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This study proposes a method for identifying the coefficients in the governing equation of a beam from the sparsely measured response of bridge displacement. The following three steps constitute the procedure: First, the full-field displacement is reconstructed by the compressive sensing theory, which is innovatively applied in the spatial domain with a proposed general basis matrix. Then, the reconstructed displacement is fitted by the B-spline surface basis functions, and the derived control points are further adopted to compute the derivatives of displacement. The genetic algorithm is finally applied to seek the coefficients of the governing equation by substituting the fitted response into the equation. The proposed method is validated numerically and experimentally, including a parametric study to evaluate the performance subjected to various practical conditions. The salient advantages of the proposed method lie in that it requires few sensors for measurement, and the derivatives of displacement are computed robustly.

By employing the modeling method of flexible multi-body systems, the linear and nonlinear dynamic models of a rotating Euler-Bernoulli beam with a flexible support are established, in which the deformation is coupled with the overall motion resulting from the flexibility of the support. The flexural vibration equation in the linear coupling model can be linearly transformed into the classical model of a rotating beam with a flexible support. The nonlinear model is applicable to rotating beams with different flexible supports. Based on the nonlinear model, the global bifurcation behavior of the rotating beam with supports of different stiffnesses is investigated by applying assumed modes method, and the results including the critical rotary speeds, the diversely buckled-equilibria, and the bifurcation type are verified by the shooting method.

In this paper, the principal resonance is investigated for a cantilever with a contact end. The cantilever is modeled as an Euler–Bernoulli beam, and the contact is modeled by the Derjaguin–Müller–Toporov theory. The problem is formulated as a linear nonautonomous partial-differential equation with a nonlinear autonomous boundary condition. The method of multiple scales is applied to determine the steady-state response. The equation of response curves is derived from the solvability condition of eliminating secular terms. The stability of steady-state responses is analyzed by using the Lyapunov-linearized stability theory. Numerical examples are presented to highlight the effects of the excitation amplitude, the damping coefficient, and the coefficients related to the contact.

Postbuckling, nonlinear bending, and nonlinear vibration analyses are presented for a simply supported Euler–Bernoulli beam resting on a two-parameter elastic foundation. The nonlinear model is introduced by using the exact expression of the curvature. Two kinds of end conditions, namely movable and immovable, are considered. The nonlinear equation of motion, including beam–foundation interaction, is derived separately for these two kinds of end conditions. The analysis uses a two-step perturbation technique to determine the postbuckling equilibrium paths of an axially loaded beam, the static large deflections of a bending beam subjected to a uniform transverse pressure, and the nonlinear frequencies of a beam with or without initial stresses. The numerical results confirm that the foundation stiffness has a significant effect on the nonlinear behavior of Euler–Bernoulli beams. The results also reveal that the end condition has a great effect on the nonlinear bending and nonlinear vibration behaviors of Euler–Bernoulli beams with or without elastic foundations.

Free vibrations of straight beams which are partially supported by an elastic foundation are analyzed. For the sake of simplicity, only the Euler–Bernoulli beam model coupled with a Winkler-type elastic foundation is considered.

This structural system can be used to study, in a rather accurate way, the dynamic response of partially embedded piles (like those used for telecommunications) when dealing with the problem of identifying their mechanical properties during operative conditions.

The study makes clear that different kinds of vibration modes may occur in the part of the beam which is supported by the continuous elastic foundation: indeed apart from the classical modes, corresponding to the dynamics of a free beam, it is possible to have vibration modes which are similar to the static deflection of a beam on an elastic support or even corresponding to rigid-body modes. For the same beam it is shown that transition between these vibration modes can appear when switching from the fundamental natural frequency to subsequent ones. This effect is the focus of the presented numerical examples.

In particular, the analytic expression of the transcendental functions governing the vibration modes, and of the coefficients of the eigenfunctions for all occurring cases, are given here — to the best of the author's knowledge — for the first time.

From the practical point of view, the reported results allow to define a suitable range of the elastic stiffness parameter such that the behavior of a partially supported beam can be conveniently approximated with that of a single-span beam, having one built-in end and the other free.

The transverse vibrations of cracked beams with rectangular cross sections resting on Pasternak and generalized elastic foundations are considered. Both the Euler–Bernoulli (EB) and Timoshenko beam (TB) theories are used. The open edge crack is represented as a rotational spring whose compliance is obtained by the fracture mechanics. By applying the compatibility conditions between the beam segments at the crack location and the boundary conditions, the characteristic equations are derived, from which the nondimensional natural frequencies are solved as the roots. Sample numerical results showing the effects of crack depth, crack location, foundation type and foundation parameters on the natural frequencies of the beam are presented. It is observed that the existence of crack reduces the natural frequencies, whereas the elasticity of the foundation increases the stiffness of the system and thus the natural frequencies. It is also observed that the type of elastic foundation has a significant effect on the natural frequencies of the cracked beam.

The beams can be considered as fundamental structural elements applied in many technical structures and equipment. In the operating regimes, these beam structures are very often loaded by the time-dependent forces which cause their undesirable dynamical behavior and the whole system is getting into critical resonant state. It is clear that the reduction of the level of unwanted vibrations or prevention of their occurrence should be one of the important objectives in the design of machine equipment and structures. To achieve these aims, the knowledge of modal properties of beam structures in relation to their internal structure is necessary. The insertion of reinforcing core into a beam body is one of the possible techniques to achieve the required changes in modal properties (mode shapes, natural frequencies) of the beam structure. The dependency of the modal properties of structurally modified beam structure on the geometric parameters and material properties of the inserted reinforcing core is studied in this paper.

In this paper, the free vibration analysis of non-uniform rotating Euler–Bernoulli beam is carried out. It is assumed that the beam has exponentially decaying circular cross-section. In order to solve the problem, the differential transform method (DTM) is utilized. Based on our knowledge, we claim that the recurrence relation presented herein is an elaborate recurrence relation which has been obtained for ordinary differential equations. Non-dimensional natural frequencies of the beam are obtained and tabulated for different values of the beam parameters such as taper ratio and rotating speed. Furthermore, the finite element method (FEM) is employed to solve the problem. Comparison of the results obtained by DTM and FEM indicates the accuracy of proposed solutions.

Treated herein is an elastic beam that is subjected to a constant load that travels continuously (back and forth) along its span. The dynamic deflection of the beam is investigated analytically so as to predict the critical and cancellation speeds. Closed-form solutions are obtained for the damped system for each load condition and superimposed to determine the total solution of the response. Unlike the classical case of load traveling at constant speed in one direction, multiple resonance peaks are observed for reciprocating load at speeds that are lower than the classical critical speed. It is also observed that resonance may not exist at the classical critical speed for simply supported beams due to the symmetry of the beam. The dynamic deflection is examined for simply supported beams to determine the speeds that cause amplification or cancellation of the free response. The current load condition may have possible application in view of its potential use for vibration suppression, as a moving vibration absorber, or for magnification, in energy harvesting. The results are interpreted in order to understand the variation of dynamic deflection and to estimate the critical speeds for different load conditions.

The modeling and analysis for mechanical response of nano-scale beams undergoing large displacements and rotations are presented. The beam element is modeled as a composite consisting of the bulk material and the surface material layer. Both Eringen nonlocal elasticity theory and Gurtin–Murdoch surface elasticity theory are adopted to formulate the moment–curvature relationship of the beam. In the formulation, the pre-existing residual stress within the bulk material, induced by the residual surface tension in the material layer, is also taken into account. The resulting moment-curvature relationship is then utilized together with Euler–Bernoulli beam theory and the elliptic integral technique to establish a set of exact algebraic equations governing the displacements and rotations at the ends of the beam. The linearized version of those equations is also established and used in the derivation of a closed-form solution of the buckling load of nano-beams under various end conditions. A discretization-free solution procedure based mainly upon Newton iterative scheme and a selected numerical quadrature is developed to solve a system of fully coupled nonlinear equations. It is demonstrated that the proposed technique yields highly accurate results comparable to the benchmark analytical solutions. In addition, the nonlocal and surface energy effects play a significant role on the predicted buckling load, post-buckling and bending responses of the nano-beam. In particular, the presence of those effects remarkably alters the overall stiffness of the beam and predicted solutions exhibit strong size-dependence when the characteristic length of the beam is comparable to the intrinsic length scale of the material surface.

The fourth-order governing equation of Euler-Bernoulli beams necessitates the employment of C¹ Hermite shape functions in finite element analysis. However, unlike the C⁰ finite element shape functions that can be easily formulated as Lagrangian polynomials in a unified manner, the Hermite shape functions of Euler-Bernoulli beam elements are usually constructed case by case. In this work, a simple and unified meshfree path is proposed to develop arbitrary order Hermite shape functions of Euler-Bernoulli beam elements. This approach is realized by proving that the Hermite reproducing kernel meshfree shape functions degenerate to the interpolatory Hermite finite element shape functions under a proper choice of support sizes. The proposed approach provides an explicit formalism for Hermite finite element shape functions, which enables an easy construction of arbitrary order Hermite beam elements. Accordingly, in addition to the conventional cubic Hermite beam elements, explicit shape functions, mass and stiffness matrices for quintic and septic Hermite beam elements are presented. Meanwhile, the frequency errors and general higher order mass matrix formulation for these Hermite beam elements are elaborated as well. These theoretical results are consistently demonstrated by numerical examples with respect to both static and free vibration analysis.

This paper proposes a modified segmented rod method (MSRM) in order to facilitate the introduction of boundary conditions in the analysis of free vibrations and buckling of Euler–Bernoulli beams. First, the basic idea of proposed modifications is presented in the case of elementary length beam, and then the idea is generalized to the full-length beam. In addition to facilitating the introduction of boundary conditions, by comparing the results of MSRM with segmented rod method (SRM) and Hencky bar-chain method (HBM), a faster convergence of MSRM results to the exact ones was observed. Although the case of uniform beam is considered in this paper, proposed MSRM may contribute to greater use of segmented beam approach in solving of various static, free vibration or buckling problems of non-uniform beams. Also, it was shown that MSRM can be easily adapted for the analysis of beams that resting on partial elastic foundation.

Trees have various forms and structures, most of which have excellent mechanical properties. Spruce has a stable structure and excellent resistance to wind and snow loads. In this paper, the vibration performance of spruce is studied by the semi-analytical method. Based on the vibration theory of Euler–Bernoulli beam, the vibration mechanical model of the spruce trunk with variable cross-section is established by using the segmental equivalence method. Then, the natural frequency and modal function of the trunk are analyzed and the displacements are solved under snow load within one hour. The results show that the first four natural frequencies are 120.417rad/s, 188.414rad/s, 237.699rad/s and 288.964rad/s, respectively. During the continuous loading stage, the displacements increase over time. During the static load stage, the displacements change in a harmonic form. The above results are verified by an experimental measurement. This paper provides a reference for the study of the vibration performance of similar structures.

Keeping in view the design of short to medium-span bridges in critical locations such as road/river crossings under the action of oppositely moving high-speed loads, this paper presents a hybrid model for the prediction of dynamic displacement employing a theoretical non-dimensional framework aided by an artificial neural network (ANN). The introduction of such a hybrid model reduces parametric space and time. The feed-forward ANN is implemented to train the comprehensive data set obtained from the non-dimensional theoretical scheme. The dimensionless input to ANN comprises the bridge and load parameters, whereas maximum dynamic displacement is the output parameter. The robustness and efficiency of the proposed surrogate ANN framework in terms of computational efficiency are compared with the conventional mode-superposition method formulated in dimensionless form. The training of test and validated data sets results in best-fit models with optimistic statistical metrics and accuracy $\le$ 6%. Moreover, the sensitivity of different parameters to the displacement response is highlighted using Pearson’s correlation and Quantile–Quantile plots. To incorporate the best-fit models into the design by bypassing the high-fidelity model, a user-friendly interface is developed, confirming the standards of the high-speed rating code (TZY2014-232 2014). Such interfaces have the potential for wide application in the preliminary design of short-to medium-span bridges under opposite moving loads in critical locations such as crossing valleys, rivers, and railroads.

In this paper, we present a deep energy method for functionally graded beams based on both Euler–Bernoulli and Timoshenko beam theory to study their mechanical and thermal properties. We consider the effect of temperature as well as porosity under saturated and unsaturated conditions. The objective function is related to the total potential energy (and boundary conditions) and minimized through neural network training. The results are validated by comparison with benchmark problems available in the literature.

Based on the multi-symplectic idea, a generalized multi-symplectic integrator method is presented to analyze the dynamic response of multi-span continuous beams with small damping coefficient. Focusing on the local conservation properties, the generalized multi-symplectic formulations are introduced and a fifteen-point implicit structure-preserving scheme is constructed to solve the first-order partial differential equations derived from the dynamic equation governing the dynamic behavior of continuous beams under moving load. From the results of the numerical experiments, it can be concluded that, for the cases considered in this paper, the structure-preserving scheme is generalized multi-symplectic if the viscous damping c ≤ 0.3751 when the continuous beam is under a constant-speed moving load and the structure-preserving scheme is generalized multi-symplectic if the viscous damping c ≤ 0.3095 when the continuous beam is under a variable-speed moving load with fixed step lengths Δt = 0.05 and Δx = 0.025. Similar to a multi-symplectic scheme, the generalized multi-symplectic scheme also has two remarkable advantages: the excellent long-time numerical behavior and the good conservation property.

The buckling and free vibration of a Euler–Bernoulli beam composed of two-directional functionally graded materials (FGMs) in thermal environment are analyzed. The material properties and temperature distributions are considered to be continuously varied along both axial and thickness directions. Such two-directional FGMs provide the basis of a promising strategy to tune the dynamic behavior of a structure in a controlled fashion, achieving tunable response as desired. The dynamic equation of the beam and relevant boundary conditions are derived based on Hamilton’s principle. The generalized differential quadrature method is used for determining the exact buckling configuration and the natural frequencies of the beam with different boundary conditions. Numerical results are presented to examine the effects of material gradations on the critical buckling temperature. It is concluded that both temperature change and material properties have significant influences on the natural frequency, which suggests that it is possible to tailor or tune the dynamic behaviors of a beam by using man-made FGMs in a complex environment.

In this study, the dynamic responses of a cracked beam layed on a visco-elastic foundation subjected to moving loads are calculated. An Euler–Bernoulli beam model is used to describe the beam behaviors. In addition, the beam has several open cracks one-sided with different depths. By using the Fourier transform, the dynamic responses of the beam are determined analytically in the frequency domain with the help of Green’s function. By coupling with the periodic supported beam model, an analytical model of the railway sleeper is developed. This model allows the fast calculation of the dynamic responses of a damaged sleeper. A dynamic computational model using the FEM method was also developed and compared to the analytical model. The results from the two methods are relatively comparable for three cases of beams without cracks, beams with one crack, and beams with two cracks.

A two-dimensional red blood cell (RBC) membrane model based on elastic and Euler–Bernoulli beam theories is introduced in the frame of immersed boundary-lattice Boltzmann method (IB-LBM). The effect of the flexible membrane is handled by the immersed boundary method in which the stress exerted by the RBC on the ambient fluid is spread onto the collocated grid points near the boundary. The fluid dynamics is obtained by solving the discrete lattice Boltzmann equation. A "ghost shape", to which the RBC returns when restoring, is introduced by prescribing a bending force along the boundary. Numerical examples involving tumbling, tank-treading and RBC aggregation in shear flow and deformation and restoration in poiseuille flow are presented to verify the method and illustrate its efficiency. As an application of the present method, a ten-RBC colony being compressed through a stenotic microvessel is studied focusing the cell–cell interaction strength. Quantitative comparisons of the pressure and velocity on specified microvessel interfaces are made between each aggregation case. It reveals that the stronger aggregation may lead to more resistance against blood flow and result in higher pressure difference at the stenosis.

A method is developed to obtain the natural frequencies of a beam with any number of changes in cross-section and applied to an n-step beam of constant depth consisting of (n+1) equal portions with the breadth changes at the steps in arithmetic progression. The procedure enabled any number of steps to be tackled without encountering any computational difficulties. Table of the first three frequencies of a clamped-free and a free-clamped stepped beam with 1, 2, 3, 4, 5, 10, 50, 100, 200 steps are tabulated. It is shown that as the number of steps is increased, the frequencies asymptotically approach the frequencies of a 'truncated' beam of constant depth and linearly varying breadth.