Elastic Beam Research Articles

Dynamic coefficient of flexural motion of beam experiencing simple support under successive moving loads

AbstractAnalytic expressions of the dynamic coefficient (DC) factor and vibrational behavior of a uniformly elastic isotropic beam with a simple boundary condition caused by accelerating masses with varying velocities are analyzed. The motion of this problem is described by a fourth‐order partial differential equation, which governs its behavior. The weighted residual method converts the governing equation into a sequence of linked second‐order differential equations to facilitate the analysis. A rewritten version of Struble's asymptotic method further simplifies the transformed governing equation. This modification aids reduction in the complexity of the equation. The closed‐form response is contrasted across three force motions: acceleration, deceleration, and uniform motion. The study thoroughly examines how different velocities and frequencies of the moving force affect the dynamic behavior of the beam. The study also examines the influence of load velocity on the DC of the beam subjected to pinned–pinned boundary conditions.

Deformation and Fracture Mechanisms of Thick Hard Roofs in Upward Mining Coalfaces: A Mechanical Model and Its Validation

The safety and efficiency of underground coal mining are threatened by thick hard roofs characterized by large overhang areas, problematic spontaneous caving, and high dynamic load upon their breakage. In this study, a mechanical model of the bearing capacity of thick hard roofs in upward mining coalfaces associated with mining activities is built based on bending theories for beams with single generalized displacement and the elastic foundation beam theory. Using this method, we analyze the deformation and fracture mechanisms of a thick hard roof during upward mining. We further derive the mechanical equations of rotational angle, bending moment, shear force, and deflection of the free overhang and coal-bearing zone in the thick hard roof and an equation for calculating the limiting span. The mechanical behaviors of the thick hard roof bearing state are analyzed under different parameters. The results show that the foundation coefficient, roof thickness, and angle of upward mining have little influence on the roof bending moment but are positively correlated to the limiting span. Roof load and overhang length have a significant influence on the roof bending moment. They are negatively and positively correlated with the limiting span, respectively. Finally, a case study is performed on the Ш601 upward mining coalface in the Zhuzhuang Coal Mine. The distribution characteristics of the bending moment of the thick hard roof at different extraction stages are analyzed. At each stage, the limiting spans of the thick hard roof upon breaking were calculated as 13.18, 18.82, and 22.50 m, respectively, being close to the on-site measured periodic weighting lengths of 13.33, 19.33 m, and 22.67 m. This close fit proves the feasibility and accuracy of the developed mechanical model. The present study offers theoretical guidance for estimating the weighting length of thick hard roofs in coalfaces and for engineering technology control in similar scenarios.

The topological dynamics of continuum lattice grid structures

Continuum lattice grid structures which consist of joined elastic beams subject to flexural deformations are ubiquitous. In this work, we establish a theoretical framework of the topological dynamics of continuum lattice grid structures, and discover the topological edge and corner modes in these structures. We rigorously identify the infinitely many topological edge states within the bandgaps via a theorem, with a clear criterion for the infinite number of topological phase transitions. Then, we obtain analytical expressions for the topological phases of bulk bands, and propose a topological index related to the topological phases that determines the existence of the edge states. The theoretical approach is directly applicable to a broad range of continuum lattice grid structures including bridge-like frames, square frames, kagome frames, continuous beams on elastic springs. The frequencies of the topological modes are precisely obtained, applicable to all the bands from low to high frequencies. Continuum lattice grid structures serve as excellent platforms for exploring various kinds of topological phases and demonstrating the topological modes at multiple frequencies on demand. Their topological dynamics has significant implications in safety assessment, structural health monitoring, and energy harvesting.

Emergent behaviors of buckling-driven elasto-active structures

Active systems of self-propelled agents, e.g., birds, fish, and bacteria, can organize their collective motion into myriad autonomous behaviors. Ubiquitous in nature and across length scales, such phenomena are also amenable to artificial settings, e.g., where brainless self-propelled robots orchestrate their movements into spatial-temporal patterns via the application of external cues or when confined within flexible boundaries. Like their natural counterparts, these approaches typically require many units to initiate collective motion, so controlling the ensuing dynamics is challenging. Here, we demonstrate a simple mechanism that leverages nonlinear elasticity to tame near-diffusive motile particles in forming structures capable of directed motion and other emergent behaviors. Our elasto-active system comprises two centimeter-sized self-propelled microbots connected with elastic beams. These microbots exert forces that suffice to buckle the beam and set the structure in motion. We first rationalize the physics of the interaction between the beam and the microbots. Then we use reduced-order models to predict the interactions of our elasto-active structures with boundaries, e.g., walls and constrictions, and demonstrate how they can exhibit remarkable emergent behaviors such as maze navigation. These findings demonstrate that allowing and understanding changes in body morphology can enhance the capabilities of active matter systems and enable the design of robotic materials capable of space exploration, adaptation, and complex interactions with their surrounding environment.

A new local resonance metamaterial for flat and cylindrical structures depended on elastic chiral spiral beams

A new local resonance metamaterial for flat and cylindrical structures depended on elastic chiral spiral beams

Experimental investigation on shear fatigue behavior of perfobond strip connectors made of high strength steel and ultra-high performance concrete

This paper explores the fatigue performance of perfobond strip connectors (PBL) made from high-strength steel (HSS) and ultra-high performance concrete (UHPC) by fabricating and testing eight push-out specimens. The failure modes, load-slip curves, stiffness degradation, and residual strength of the PBLs are presented and discussed. The experimental results show that fatigue loads caused connector’s crack formation and stiffness degradation, leading to concrete dowel shear fracture and transeverse rebar bending-shear rupture damage. Compared to the static performance, the fatigue residual strength of the PBL subjected to 0.5 × 106, 1.25 × 106, and 2.0 × 106 loading cycles decreases by 1.2 %, 3.6 %, and 14.5 %, respectively, with corresponding reductions in ductility of 14.7 %, 19.4 %, and 4.7 %. Under the same loading cycles, the fatigue residual strength and ductility of the PBL with a maximum repeated load of 0.65Pu are 13.0 % and 3.9 %, respectively, smaller than those of 0.55Pu. A semi-empirical model based on the elastic foundation beam theory and linear cumulative damage model is developed to predict the residual strength of the PBLs using HSS and UHPC. The accuracy and applicability of the model are calibrated using the experimental results. The findings of this study can serve as an experimental foundation and theoretical reference for the fatigue design of PBLs made with high-strength materials.

Three-dimensional numerical analysis of the effect of foundation stiffness on soil-structure interaction

Stiffness-related geometric characteristics are closely linked to the performance of structural systems. The influence of foundation stiffness on the soil-structure interaction (SSI) mechanism was investigated by numerical modeling of different building foundations. Numerical tools (PLAXIS 3D and TQS/CAD) were used for decoupling analyses of the interaction between the structural components. Results obtained from instruments installed in a building case study were used to refine the numerical calculations. Linear elastic constitutive models and beam elements were useful in reducing computational processes and ensuring study feasibility. The simulation results provided a satisfactory representation of field observations. Interaction between piles and the environment governs the relationship between foundation stiffness and uniform settlement. Rigid pile groups are achieved with greater pile length, diameter and spacing. This study highlights the different effects of pile geometric parameters on loading and unloading rates in pillars, demonstrating that the SSI mechanism is more sensitive to less rigid foundations, particularly those with greater pile embedment. It was concluded that numerical simulations to predict settlement as realistically as possible require understanding and selecting the most appropriate model and type of calculation.

Problem of Three-Point Bending of an Elastic Beam from Porous Metal

Using numerical methods, we construct a solution to a physically and geometrically nonlinear problem of three-point bending of an elastic beam, made of porous metal, with rectangular cross-section. Unlike the classical version of the problem for a homogeneous beam, the heterogeneity over the cross-section due to material compaction because of the collapse of pores, which occurs in the compression zone at sufficiently large deflections, is taken into account. To describe the elastic state of a porous metal, the stress – strain diagram of a bimodular medium is used. The results of computations of strong bending of a beam, made of the low-porosity aluminum foam, are presented. These results demonstrate the difference between the obtained solution and similar solutions for beams, made of homogeneous porous and compacted material.

Fast and Compact Partial Differential Equation (PDE)-Based Dynamic Reconstruction of Extended Position-Based Dynamics (XPBD) Deformation Simulation

Dynamic simulation is widely applied in the real-time and realistic physical simulation field. How to achieve natural dynamic simulation results in real-time with small data sizes is an important and long-standing topic. In this paper, we propose a dynamic reconstruction and interpolation method grounded in physical principles for simulating dynamic deformations. This method replaces the deformation forces of the widely used eXtended Position-Based Dynamics (XPBD), which are traditionally derived from the gradient of the energy potential defined by the constraint function, with the elastic beam bending forces to more accurately represent the underlying deformation physics. By doing so, it establishes a mathematical model based on dynamic partial differential equations (PDE) for reconstruction, which are the differential equations involving both the parametric variable u and the time variable t. This model also considers the inertia forces caused by acceleration. The analytical solution to this model is then integrated with the XPBD framework, built upon Newton’s equations of motion. This integration reduces the number of design variables and data sizes, enhances simulation efficiency, achieves good reconstruction accuracy, and makes deformation simulation more capable. The experiment carried out in this paper demonstrates that deformed shapes at about half of the keyframes simulated by XPBD can be reconstructed by the proposed PDE-based dynamic reconstruction algorithm quickly and accurately with a compact and analytical representation, which outperforms static B-spline-based representation and interpolation, greatly shortens the XPBD simulation time, and represents deformed shapes with much smaller data sizes while maintaining good accuracy. Furthermore, the proposed PDE-based dynamic reconstruction algorithm can generate continuous deformation shapes, which cannot be generated by XPBD, to raise the capacity of deformation simulation.

A size‐dependent model of strain gradient elastic laminated micro‐beams with a weak adhesive layer

AbstractA size‐dependent model for a laminated micro‐beam with a soft adhesive is developed in the framework of strain gradient elasticity theory. The layered beam is constituted by two Euler–Bernoulli strain gradient elastic isotropic beams, joined through a strain gradient spring‐type contact law at the adhesive level. The governing bending and extensional equations and boundary conditions are obtained by using the variational principle. The differential system shows a coupling between the flexural and axial behaviors of the upper and lower beams, due to the presence of interface terms related to shear and peeling stresses. Two benchmark problems have been presented through their closed‐form solutions, namely a simply‐supported laminated beam subjected to a uniform distributed load and a mode 2‐type loading configuration of a layered axially deformable beam. Size effects and non‐local phenomena, due to high strain concentrations, are highlighted. Though simple in their features, the examples prove the efficiency of the proposed approach in designing micro‐scale layered beams.

Analysis of the underlying shield tunnel deformation caused by a newly built channel

As the development of underground space progresses, newly constructed waterways inevitably intersect with existing tunnels, posing significant challenges to the safety of underlying tunnels and making deformation control a critical issue in engineering design and construction. This study investigates the impact of waterway foundation pit excavation on the deformation of underlying tunnels. The research considers not only the vertical unloading at the bottom of the pit but also the horizontal unloading from the four side walls, which induces additional vertical stress fields in the underlying soil. These additional stresses are applied to the tunnel structure, modeled as a Pasternak elastic foundation beam, and the vertical tunnel deformation due to excavation unloading is calculated using the finite difference method. To validate the analytical results, a three-dimensional numerical model is employed. Furthermore, the study analyzes deformation trends under varying excavation sizes and the relative positions of the foundation pit and tunnel. The findings demonstrate that the proposed method provides a highly accurate, cost-effective, and efficient solution for predicting tunnel deformation during the excavation of small foundation pits. This research offers valuable insights for the design and assessment of underground structures in urban environments.

A projection-based time-segmented reduced order model for fluid-structure interactions

In this paper, a type of novel projection-based, time-segmented reduced order model (ROM) is proposed for dynamic fluid-structure interaction (FSI) problems based upon the arbitrary Lagrangian–Eulerian (ALE)-finite element method (FEM) in a monolithic frame, where spatially, each variable is separated from others in terms of their attribution (fluid/structure), category (velocity/pressure) and component (two/three dimension) while temporally, the proper orthogonal decomposition (POD) bases are constructed in some deliberately partitioned time segments tailored through extensive numerical trials. By the combination of spatial and temporal decompositions, the developed ROM approach enables prolonged simulations under prescribed accuracy thresholds. Numerical experiments are carried out to compare numerical performances of the proposed ROM with corresponding full-order model (FOM) by solving a two-dimensional FSI benchmark problem that involves a vibrating elastic beam in the fluid, where the performance of offline ROM on perturbed physical parameters in the online phase is investigated as well. Extensive numerical results demonstrate that the proposed ROM has a comparable accuracy to while much higher efficiency than the FOM. The developed ROM approach is dimension-independent and can be seamlessly extended to solve high dimensional FSI problems.

Harvesting Broadband Breeze Vibration Energy via Elastic Bistable Triboelectric Nanogenerator for In Situ, Self‐Powered Monitoring of Power Transmission Lines

AbstractTriboelectric nanogenerator (TENG), as a new technique for energy capture, has unique advantages in harvesting high‐entropy energy. In view of the wind‐induced vibration characteristics of power transmission lines, the symmetric elastic bistable triboelectric nanogenerator (EB‐TENG) is introduced in this work. The core component of the EB‐TENG is the elastic bistable cantilever beam structure, which is designed by introducing an elastic perturbation structure to the conventional cantilever beam. This beam results in several sub‐resonance frequencies in addition to the natural frequencies, which broadens the frequency band of the EB‐TENG. In addition, the EB‐TENG adopts a symmetrical cantilever beam to ensure effective harvesting of vibration energy, even in situations of self‐tilting and external tilting vibration. The experimental outcomes confirm that the EB‐TENG can significantly harvest vibration energy in the 7–60 Hz frequency range and delivers the maximum peak power of 2.846 mW, which meets the major vibration frequency range of power transmission lines under the action of breeze. Finally, self‐powered strategy based on EB‐TENG online monitoring devices (such as tower obstruction lights, temperature and humidity sensors, and line high temperature wireless alarms) is constructed. This work helps promote the application of in‐situ self‐powered monitoring based on TENG in smart transmission lines.

Piezoelectric energy harvesting from walking motion of a passive biped robot model with flexible legs

Passive dynamic walking biped robots utilize the inherent dynamics of their structure to achieve walking motion without continuous actuation, enhancing energy efficiency and stability. Replacing conventional rigid legs with flexible elastic beams in the simplest passive walker will be associated with undesirable vibrations. Despite the positive effect of the damper in reducing vibrations, the current research has used the piezoelectric energy harvesting with the aim of reducing the wasted energy of the gait cycle and improving its stability range. For this purpose, the electromechanical Lagrange equations are determined by applying Hamilton's principle and the assumed mode method and then, the new definition of the step function is obtained through numerical methods, solving a boundary value problem to establish proper initial conditions for stable walking. Numerical simulations indicate that the piezoelectric energy harvester can create a stable period-one gait cycle by optimizing the length of the piezo layers attached to the undamped elastic legs despite the presence of small vibrations. Additional results are presented in bifurcation diagrams, investigating the effect of important physical and structural parameters such as slope angle, length and thickness of the piezo layer.

Design and kinematics analysis of a cable-stayed notch manipulator for transluminal endoscopic surgery

The friction between the joints of the continuum manipulator with discrete joints brings great difficulties to kinematic modeling. The traditional driving wire arrangement limits the load capacity of the manipulator. A cable-stayed notch manipulator for transluminal endoscopic surgery is proposed, and a driving force coupling kinematic mode is established. The manipulator is fabricated from a superelastic Nitinol tube with bilaterally cut rectangular notches and is actuated by a stay cable. By applying the comprehensive elliptic integral solution (CEIS) for large deformation beams, the bending angle of each elastic beam is obtained, and the kinematics from the driving space to the joint space is formed. According to the bending angle of each elastic beam, the expression of the manipulator in Cartesian space can be obtained by geometric analysis. The kinematics from the joint space to the Cartesian space is established. The outer diameter of the manipulator is only 3.5 mm, and the inner diameter can reach 2 mm, allowing instruments to pass through. The maximum error of the manipulator movement is less than 5%. The load capacity of the manipulator has been verified through the stiffness experiments, and the maximum load of the manipulator can reach 400 g. The cable-stayed notch manipulator can be accurately modeled on the base of CEIS, and its motion accuracy can meet the needs of engineering applications. The compact size and excellent load capacity of the manipulator make it potential for application in transluminal endoscopic surgical robots.

Computation of dynamic deflection in thin elastic beam via symmetries

The deflection profiles governed by Euler Bernoulli's fourth-order equations under varied applied loads are investigated in this research. This study provides essential insights for engineers designing aircraft components, bridges, and similar structures, ensuring system safety and efficiency. The investigation emphasizes critical factors such as amplitude and frequency, load history, and material properties. Initially, conservation laws of the equations with applied loads are derived by expressing them in the Euler-Lagrange form, where the resultant conservation laws satisfy the divergence expression. The association between symmetries and conservation laws is demonstrated, followed by the application of double reduction theory, which reduces both the variables and the order of the equation. Graphical representations of the outcomes illustrate the impact of load variations on the beam's deflection profiles. These visual aids facilitate a deeper understanding of the influence of different loading conditions. A comparison between varying loads is presented, showcasing the impact of these variations on structural behavior. The findings are crucial for enhancing structural design and ensuring safety under varied loading conditions, showcasing the novelties in the analytical approach and the practical applications of the derived results.

A 3-DOF spreading precision positional stage for ductile end-face fly-cutting of quartz glass

Ductile removal is widely employed to eliminate subsurface damage in brittle materials. Achieving this requires the cutting depth to be set extremely low, presenting significant challenges for error compensation and precise feeding of the machine tool. In this paper, a novel three-degree-of-freedom spreading precision positioning stage is developed to mitigate the effects of workpiece deflection errors and feed resolution on the depth of cut during the end-face fly-cutting process. First, a bridge and half-bridge composite structure is designed to facilitate the planar spreading of the spatial motion mechanism. A mathematical model of the composite structure is developed based on elastic beam theory. Second, the effects of various structural parameters on the amplification ratio of the structure are investigated. The accuracy of the theoretical model is verified by finite element analysis. Finally, ductile fly-cutting experiments on quartz glass are conducted using a precision 5-axis machine tool.

Harnessing centrifugal and Euler forces for tunable buckling of a rotating elastica

We investigate the geometrically nonlinear deformation and buckling of a slender elastic beam subject to time-dependent ‘fictitious’ (non-inertial) forces arising from unsteady rotation. Using a rotary apparatus that accurately imposes an angular acceleration around a fixed axis, we demonstrate that dynamically coupled centrifugal and Euler forces can produce tunable structural deformations. Specifically, by systematically varying the acceleration ramp in a highly automated experimental setup, we show how the buckling onset of a cantilevered beam can be precisely tuned and its deformation direction selected. In a second configuration, we demonstrate that Euler forces can cause a pre-arched beam to snap-through, on demand, between its two stable states. We also formulate a theoretical model rooted in Euler’s elastica that rationalizes the problem and provides predictions in excellent quantitative agreement with the experimental data. Our findings demonstrate an innovative approach to the programmable actuation of slender rotating structures, where complex loading fields can be produced by controlling a single input parameter, the angular position of a rotating system. The ability to predict and control the buckling behaviors under such non-trivial loading conditions opens avenues for designing devices based on rotational fictitious forces.

A sandwich-type Winkler-Pasternak double elastic foundation beam model for analysis of nut column load transfer systems of shiplifts

The rack and pinion shiflifts feature its capability of fortification against extreme accidents of unbalance of ship chamber. This requires the load transfer systems of nut columns to be able to transfer the huge and centralized loads from the safety mechanisms to the tower columns in a distributive bearing way. Aiming at exploring a design-oriented computation method for the nut column load transfer systems (NCLTSs), a sandwich-type semi-infinite coupling Winkler-Pasternak double elastic foundation beam model (DEFBM) has been established, with a distributive assumption of the axial constrain force on the interfaces between the nut columns, mortar beds, adjusting beams and the second stage concrete based on the physical model test, has been proposed. A set of formulars for describing the distribution of deflections, internal forces and stress of nut columns and adjusting beams have been derived. A case study of the Three-Gorges shiplift has been conducted to demonstrate the approach of calculation of internal forces and stress by applying the formulas based on sandwich-type Winkler-Pasternak DEFBM, showing the convenience and rationality of the method to parametric layout design of the nut columns and adjusting beams in NCLTSs. Not only to NCLTSs and rack load transfer systems of shiplifts, the sandwich-type Winkler-Pasternak DEFBM also supplies the reference to the modeling and analysis of the rail-embedment structural systems broadly applied in hydraulic engineering.

ANALYTICAL DESCRIPTION AND ALGORITHM OF CALCULATIONS OF MECHANICS OF STRUCTURES WITH BERNOULLI – EULER BEAMS USING SCAS KIDY

The problems of developing a universal analytical description and algorithm for automatic computer calculations of dynamics, statics and kinetostatics of mechanical models of structures including a beam lattice are considered. This can be calculations of transient processes, steady-state free and forced oscillations, determination of equilibrium positions and stress-strain states under static and dynamic loads, etc. The structure itself can be flat or spa- tial, stationary or moving on a plane or in space. Various instruments and devices can be attached to it. Arbitrary connections can be taken into ac- count. It is shown how it is possible to succinctly, using a special language for preparing computer data, analytically describe a part of a structure rep- resenting a beam lattice. Based on the theory of elasticity of Bernoulli-Euler beams, 2 forms of the canonical representation of the potential energy of an elastic beam are obtained. This makes it possible to introduce into the language of description of mechanical models of the special computer algebra system KiDyM (SCAS KiDyM) a new beam element, for which the position of the coordinate systems associated with the extreme sections is indi- cated. The positions of these sections are determined by lattice nodes, like solid bodies. Angular and linear coordinates of such solid bodies give gen- eralized coordinates of the mechanical model. An algorithm has been developed for the formation of elements adopted to describe mechanical models in SCAS KiDyM. Thus, the elastic structure of the mechanical model is formed. Using the available tools in this program, equations of dynamics and statics are automatically constructed, that is, a mathematical model is formed, and dynamic and static calculations can be carried out. The proposed method is demonstrated in detail on the example of calculating the deformation of an elastic lattice, with is the basis of a UAV (unmanned aerial vehi- cle). The results are compared with calculations using the ANSYS program