predicting EI=10,400 pN μm(2) and kGA=247 pN. Assuming a homogenous rod, the shear modulus (G=1.9 kPa) was four orders of magnitude less than Young's modulus (E=14.1 MPa), indicating that significant shear deformation occurs within deflected kinocilia. When analyzed as an Euler-Bernoulli beam, which neglects translational shear, EI increased linearly with kinocilium height, giving underestimates of EI for shorter kinocilia. Second, we measured the rotational stiffness of the kinocilium insertion (κ) into the hair cell's apical surface. Following BAPTA treatment to break the kinocilial links, the kinocilia remained upright, and κ was measured as 177±47 pN μm rad(-1). The mechanical parameters we quantified are important for understanding how forces arising from head movement are transduced and encoded by hair cells.

New formulation for vibration analysis of Timoshenko beam with double-sided cracks

It is the intention of this study to synthesize the effects of double-edge cracks on the dynamic characteristics of a beam. The stiffness matrix is first determined for a Timoshenko beam containing two same-line edge cracks. The presented model is then developed for elements with two parallel double-sided cracks, considering the interaction between the stress fields of adjacent cracks. Finally, a finite element code is implemented, to examine the influence of depth and location of double cracks, on the natural frequencies of the damaged system.

Slope-deflection equations for stability and second-order analysis of Timoshenko beam–column structures with semi-rigid connections

Slope-deflection equations for stability and second-order analysis of Timoshenko beam–column structures with semi-rigid connections

Vibration analysis of Timoshenko beams under uniform partially distributed moving masses

The current article presents an analytical approach, as well as a calculation method for determining the dynamic response of Timoshenko beams under uniform partially distributed moving masses, using the so-called discrete element technique (DET). It is shown that the proposed methodology offers a compact and computationally efficient way of conducting parametric studies to evaluate the dynamic response of the beam-like structures with arbitrary boundary conditions such as railway and road bridges. First, a formulation is presented in a matrix form for beams under partially distributed masses. The results have been validated against analytical formulations, finite difference, and finite element results. A number of parametric studies were conducted to assess the effects of the moving mass velocity and moving mass length on the beam deflection. A major aspect of the work is the study of successive travelling masses for which the transient effects are important. Both analytical and DET results showed that the critical speeds were affected for successive moving masses. Finally, the effects of rotary inertia were studied for a range of inertia values to quantify the error arising from using the Bernoulli-Euler formulation for beams with large inertias.

Impact of the 7.62-mm APM2 projectile against the edge of a metallic target

The core of the 7.62-mm, armor-piercing APM2 projectile is very strong, but also brittle. A combined analytical, numerical and experimental investigation examined the conditions to fracture the core. The analytical model is based on a Timoshenko beam analysis, where the projectile is idealized as a semi-infinite cylinder. A 3-D numerical model of the core, impacting against the edge of a metallic plate, was then used to investigate transient effects, the influence of constitutive model assumptions, and the influence of target properties on the magnitude of projectile bending strains. Results are then compared to experimental data. Numerical simulations of the full (jacketed) projectile were also investigated, and it is demonstrated that the erosion strain of the jacket is a critical parameter for successful simulations.

Comparison of the In-Plane Natural Frequencies of Symmetric Cross-Ply Laminated Beams Based on the Bernoulli-Euler and Timoshenko Beam Theories

The in-plane free vibration problem of symmetric cross-ply laminated beams is studied based on the transfer matrix method. Distributed parameter model is used in the mathematical formulation. The rotary inertia, the shear and extensional deformation effects, are considered for the Timoshenko's beam analysis. These effects are neglected in the Bernoulli-Euler analysis. The exact overall dynamic transfer matrix of the beam is obtained by making use of the numerical algorithm available in the literature. In order to obtain detailed knowledge about the effects of the rotary inertia, shear and axial deformations on the first six non-dimensional frequencies, the results of Timoshenko and Bernoulli-Euler theories are compared with each other for length-to-thickness ratios from ranging 3 to 20. Fixed-fixed, fixed-simple, and fixed-free boundary conditions are considered for three values of the thickness-to-width ratios of rectangular section (2, 1, 0.5).

Dynamic Stability of Timoshenko Beams by Finite Element Method

A Finite Element model is developed for the stability analysis of Timoshenko beam subjected to periodic axial loads. The effect of the shear deformation on the static buckling loads is studied by finite element method. The results obtained show excellent agreement with those obtained by other analytical methods for the first three buckling loads. The effect of shear deformation and for the first time the effect of rotary inertia on the regions of dynamic instability are investigated. The elastic stiffness, geometric stiffness, and inertia matrices are developed and presented in this paper for a Timoshenko beam. The matrix equation for the dynamic stability analysis is derived and solved for hinged-hinged and cantilevered Timoshenko beams and the results are presented. Values of critical loads for beams with various shear parameters are presented in a graphical form. First four regions of dynamic instability for different values of rotary inertia parameters are presented. As the rotary inertia parameter increases the regions of instability get closer to each other and the width of the regions increases thus making the beam more sensitive to periodic forces.

Timoshenko Beam Dynamics

The general problem of Timoshenko beam analysis is solved using the Laplace transform method. Time-dependent boundary and normal loads are considered. It is established that the integrands of the inversion integrals are always single-valued for beams of finite length and modal solutions can always be obtained using the residue theorem.