Elastic Beam Model Research Articles

Main Roof Breakage and Vibration Induced Coal Burst Occurring in Longwall Roadways

Rock burst is one major threat to mining safety and economy. Rock burst occurring in the longwall mining roadway accounts for 85% of the total amount of burst events. This paper investigates the causality mechanism of rock burst in longwall roadways by establishing a finite elastic beam model in the working face based on the elastic foundation theory. The breakage process of the main roof and related dynamic effects are analysed. The result shows that the movement of the main roof shows free vibration under certain damping resistance. It is also found that the roof dominant vibration frequency increases with the increase in the thickness and elastic modulus of the roof. During roof vibration, the vertical stress applied on the coal mass is unloaded. The destressing of the roof-coal interface causes the coal mass in the roadway rib to slip into the roadway under the horizontal ground stress, resulting in rock burst. The possibility of rock burst increases with increase in the strength and thickness of the roof and horizontal ground stress within the coal mass. This mechanism explains the occurrence of rock burst in the mining roadway; it provides the fundamental theory for the prevention and controlling technologies of longwall roadway rock burst.

The double elastic foundation beam model for analysis of load transfer systems of racks and nut columns in shiplifts

The load transfer systems of racks and nut columns are important components of the rack and pinion vertical shiplifts which bear the ship chambers and transfer the ship chamber loads in special cases to the tower columns. Aiming at exploring a design based calculation method for the nut column, racks and the associated second stage embedded parts, this paper proposes the double elastic foundation beam model for both the rack load transfer systems and nut column load transfer systems by which the formulas for describing the axial distribution of internal forces and deflections of the rack, column nut and the associated second stage embedded parts are derived. The calculation method is demonstrated by the description of the application in the Three-Gorges shiplift.

Longitudinal Deformation of Deep Shield Tunnels Caused by Upper Load Reduction.

Above-crossing excavations may cause uplift damages on existing shield tunnels. Therefore, to accurately calculate the deformation of shield tunnels is very necessary for geotechnical engineers. At present, the single-sided elastic foundation beam model is usually used in longitudinal deformation calculations for shield tunnels, which overestimates the uplift of deep shield tunnels. Because of the existence of the ground arch, deep shield tunnels are subjected to two-sided foundation reaction forces. Therefore, this paper proposes a partial missing double-sided elastic foundation beam model and the related fourth-order partial differential equations. In this model, the shield tunnel is subjected to double Winkler foundation springs and is simply considered a Euler–Bernoulli beam. A two-stage analysis method is used to solve the problem. First, the vertical unloading stress due to the above-crossing tunnelling at the tunnel location is calculated through Mindlin’s solution. Second, the deformation response of the beam subjected to an unloading stress is calculated by the finite difference method. Two engineering cases are used to verify the research. The results indicate that the proposed model is more accurate than traditional models in predicting the maximum uplift value, which is basically consistent with the observations. Due to the existence of segment staggering, the longitudinal influence range of the calculation by two models is larger than the actual measurement.

Analytical solutions considering tangential contact conditions for circular lined tunnels under longitudinally propagating shear waves

Analytical solutions based on the elastic foundation beam model for the bending response of circular tunnels under longitudinally propagating shear waves are studied considering the tangential contact conditions at the ground-tunnel interface. First, two kinds of new analytical solutions based on Timoshenko beams resting on Winkler or Pasternak foundations are developed considering different contact conditions at the ground-tunnel interface. The contact conditions are considered by introducing a spring-type flexibility coefficient in the force–displacement relationship. Then, the new analytical solutions, as well as the traditional solutions using Euler or Timoshenko beams on Winkler or Pasternak foundations, are compared with the dynamic numerical results. The new analytical solutions exhibit good agreement with the dynamic numerical results for different contact conditions. However, due to the neglect of the tangential interaction between the tunnel and surrounding ground, traditional solutions are only suitable for predicting the seismic response of tunnels in the full-slip condition. Finally, a parametric study is conducted to investigate the effects of the earthquake frequency, ground condition, lining thickness and contact condition. The proposed analytical solutions can be used to predict the seismic response of circular tunnels under longitudinally propagating shear waves in engineering design.

Estimation of Wheelset Natural Vibration Characteristics Based on Transfer Matrix Method with Various Elastic Beam Models

The elastic vibration of the wheelset is a potential factor inducing wheel-rail defects. It is important to understand the natural vibration characteristics of the flexible wheelset for slowing down the defect growth. To estimate the elastic free vibration of the railway wheelset with the multidiameter axle, the transfer matrix method (TMM) is applied. The transfer matrices of four types of elastic beam models are derived including the Euler–Bernoulli beam, Timoshenko beam, elastic beam without mass and shearing stiffness, and massless elastic beam with shearing stiffness. For each type, the simplified model and detailed models of the flexible wheelset are developed. Both bending and torsional modes are compared with that of the finite element (FE) model. For the wheelset bending modes, if the wheel axle is modelled as the Euler–Bernoulli beam and Timoshenko beam, the natural frequencies can be reflected accurately, especially for the latter one. Due to the lower solving accuracy, the massless beam models are not applicable for the analysis of natural characteristics of the wheelset. The increase of the dividing segment number of the flexible axle is helpful to improve the modal solving accuracy, while the computation effort is almost kept in the same level. For the torsional vibration mode, it mainly depends on the axle torsional stiffness and wheel inertia rather than axle torsional inertia.

Euler\u2013Bernoulli elastic beam models of Eringen\u2019s differential nonlocal type revisited within a mathbf{C }^{0}-continuous displacement framework

A theory of the Erigen’s differential nonlocal beams of (isotropic) elastic material is prospected independent of the original integral formulation. The beam problem is addressed within a C^{(0)}-continuous displacement framework admitting slope discontinuities of the deflected beam axis with the formation of bending hinges at every cross section where a transverse concentrated external force is applied, either a load or a reaction. Concepts sparsely known from the literature are in this paper used within a more general context, in which the beam is envisioned as a macro-beam whose microstructure is able to take on a size dependent initial curvature dictated by the loading and constraint conditions. Indeed, initial curvature seems to be an effective analytical tool to inject size effects into micro- and nano-beams. The proposed theory is applied to a set of benchmark beam problems showing that a softening behaviour is always predicted without the appearance of paradoxical situations. Comparisons with other theories are also presented.

Analytical Solutions for the Mechanical Responses of Shallow Double-Arched Tunnel Subjected to Symmetric Loads

Double-arched tunnel is a special kind of tunnel structure which is widely adopted for shallow rock tunnels. The internal forces within a double-arched tunnel are difficult to be determined due to its special geometry and complex interactions with surrounding rock masses. This paper presents a set of analytical solutions for determining the internal forces within the lining structure of shallow double-arched tunnel subjected to symmetric external loads. The double-arched tunnel is decomposed into three main parts, i.e., the arch rings, the side walls, and the middle wall. The force method and the elastic foundation beam model are employed for the arch rings and the side walls, respectively, while the middle wall is treated as a cantilever beam fixed at the bottom. Analytical solutions for the internal forces of the three parts are derived separately, which are continuous at the conjunctions of different parts. The derived analytical solutions are verified by comparing with the FEM simulation results. Finally, parametric studies are performed to investigate the influences of burial depth, elastic resistance coefficient, opening angle, and tunnel span on the internal forces based on which some recommendations are provided for the construction and design of the double-arched tunnels. The derived analytical solutions provide fast estimations for the internal forces and deformation of the double-arched lining structures, which will be a useful tool for design optimization.

A simplified analytical solution for deformation behavior of existing tunnels subjected to influences of landslides

Current scholars have made more efforts to investigate the deformation of landslide caused by tunnel excavation, but there are few studies on the mechanical properties of existing tunnels subjected to influences of landslides. Furthermore, very little attention is paid to the simplified analytical solution for landslide-tunnel interaction and current studies still remain on the monitoring and numerical modeling of engineering. A simplified analytical method is proposed in this paper to evaluate on the deformation of existing tunnels induced by landslides. Firstly, the landslide thrust sliding forces are calculated based on the transferring coefficient method and the limit equilibrium method. Secondly, imposing the thrust sliding force on the existing tunnels, the elastic foundation beam model is established for predicting the landslide-tunnel interaction. Then, the internal force and displacement for the existing tunnels caused by the landslides are solved by the transfer matrix method. In addition, the three-dimensional numerical simulation results are compared with the simplified analytical solution and good consistency is obtained. Furthermore, the parametric analysis for influences of the cohesive force, the internal friction angle, and the unit weight of landslide body is conducted on the thrust sliding force. The influences of the thrust sliding force, the foundation coefficient of landslide and sliding bed, the tunnel lining stiffness, and the tunnel length on the existing tunnels are also analyzed with the parametric analyses by the simplified analytical solution. The paper contributes to the understanding of interaction mechanics for landslides as a potential risk in preliminary design of tunnels.

Analysis of the Vibration of the Ground Surface by Using the Layered Soil: Viscoelastic Euler Beam Model due to the Moving Load

Moving loads will have a certain impact on the safety of the structures. Since concrete is a viscoelastic material, the elastic concrete model cannot describe its viscoelastic characteristics under moving loads. It is necessary to establish a model that can describe the viscoelastic characteristics of concrete materials. In addition, the layered of the soil is also an important factor affecting the propagation of subway vibration waves. Considering the effects of the properties of the concrete material of the subway tunnel structure and the layered soil foundation as well as the load velocity on the vibration of the ground surface caused by the moving load, the standard linear elastic solid Euler beam model is described for the subway tunnel structure in this paper. The equivalent stiffness of the layered soil-viscoelastic beam coupling system subjected to a moving load is formed by using the transmission and reflection matrix (TRM) method. The numerical solution of ground surface displacement caused by subway tunnel in time-space domain is obtained by IFFT algorithm. The correctness of the algorithm is verified by comparing with the reference results. Numerical results show that, with the increase of the viscous coefficient of the viscoelastic Euler beam, the vibration amplitude of the ground surface will decrease. Up to a certain value of the increasing the viscous coefficient of the Euler beam, it will have little effect on the vibration amplitude of the ground surface. Therefore, the standard solid model of viscoelastic Euler beam can well describe the creep and relaxation of materials. The model of viscoelastic beam is reasonable for the working condition of subway tunnel concrete structure. At high speed of moving load, the maximum value of ground surface displacement spectrum will appear at the smaller frequency domain and the maximum value of displacement spectrum will also increase for the soft layer soil, while it is opposite to that of the stiffer layer soil.

Piezoelectrically induced nonlinear resonances for dynamic morphing of lightweight panels

A rich variety of resonance scenarios induced in thin elastic plates by periodic axial strains is investigated to describe the morphing capability of flexible, ultra-lightweight panels. The strain-induced excitation is provided by embedded piezoelectric strips actuated by time-varying voltages. The study deals with plates characterized by geometric and mechanical symmetry entailing cylindrical motions so that an ad hoc approximate nonlinear model of elastic beams is adopted to formulate the incremental equation of motion about the nonlinear equilibrium under dead loads. An asymptotic approach based on the method of multiple scales is employed to study the plate dynamics and, primarily, to investigate the effect of the plate slenderness on the nonlinearity of the lowest normal modes. A bifurcation analysis is carried out to investigate the wide array of resonances induced by the harmonic axial excitation, acting in nontrivial equilibrium states, which include primary, super-harmonic, sub-harmonic, and principal parametric resonances, respectively. For selected slendernesses of the plate, corresponding to nonlinear softening, hardening and quasi-linear behaviors, respectively, comprehensive parametric studies on the effect of the PZT cross-sectional size, the plate damping, and the piezoelectric coupling coefficient are carried out to identify the threshold voltages triggering the different resonances.

Thermal Vibration of Zinc Oxide Nanowires by using Nonlocal Finite Element Method

Zinc oxide nanowires (ZnO NWs) can be used in some NEMS applications due to their remarkable chemical, physical, mechanical and thermal resistance properties. In terms of the suitability of such NEMS organizations, a correct mechanical model and design of ZnO NWs should also be established under different effects. In this study, thermal vibration analyses of elastic beam models of ZnO NWs are examined based on Eringen's nonlocal elasticity theory. The resulting equation of motion is solved with a finite element formulation developed for the atomic size-effect and thermal environment. The vibration frequencies of ZnO NWs with different boundary conditions are calculated under nonlocal parameter and temperature change values ​​and numerical results were discussed.

Comments on the large deformation elastic beam model developed by D.Y. Gao

In this comment mathematical formulations concerning the nonlinear beam model published in Gao [1] are analysed. The beam is subjected to vertical and axial loading (at its right end). This model can be especially used to study post-buckling problems, but in this paper we focus only on convex cases. However, some inconsistency between properties of bending problems and their convexity bounds was discovered by the authors of this comment. This is concerned with definition of an integral constant, which is in Gao [1] not strictly determined, only later in Santos and Gao [3]. In this comment there is proposed an alternative definition of this constant which should solve the mentioned discrepancy. Except this one, no in-depth revision of the considered beam model is intended here.

Analysis of mechanical behavior of a pipe-roof based on model of anisotropic plate on elastic foundation

According to the equivalent stiffness method of elastic theory, pipe roofs can be regarded as stiffening ribs and reinforced rock and soil can be regarded as an isotropic plate. Hence, pipe roofs and reinforced rock and soil in the local area of tunnel’s arch crown can be considered as an anisotropic plate on an elastic foundation with specific constraints on the four edges. To improve the traditional elastic foundation beam model for pipe roofs, an anisotropic plate model of a pipe roof on an elastic foundation is established. The hyperbolic model is adopted for the foundation reaction. The mechanical model of a pipe roof is extended from one to two dimensions and solved by finite element method via the COMSOL partial differential equation (PDE) module. The model is used to analyze the mechanical behavior of a pipe-roof for a shallow-buried and soft rock highway tunnel. Meanwhile, the calculated results of the hyperbolic and traditional models (Pasternak model and Winkler model) are compared and analyzed. The results show that the following: (1) The stress and deformation of pipe roofs mainly appear in the section no more than 6 m ahead of the working face. The pipe roof plays a role of a supporting beam to improve the stabilization of the working face. It can effectively transfer the upper load to the un-excavated section and better distribute the stress of surrounding rock. (2) The 2D pipe-roof model overcomes defects of traditional models that cannot simulate the plane stress of the reinforced area and provides a reference for the design and construction of pipe-roofs. The longitudinal bending moment (My ) is about 14 times the size of the lateral bending moment (Mx ), which indicates that the pipe-roof has an obvious effect on My along the tunnel. Mx however cannot be neglected. (3) The maximum deflection difference between the hyperbolic model and Pasternak model is about 5%, and the calculated results are in good agreement with each other. The hyperbolic model considers the monotonous and bounded characteristics of sub-grade reaction varying with the displacement. The basis of the model is verified and extent of its accuracy is demonstrated.

Measuring the location and width of the Antarctic grounding zone using CryoSat-2

Abstract. We present the results of mapping the limit of the tidal flexure (point F) and hydrostatic equilibrium (point H) of the grounding zone of Antarctic ice shelves from CryoSat-2 standard and swath elevation data. Overall we were able to map 31 % of the grounding zone of the Antarctic floating ice shelves and outlet glaciers. We obtain near-complete coverage of the Filchner–Ronne Ice Shelf. Here we manage to map areas of Support Force Glacier and the Doake Ice Rumples, which have previously only been mapped using break-in-slope methods. Over the Ross Ice Shelf, Dronning Maud Land and the Antarctic Peninsula, we obtained partial coverage, and we could not map a continuous grounding zone for the Amery Ice Shelf and the Amundsen Sea sector. Tidal amplitude and distance south (i.e. across-track spacing) are controlling factors in the quality of the coverage and performance of the approach. The location of the point F agrees well with previous observations that used differential satellite radar interferometry (DInSAR) and ICESat-1, with an average landward bias of 0.1 and 0.6 km and standard deviation of 1.1 and 1.5 km for DInSAR and ICESat measurements, respectively. We also compared the results directly with DInSAR interferograms from the Sentinel-1 satellites, acquired over the Evans Ice Stream and the Carlson Inlet (Ronne Ice Shelf), and found good agreement with the mapped points F and H. We also present the results of the spatial distribution of the grounding zone width (the distance between points F and H) and used a simple elastic beam model, along with ice thickness calculations, to calculate an effective Young modulus of ice of E=1.4±0.9 GPa.

Coupling effects of surface energy, strain gradient, and inertia gradient on the vibration behavior of small-scale beams

This paper develops three surface energy-enriched gradient elastic beam models, respectively, in the contexts of Euler-Bernoulli, Timoshenko, and Levinson-Brickford-Reddy kinematic hypotheses to investigate the coupling effects of surface energy, strain gradient, and inertia gradient on the vibration behavior of small-scale beams. Modified strain-inertia gradient elasticity theory and Gurtin-Murdoch surface elasticity theory are combined to capture three types of small-scale effects. The equations of motion and consistent boundary conditions are derived by using the Euler-Lagrange variational principle. To analyze the general free vibration problems of three non-classical beam models, we construct the related differential quadrature finite elements according to two kinds of differential quadrature-based geometric mapping schemes. The efficacy of our theoretical models and numerical solution methods is established by comparing the degenerated results with the reported ones. Finally, we apply the newly developed models to investigate the size-dependent behavior of freely vibrating small-scale beams. It is revealed that the coupling effects of three physical factors can result in not only the stiffening or softening characteristic of vibration frequencies but also the significant change in the vibration mode shapes. Besides, the introduction of the strain gradient effect may induce a boundary layer for small-scale beams with at least one end clamped.

Mechanical Behavior in Perfobond Rib Shear Connector with UHPC-Steel Composite Structure with Coarse Aggregate

Using experimental and numerical analysis, this paper aims at investigating the mechanical behavior in Perfobond rib (PBL) shear connectors with ultrahigh-performance concrete (UHPC)-steel composite structures. Twelve push-out specimens fabricated according to the design used for the connectors in the UHPC-steel composite structures in bridges have been investigated. The main objective of this paper was to discuss the mechanism of failure and the influence of different parameters on PBL shear connectors mechanical properties, including the diameter of transverse rebar, hole spacing and number of holes. The results showed that the failure mode of this type is different from that of conventional concrete specimens. During the failure of specimens, there are few cracks and the overall stiffness still maintain a high level, and reveal that a balance between the size of transverse rebar and the diameter of the hole. In addition, the basic form of the ultimate bearing capacity of existing PBL shearing bonds is summarized, and a new concept of steel fiber shearing in UHPC is proposed. Finally, based on the elastic foundation beam model, the full curve calculation formula of UHPC single-hole PBL shear bond is derived by using Timoshenko beam element. The formula calculation results are in good agreement with the experimental values.

Strata Movement Model of Filling Coal Mining Based on Two-Parameter Elastic Foundation

To avoid the discontinuity defect of the Winkler elastic foundation beam model, a two-parameter elastic foundation model of Pasternak reflecting the continuity of the waste foundation was applied in the regular analysis of strata movement in waste filling coal mining. A two-parameter elastic foundation beam strata movement mechanical model was established. The formulae to calculate the shear force, bending moment, deflection and foundation reaction force of roof strata were obtained using elastic foundation beam theory. The two-parameter elastic foundation beam model and the Winkler elastic foundation beam model were applied to an engineering example to analyze the influence of different waste foundation coefficient k and shear stiffness Gp on the force and deformation of roof strata respectively, and the results were compared accordingly. The results show that the shear stiffness has a certain influence on the force state and subsidence of the roof strata, when waste foundation coefficient is smaller, the difference between the two elastic foundation models is greater. Considering the continuity of the waste filling material, the subsidence of the roof beam obtained by the two-parameter foundation model is smaller than that by the Winkler model. The prediction of the development height of water flowing fractured zone and surface subsidence by the new model is more in line with the engineering practice.

A 3-dimensional elastic beam model for form-finding of bending-active gridshells

In this paper, we present a 3-dimensional elastic beam model for the form-finding and analysis of elastic gridshells subjected to bending deformation at the self-equilibrium state. Although the axial, bending, and torsional strains of the beam elements are small, the curved beams connected by hinge joints are subjected to large-deformation. The directions and rotation angles of the unit normal vectors at the nodes of the curved surfaces in addition to the translational displacements are chosen as variables. Based on the 3-dimensional elastic beam model, deformation of an element is derived from only the local geometrical relations between the orientations of elements and the unit normal vectors at nodes without resorting to a large rotation formulation in the 3-dimensional space. Deformation of a gridshell with hinge joints is also modeled using the unit normal vectors of the surface. An energy-based formulation is used for deriving the residual forces at the nodes, and the proposed model is implemented within dynamic relaxation method for form-finding and analysis of gridshells. The accuracy of the proposed method using dynamic relaxation method is confirmed in comparison to the results by finite element analysis. The results are also compared with those by optimization approach for minimizing the total potential energy derived using the proposed formulation.

JuSFEM: A Julia-based open-source package of parallel Smoothed Finite Element Method (S-FEM) for elastic problems

The Smoothed Finite Element Method (S-FEM) proposed by Liu G.R. can achieve more accurate results than the conventional FEM. Currently, much commercial software and many open-source packages have been developed to analyze various science and engineering problems using the FEM. However, there is little work focusing on designing and developing software or packages for the S-FEM. In this paper, we design and implement an open-source package of the parallel S-FEM for elastic problems by utilizing the Julia language on multi-core CPU. The Julia language is a fast, easy-to-use, and open-source programming language that was originally designed for high-performance computing. We term our package as juSFEM. To the best of the authors’ knowledge, juSFEM is the first package of parallel S-FEM developed with the Julia language. To verify the correctness and evaluate the efficiency of juSFEM, two groups of benchmark tests are conducted. The benchmark results show that (1) juSFEM can achieve accurate results when compared to commercial FEM software ABAQUS, and (2) juSFEM only requires 543 s to calculate the displacements of a 3D elastic cantilever beam model which is composed of approximately 2 million tetrahedral elements, while in contrast the commercial FEM software needs 930 s for the same calculation model; (3) the parallel juSFEM executed on the 24-core CPU is approximately 20× faster than the corresponding serial version. Moreover, the structure and function of juSFEM are easily modularized, and the code in juSFEM is clear and readable, which is convenient for further development.

Nonlinear secondary resonance of FG porous silicon nanobeams under periodic hard excitations based on surface elasticity theory

To impart desirable material properties, functionally graded (FG) porous silicon has been produced in which the porosity changes gradually across the material volume. The prime objective of this work is to predict the influence of the surface free energy on the nonlinear secondary resonance of FG porous silicon nanobeams under external hard excitations. On the basis of the closed-cell Gaussian-random field scheme, the mechanical properties of the FG porous material are achieved corresponding to the uniform and three different FG patterns of porosity dispersion. The Gurtin–Murdoch theory of elasticity is implemented into the classical beam theory to construct a surface elastic beam model. Thereafter, with the aid of the method of multiple time-scales together with the Galerkin technique, the size-dependent nonlinear differential equations of motion are solved corresponding to various immovable boundary conditions and porosity dispersion patterns. The frequency response and amplitude response associated with the both subharmonic and superharmonic hard excitations are obtained including multiple vibration modes and interactions between them. It is found that for the subharmonic excitation, the nanobeam is excited within a specific range of the excitation amplitude, and this range shifts to higher excitation amplitude by incorporating the surface free energy effects. For the superharmonic excitation, by taking surface stress effect into account, the excitation amplitude associated with the peak of the vibration amplitude enhances. Moreover, in the subharmonic case, it is demonstrated that by increasing the porosity coefficient, the value of the excitation frequency at the joint point of the two branches of the frequency-response curve reduces. In the superharmonic case, it is revealed that an increment in the value of porosity coefficient leads to decrease the peak of the oscillation amplitude and the associated excitation frequency.