Double-beam System Research Articles

The Effect of a Fully Asymmetric Discontinuity in the Elastic Layer of a Geometrically Nonlinear Double-Beam Coupled Mechanical System

This paper presents the effect of an antisymmetric discontinuity in a Winkler nonlinear elastic layer and compares it with the case of a double Timoshenko beam system without discontinuities. The effects of geometric nonlinearity are considered, and the obtained results represent a time-domain analysis. A modified p-version finite element method is applied to analyse the vibrations of mechanical systems with discontinuities. The main contribution of this work is a comparative analysis of a double beam system without discontinuities and a double beam system with an antisymmetric discontinuity in the Winkler elastic layer. The study demonstrates significant deviations in the beam response under a concentrated periodic external force when an antisymmetric discontinuity is present. Its qualitative and quantitative characteristics are illustrated through time histories, showing amplitude variations in the steady-state oscillation regime. Forced vibrations in the time domain are analysed using the Newmark direct integration method. The cases of deviations in the antisymmetric model are explained, highlighting their potential applications in technical practice as well as in the field of deformable bodies and structures.

Boundary element method for buckling analysis of elastically connected double-beam system

Studies of Double-beam systems have received significant attention from researchers due to their wide applications in civil and mechanical engineering. Commonly, finite element method or exact solutions have been employed to solve double-beam systems, with few others considering buckling of axially loaded double-beam systems with classical boundary conditions. This paper presents a novel formulation of the Boundary Element Method (BEM) to determine the critical buckling load of the double-beam system elastically connected by a Winkler elastic layer with generalized boundary conditions. Based on the Euler-Bernoulli beam theory, the double-beam system is composed of two identical beams. This paper provides a detailed discussion of each step involved in the BEM, including the fundamental solution, boundary integral equations, and algebraic system. Examples considering different boundary conditions, material properties, and load cases are done and compared to corresponding analytical solution. The results show excellent agreement between the BEM approach and the analytical solution, confirming the accuracy and effectiveness of the technique.

Selection of optimal intensity measures for seismic performance evaluation of underground utility tunnel and internal pipeline system

Seismic intensity measures (IMs), which are closely related to both earthquake hazards and structural responses, play a critical role in the performance-based earthquake engineering (PBEE) framework. This paper aims to investigate and define the most representative IMs for probabilistic seismic demand model (PSDM) for the longitudinal utility tunnel and internal pipeline system. Similar to the requirements by the Chinese Code for seismic design of urban rail transit structures, the utility tunnel in this study is assumed being buried in four typical engineering site classes for the analyses. Soil-tunnel-pipeline interactions are simulated using a double-beam system resting on a nonlinear foundation. Soil-tunnel interaction, tunnel-pipeline interaction and joint connections are modeled as nonlinear springs with different mechanical properties. A series of 17 pairs of outcrop ground-motion records and 27 commonly used IMs are selected in the analyses. Engineering demand parameter (EDP) is defined based on the maximum joint opening and the maximum strain of the utility tunnel and internal pipeline, respectively. The selected IMs for predicting the seismic responses of tunnel-pipeline systems are tested using different criteria, i.e., correction, efficiency, practicality, and proficiency. The final decision-making procedure employs a fuzzy multi-criteria decision method. The numerical results indicate that: (1) The velocity-related IMs are the most representative IMs based on the four criteria for structures buried in four sites of different conditions; (2) The optimal seismic IMs vary with different criteria, site classes and structure types. According to the fuzzy multi-criteria decision approach, optimal IMs for the tunnel-pipeline system are PGV, PGV, EPV and HI for site classes I – IV, respectively. The proposed optimal IMs can be utilized to construct seismic fragility curves of utility tunnel, internal pipeline and tunnel-pipeline systems in four different site classes.

Analysis of a double Timoshenko beam model

In this work we consider a double beam system modeled in the theory of Timoshenko. An existence and uniqueness result is achieved by using the standard theory of linear semigroup. The exponential stability is also proved. Then, fully discrete approximations are introduced and a prior error estimates are shown. Finally, some numerical simulations are presented.

Dynamic modeling and parameter sensitivity analysis of AUV by using the POD method and the HB-AFT method

Autonomous Underwater Vehicle (AUV) parameterization is a geometric modification method, and can quickly obtain excellent structure with low vibration and noise performance. Blindness and inefficiency are common outcomes of the lack of an efficient parameter selection approach. But not enough research has been done in this area. In this paper, the finite element (FE) method is used to establish the propeller–shaft–shell double beam system with ball bearings, which is more similar to the real model. The basic theory of the proper orthogonal decomposition (POD) method is introduced in detail, and the original full order model (FOM) is reduced to 4 DOFs. It is found that the accuracy of the reduced order model (ROM) is high enough and the calculation time is greatly reduced. The semi-numerical semi-analytical expression of the ROM system is solved by the harmonic balance and alternating frequency/time domain (HB-AFT) method, whose results are in good agreement with those by the Runge–Kutta method. The sensitivities of the shell responses to the spring stiffness are discussed. The AUV setup is designed to analysis of numerical and experimental results. These results in this paper can provide an advanced theoretical basis for the structural parameter design of the AUV.

Numerical Analysis of Cracked Double-Beam Systems

Based on elasticity theory, this paper discusses the static analysis of a cracked double-beam system in the presence of a Winkler-type medium. It is further assumed that the double-beam system is constrained at both ends by elastically flexible springs with transverse and rotational stiffness. Using a variational formulation, the governing static equations are derived and solved using analytical and numerical approaches. In the first approach, closed-form solutions for the displacement functions are obtained based on the Euler–Bernoulli beam theory. In the second approach, the Cell Discretisation Method (CDM) is performed, whereby the two beams are reduced to a set of rigid bars connected by elastic constraints, in which the flexural stiffness of the bars is concentrated. The resulting stiffness matrix is easily deduced, and the governing equations of the static problem can be immediately solved. A comparative analysis is performed to verify the accuracy and validity of the proposed method. The study focuses on the effect of various parameters, including crack depth and position, boundary conditions, elastic medium and slenderness. The validity of the proposed analysis is confirmed by comparing the current results with those obtained from other approaches. In particular, the results obtained by closed-form solution and CDM are compared with the Finite Element Method (FEM). The accuracy of the results was assessed by making comparisons with results found in the literature and reported in the bibliography. It was shown that the proposed algorithm provides a simple and powerful tool for dealing with the static analysis of a double-beam system. Finally, some concluding remarks are made.

Effects of Kerr-type viscoelastic coupling layer with different stiffnesses on stochastic stability of Rayleigh double beams

This paper discusses the stochastic stability of a double Rayleigh beam system connected by a Kerr-type three-parameter elastic layer with two different stiffnesses, under compressive axial loads. The beams are modeled using the Rayleigh beam theory, and the axial forces consist of a constant component and a time-dependent stochastic function. The study investigates the almost-sure and moment stability of the double beam system subjected to stochastic compressive axial loading, utilizing the Lyapunov exponent and moment Lyapunov exponents. In the case of weak noise excitations, a singular perturbation method is employed to derive second-order expansions of the moment Lyapunov exponent and the Lyapunov exponent. Monte Carlo simulation is included to validate the obtained results. A numerical study is conducted for selected parameters, and the almost-sure and moment stability in the first and second perturbation are graphically presented. The results provide insights into the influences of different stiffnesses, damping, and the shear parameter of the Kerr-type layer on the stochastic stability of the coupled Rayleigh beam mechanical system, considering the effects of rotational inertia. It is quantitatively and qualitatively determined that reducing the stiffness of one part of the Kerr-type layer leads to a decrease in the stable region of stochastic stability, while reducing the shear parameter results in an increase in the stable region of stochastic stability. Furthermore, a quantitative relationship between different dampings of the Kerr-type viscoelastic layer is determined, where increasing either of the two damping parameters leads to an expansion of the stable region of stochastic stability. The inclusion of rotational inertia effects through the Rayleigh beam theory contributes to more accurate approximations of the obtained solutions.

Vibration analysis of partially viscoelastic connected double-beam system with variable cross section

Vibration analysis of partially viscoelastic connected double-beam system with variable cross section

Vibrations of porous functionally graded CNT reinforced viscoelastic beams connected via a viscoelastic layer

This work analyses the vibrational response of porous viscoelastic functionally graded (FG) carbon nanotube (CNT) reinforced double beams. The beams are simply supported, connected through a viscoelastic layer, where four functionally graded patterns of CNT in the thickness directions (uniformly distributed (UD), FG-V, FG-O, and FG-X) are considered. The porosity models considered are UD, PO-V and PO-X. After formulating the coupled equations of motion via an energy-based method, and incorporating the effects of the viscoelastic beams and viscoelastic layers, the equations are solved using an assumed series expansion technique. The equations and solution methodology are verified utilising simplified models from existing literature and also through development of a finite element method (FEM) based simplified model; very good agreement has been achieved. Effects of the stiffness and viscous coefficients of the viscoelastic layer, the structural viscosity parameter of the porous CNT reinforced double beams, the porosity coefficient, and porosity models are examined. Overall, the results indicate that the stiffness and viscous coefficients of the viscoelastic layer have increasing effects on the real and imaginary components of second series fundamental transverse natural frequency of the double beam system, respectively. An increase in the structural viscosity parameter of the porous CNT reinforced double beams leads to an increase in the imaginary components of the fundamental frequencies of both the series. Moreover, a range of different effects on the real and imaginary components of the natural frequencies were noted while varying the porosity coefficient and models. The FG-X CNT porous reinforced viscoelastic double beams with a viscoelastic layer have the highest natural frequencies of all the cases. The study demonstrates the importance of the porosity, viscoelastic layer and viscoelastic material properties on the dynamics of the system.

Variational principles for a double Rayleigh beam system undergoing vibrations and connected by a nonlinear Winkler–Pasternak elastic layer

Abstract Variational principles and variationally consistent boundary conditions are derived for a system of double Rayleigh beams undergoing vibrations and subject to axial loads. The elastic layer connecting the beams are modelled as a three-parameter nonlinear Winkler–Pasternak layer with the Winkler layer having linear and nonlinear components and Pasternak layer having only a linear component. Variational principles are derived for the forced and freely vibrating double beam system using a semi-inverse approach. Hamilton’s principle for the system is given and the Rayleigh quotients are derived for the vibration frequency of the freely vibrating system and for the buckling load. Natural and geometric variationally consistent boundary conditions are derived which leads to a set of coupled boundary conditions due to the presence of Pasternak layer connecting the beams.

The exact spectral element modeling and vibration analysis of the acoustic black hole double-beam system

In this paper, we present a study of vibration characteristics of the double-beam structures combined with the concept of the acoustic black hole (ABH), which is an effective technique for vibration and noise control. In the proposed double-beam structure, the ABH beam as the vibration damper is attached to the primary uniform beam by a range of translational and rotational springs. We formulate the closed-form spectral element matrix of the double-beam structure and calculate the natural frequencies and mode shapes of the system. The results demonstrate the ABH effect of increasing the modal damping ratios. We also study the power flow and mechanical intensity of the system to offer physical insight into the vibration suppression mechanism of the ABH. A detailed parametric analysis of both translational and rotational springs is carried out. The investigation contributes to the exact dynamic modelling and analysis of the double-beam system containing ABH elements. Furthermore, the proposed ABH double-beam structure shows great potential for vibration control and energy harvesting.

A closed-form solution of forced vibration of a double-curved-beam system by means of the Green's function method

Double-curved-beam systems (DCBSs) are usually seen in many engineering fields. Compared to straight double-beam systems, DCBSs are more efficient in noise and vibration control. This paper aims to obtain a closed-form solution of steady-state forced vibration of a DCBS. The classical Euler-Bernoulli beam model is employed in this work to model vibration equations of the DCBS. Green's function and Laplace transform methods are successively used to obtain the fundamental solution of vibration equations of the DCBS. The fundamental solution is the general solution and can be used for any boundary conditions. In the numerical result section, the present solution is verified by comparing its results with some results in references. Effects of some important geometric and physical parameters on vibration responses and interaction between the stiffness of the elastic layer and the DCBS are discussed. Results show that the DCBS can be degenerated to a straight double-beam system by setting two radii of curvature of the beams to infinity, and the DCBS can also be reduced to a double-beam system whose one upper or lower beam is a straight beam and the other is a curved beam.

Protection of pipeline below pavement subjected to traffic induced dynamic response

Failure of pipelines below road pavement results to the disruption of both the traffic movement and the consumers of the pipelines. Intermediate safeguard layer can be used to protect the pipeline from heavy traffic loads. The present study proposed analytical solutions to obtain the dynamic response of buried pipe below road pavement with and without considering safeguard based on the concept of triple and double beam system respectively. Pavement layer, safeguard and the pipeline are considered as Euler Bernoulli’s beam. Advanced soil model is used (viscoelastic foundation with shear interaction between springs) to model the surrounding soil. Self-weight of soil is also considered in the present study. The obtained governing coupled differential equations are solved adopting finite sine Fourier transform, Laplace transform and their inverse transformation. The proposed formulation is initially verified with the past numerical and analytical studies and then validated with the three-dimensional finite element based numerical analysis. From parametric study it is perceived that the stability of the pipe can be significantly increased by providing intermediate barrier. Further, pipe deformation is increases with increasing traffic loads. At very high-speed range (> 60 m s−1), pipe deformation is significantly rises with increasing traffic speed. The present study can be useful in preliminary design stage before performing rigorous and expensive numerical or experimental study.

Dynamic analysis of buried pipeline with and without barrier system subjected to underground detonation

Failure of pipe networks due to blast loads resulting from terrorist attacks or construction facilities, may cause economic loss, environmental pollution, source of firing or even it may lead to a disaster. The present work develops a closed-form solution of buried pipe with barrier system subjected to subsurface detonation. The solution is derived based on the concept of double-beam system. Euler Bernoulli’s beams are used to simulate the buried pipe and the barrier system. Soil is idealized as viscoelastic foundation along with shear interaction between discrete Winkler springs (advanced soil model). The finite Sine-Fourier transform is employed to solve the coupled partial differential equations. The solution is validated with past studies. A parametric study is conducted to investigate the influence of TNT charge weight, pipe material, damping ratio and TNT offset on the response of buried pipe with and without barrier system. Further a statistical analysis is carried out to get the significant soil and pipe input parameters. It is perceived that peak pipe displacements for both the cases (with and without barrier) are increases with increasing the weight of TNT charge and decreases with increasing the damping ratio and TNT offset. The deformation of pipe also varies with pipe material. Pipe safety against blast loads can be ensured by providing suitable barrier layer. The present study can be utilized in preliminary design stage as an alternative to expensive numerical analysis or field study.

Flexural Wave Propagation in a Double-Beam System Interconnected by Local Resonators with Two Degrees of Freedom

The flexural band gaps in a double-beam system on elastic foundations periodically interconnected by local resonators with two degrees of freedom were investigated using the plane wave expansion method. The transmission property of the finite periodic system was examined by the finite-element method to verify the existence of the band gaps. The mechanism of band gap formation was further studied according to the eigenmodes at the edges of band gaps and the transverse deformation patterns. The technique of broadening the band gaps was also proposed. The results indicated that significant attenuation still appears outside the band gaps for the heterolateral transmission due to the counteraction of the approximate symmetric and antisymmetric flexural bands. The band gaps can be effectively broadened, and the propagating waves are even completely eliminated in the last pass band by only adding a spring. It was also observed that increasing the distance between the left and right attached points and the gravity of the resonator gradually from the asymmetric distribution to the symmetric distribution makes two band gaps coupled to form a superwide gap. The presented results emphasize the promising potential of the double-beam system in controlling the propagation of the flexural wave.

Forced Vibration Analysis of Euler-Bernoulli Double-Beam Systems by Means of Green’s Functions

Double-curved-beam (DCB) systems are usually seen in many engineering fields. Compared to straight double-beam systems, DCB systems are more efficiency in noise and vibration control problems. This work aims to obtain closed-form solutions of steady-state forced vibrations of DCBs. The classical Euler-Bernoulli curved beam (ECB) model is employed in this work to modelling vibration equations of the DCB systems. The Green’s function and Laplace transform methods are successively used to obtain fundamental solutions for the vibration equations of the DCB systems. In this work, the fundamental solutions are general solutions and can be used for any boundary conditions. In numerical section, the present solutions are verified by comparing to some results in references. Effects of some important geometric and physical parameters on vibration responses and the interaction between the stiffness of the elastic layer and the DCB system are discussed. The results show that the DCB system can be degenerated to the straight double-beam system by setting two radiuses to infinity, and the DCB system also can be reduced to a double-beam system whose one upper or lower beam is a straight beam and the other is a curved beam.

Transverse forced nonlinear vibration analysis of a double-beam system with a supporting nonlinearity

To exploit the potential application of supporting nonlinearity in marine engineering, an attempt is made to establish the transverse forced vibration analysis model of a double-beam system supported by a spring-mass system that is nonlinear. This kind of vibration system consists of two beam sections, boundary supports, a coupling component, and a nonlinear spring-mass arrangement. The variational approach and the generalized Hamiltonian concept are used to develop the governing equations of such a double-beam system. The Galerkin truncation method (GTM) is a technique for obtaining the governing equations’ residual equations. By solving the associated residual equations numerically, the nonlinear responses of the double-beam system can be figured out. The GTM has good solidity and correctness in the prediction of the vibration system’s forced transverse vibration. The dynamic responses of the double-beam structure supported by a spring-mass system that is nonlinear are subtle to their initial calculation values. Appropriate parameters of the nonlinear support will subdue the level of vibration at the boundary of the double-beam system. In contrast, unsuitable parameters of the nonlinear support motivate complex dynamic responses of the double-beam system and harmfully influence the vibration repression at the boundary of the vibration system.

Progressive collapse mechanisms of multi-story steel-framed modular structures under module removal scenarios

Due to different assembly requirements and novel integration strategies, the modular structures may have undesirable loading mechanisms and distinct structural behaviors. However, there are limited data on the progressive collapse performances of modular structures, which impedes their industrial applications. This study numerically investigates the structural robustness of multi-story steel-framed modular structures against progressive collapse by using pushdown analysis. A six-story structure with five modules per floor is designed and established. The applied load, failure process and load redistribution mechanisms are studied under module removal scenarios. Parametric analyses are also conducted to investigate influences of cross-sectional dimensions of frame members, number of stories and bays, arrangements of bracings, and module removal scenarios on the robustness of the modular structures. The results show that the prototype modular structure designed with Hong Kong design code can withstand collapse under module removal scenarios. The load redistribution mechanisms among columns are significantly influenced by the overturning action and bracing effects. The ceiling beam in the lower module and the floor beam in the upper module form a double beam system. The column section, beam section, arrangement of bracings and number of stories significantly influence the robustness of the modular structure. Finally, a design method based on buckling load equation for columns is proposed to estimate the progressive collapse resisting capacity of modular structures with good accuracy and efficiency.

Buckling and postbuckling behaviors of symmetric/asymmetric double-beam systems

Double-beam systems composed of two interconnected parallel beams have become basic mechanical components of modern engineering structures. Focusing on the asymmetric features in boundary conditions and other structural parameters, this paper studies the buckling and postbuckling behaviors of a double-beam system supported on an elastic foundation. The nonlinear governing differential equations of the system are derived by using the Hamilton's principle, and discretized via the differential quadrature method. Then the pseudo-arc length algorithm is adopted to overcome difficulties of numerical continuation around bifurcation points. Compared with single beam systems, double-beam systems exhibit some peculiarities like multivalued critical buckling loads, in-phase and out-of-phase postbuckling paths. Based on the proposed procedures, these buckling and postbuckling behaviors are investigated in detail. For symmetric boundary cases of the two sub-beams, the multivalued critical buckling loads of the system compose a cluster of critical curves that intersect at a point determined only by the critical loads of single beams. Hence a simple criterion based on single beam critical loads is proposed to quickly estimate whether or not the axial loads would trigger the buckling of the system. However for asymmetric boundary cases, the cluster of critical curves locate in a region that depends on the intrinsic double-beam characteristics, and the criterion is revised accordingly. Also for asymmetric systems, some unique features of the out-of-phase postbuckling paths are detected. It is found that their stabilities are quite different from those of symmetric systems, and that strengthening the system stiffness may even trigger the out-of-phase buckling more easily.

Exact Closed-Form Solutions for Free Vibration of Double-Beam Systems Interconnected by Elastic Supports Under Axial Forces

This paper aims to present the exact closed-form solutions for the free vibration of double-beam systems composed of two parallel beams connected by an arbitrary number of discrete elastic supports. The general solutions of the mode shapes of the double-beam system are derived employing the Laplace transform method from a perspective of the entire domain of beams without enforcement of any segmentation. A unified strategy applied to various boundary conditions is proposed to determine the independent constants involved in the general solutions, as well as the frequency equation. Numerical calculations are performed to verify the present solutions by comparing the results from the previous literature and finite element simulation, and to discuss the effects of support parameters (stiffness, location, and number) on the modal characteristics of the double-beam system in detail. Outcomes show that the support location plays a pivotal role in regulating the modal characteristics of the double-beam system; for each-order mode, there are one or more potential optimal positions to maximize the effect of the elastic support. The mode veering phenomenon is detected as the support parameters change. It is highlighted that, by introducing an amplitude similarity index, the proximity degree for the mode shapes of the two beams influenced by the support parameters can be evaluated quantitatively. The present analysis is greatly helpful to the optimal design, health monitoring, and vibration control of the double-beam system.