Equivalent Stiffness Method Research Articles

Seismic Performance Analysis of Ancient Palace-Style Timber Structure Considering Column Rocking Effect

ABSTRACT The seismic response of palace-style timber structures is closely related to the rocking characteristics of column. To accurately describe the dynamic characteristics of timber structures, a nonlinear rotational spring is employed to simulate the rocking behavior of column base joints under seismic excitation. Firstly, a spring-rigid rod analysis model that comprehensively considers the nonlinear properties of column base joints, mortise-tenon joints, and bracket set joints is proposed, and a simplified lumped mass model is obtained through centroid method and equivalent stiffness method. The accuracy of the simplified model is verified through shake table tests, and nonlinear dynamic response time-history analysis of the structure is conducted. Based on the verified simplified model, the influence of tenon-mortise joint reinforcement on the seismic displacement response of the structure was further investigated. It was found that carbon fiber sheets reinforcement increased the initial stiffness of the joints by 1.45 times and reduced the displacement response of the structure by 15%. Additionally, the optimal stiffness enhancement of column frame mortise-tenon joints range from 4 to 6 times.

An Equivalent Linear Method to Predict Nonlinear Bending Mechanics of Dredging Floating Hose String

Dredging hoses are flexible and are particularly suitable for slurry transportations for mud or sand in dredging projects. To achieve sufficient bending stiffness and to prevent the pipe body from collapsing, this type of hose segment is a composite structure that is embedded with several cord reinforcement layers and steel wires in its rubber layer. To quickly evaluate the nonlinear bending mechanical properties of rubber hoses, this study proposes the equivalent stiffness method of linear superposition, which is verified by test data and numerical results. The results show that the equivalent bending stiffness method proposed in this study is in good agreement with numerical and experimental results. Then, by comparing the calculation results of the hose string, it was demonstrated that the linear stiffness superposition method proposed in this study can also accurately predict the bending mechanical behavior characteristics of string hose, and provide reliable guidance for hose design in practice.

Free vibration analysis of corrugated plate based on improved Fourier series method

Corrugated plates are widely used in ships, underwater vehicles and other engineering fields due to their good bending stiffness and strength characteristics. The properties of corrugated plates are determined by their corrugated shape, wavelength, and wave height, which affect each direction differently. In order to model these properties quickly and accurately, this paper employs an improved Fourier series method. The method is based on the Kirchhoff plate theory and equivalent stiffness method, expanding the displacement of the corrugated plate using Fourier series. By applying the energy principle, the free vibration characteristics of the corrugated plate can be determined. This approach allows for efficient and comprehensive multi-parameter control modeling of corrugated plates. The influence of structural form, material parameters and convergence on the vibration characteristics of corrugated plate is analyzed. This paper lays a foundation for the subsequent dynamic study of sandwich corrugated plate structure.

Lateral Dynamic Response of Helical Pile in Viscoelastic Foundation Considering Shear Deformation

Helical piles are a new type of pile that has good application prospects, and researchers have carried out an in-depth investigation into their vertical uplift and compressive bearing capacity. However, there is relatively little research on the dynamic bearing characteristics of helical piles. Therefore, the lateral vibration of a helical pile embedded in the viscoelastic foundation is systematically studied in this article. Utilizing the equivalent stiffness method to transform a helical pile into a cylindrical pile of special diameter, the lateral vibration model of the helical pile considering shear deformation is established based on the Winkler foundation model and the Timoshenko beam theory. The analytical solutions for the lateral dynamic displacement, bending moment, and shear force of the helical pile are strictly derived, and the rationality of the present solutions is also verified by comparing them with existing solutions. Based on the present solutions, a parametric study is carried out to investigate the influence of the pile and soil properties on the lateral dynamic response of the helical pile. It is found that the load excitation frequency and pile–soil stiffness ratio have a significant influence on the lateral dynamic displacement, bending moment, and shear force of the helical pile with space and time response.

Natural frequencies of prismatic and tapered composite girders with corrugated steel webs

The present study proposed an analytical formula to predict the natural frequencies of prismatic and tapered composite girders with corrugated steel webs (CSWs), and rigorously provides the derivation process. It is experimentally and numerically verified that the proposed formula is more applicable for predicting the natural frequencies of girders with CSWs, in comparison to the traditional formula based on Euler-Bernoulli beam theory. The study demonstrates that the traditional formula used for determining the natural frequencies of prismatic girders with CSWs leads to significant overestimation, mainly because it ignores the effects of shear deformation. The finite element (FE) simulation suggests that the free vibration mode shapes of tapered girders with CSWs exhibit sinusoidal patterns. Consequently, an equivalent stiffness method has been developed for accurately and conveniently estimating the natural frequencies of tapered girders with CSWs. This paper establishes a theoretical foundation for modifying the natural frequency calculation formula of prismatic and tapered girders with CSWs in current specifications, the research findings hold significant implications for the structural dynamic analysis and design of such girders.

Development and validation of a rotating blade-casing rubbing model by considering the blade deformation and abradable coating

Aiming at the phenomenon that blade bending deformation and coating wear are rarely considered in existing rubbing force models, a blade-casing rubbing model based on the Archard wear model was established in this paper. First, the bending deformation of the blade was deduced according to the torque balance relationship, the equivalent deformation of the coated casing was analysed using the equivalent stiffness method, and the clearance between the blade and casing was calculated with the consideration of coating wear based on the Archard model. Second, the influences of the casing stiffness, blade size, and Young's modulus of the coating on the rubbing force were analysed and compared with the existing rubbing force models. Subsequently, a rubbing force characterisation test bench was built considering the AlSi-polyester coating. A strain gauge was used to analyse the blade deformation, the rubbing force was compensated based on the acceleration sensor, and a grating displacement sensor was used to measure the penetration depth, which made the relative error between the experimental and simulation results was less than 15%. Finally, the rubbing force model was applied to the rotor–blade–casing system, and the influence of coating and coating wear on the dynamic characteristics of the rotor system is analyzed. The results show that the model proposed in this paper is more suitable for the rub-impact faults of the elastic blade with an abradable coating on the casing surface. The rubbing force model considering the coating wear is beneficial to accurately predict the rubbing force.

Stability Study of the Roof Plate of the Yuanjue Cave Based on the Equivalent Support Stiffness Method

As precious cultural heritage sites, the state of preservation of cave temples is closely related to the geological and climatic conditions in which they are located. This paper constructed an analytical method of sized slate stability based on the equivalent support stiffness method. The stability analysis of the roof slab of Yuanjue Cave was carried out by establishing a three-dimensional numerical calculation model. Through comparative analysis of the results of stress and displacement fields under different conditions, the stress and deformation characteristics of the roof slab of Yuanjue Cave were revealed, as well as the study of the main factors affecting the stability of the roof slab of Yuanjue Cave and the key slate to be monitored. The main research results are as follows. The stress deformation of the roof plate of Yuanjue cave is mainly divided into the initial uniform change stage, the medium-term stable change stage or the medium-term accelerated change stage, and the later rapid change stage. With the increase in the number of overhanging and broken slates and the increase in the damage factor of cracked slates, the vertical stress extremum of the stones increases continuously, and the equivalent support stiffness decreases, which aggravates the uneven stress deformation of the roof of the Yuanjue Cave. This study provides a reliable reference basis for the stability analysis and evaluation of the roof slab of a large number of cave temples existing in the Sichuan and Chongqing areas in China.

Energy scattering of hybrid FE-SEA model with nonlinear joints

Nonlinearity in structures causes energy scattering between different frequency ranges. This paper proposes a new hybrid FE-SEA formulation to analyse complex structures with nonlinear deterministic components. The formulation is derived in two stages. First the governing equation for deterministic components in time domain is transformed to frequency domain using the method of harmonic balance and the Galerkin method. Equivalent stiffness method is then applied in frequency domain to linearise the nonlinear terms. The second process is regarded to the derivation of the hybrid model. Based on a proposed nonlinear energy transfer model and the linearisation in the first stage, the linearised hybrid FE-SEA governing equation is derived, and the iterative procedure is given. For validation purpose, nonlinear Lagrange–Rayleigh–Ritz method plus Monte Carlo simulation is used as the benchmark model. Two case studies are applied for validation and investigation of the coupling loss factor (CLF). The coupling between subsystems at different frequencies can be reflected by the CLF. Good agreements between the benchmark model and proposed hybrid model can be observed.

Energy Flow Analysis Model of High-Frequency Vibration Response for Plates with Free Layer Damping Treatment

The energy flow analysis (EFA) model is developed for predicting the vibrational response of plates with free layer damping (FLD) treatment under high-frequency excitation. For the plate with fully free layer damping (FFLD) treatment, the energy density equation of the laminated plate with high damping is deduced by combining the equivalent complex stiffness method with the EFA theory. For the plate with partially free layer damping (PFLD) treatment, the relationship between energy density and energy intensity is obtained based on the wave theory of coupled plate by analyzing the energy transfer relationship at the damping boundary. The high-frequency energy density response of the structure with FLD treatment is solved through the energy finite element analysis (EFEA) method. To verify the developed EFEA model, several numerical analyses are performed for the plate with FFLD and PFLD treatments. The numerical results demonstrate that the obtained EFEA solutions consistently converge to the time-averaged modal analytical solutions at various analysis frequencies. Finally, the influences of the loss factor and the thickness of the damping layer on the resulting energy density are investigated in detail.

A fish-skeleton-like metastructure isolator for low-frequency vibration isolation

Vibration isolation for vibration-isolated objects in low-frequency vibration environment has been widely concerned by researchers. In this paper, we designed a local resonant (LR) fish-skeleton-like metastructure isolator with cantilever beam oscillators for low-frequency vibration isolation. Based on the theories of wave dynamics and structural dynamics, we analytically obtained the bandgap characteristics and transmissibility of the metastructure by using the equivalent mass and stiffness method, and carried out simulations through the finite element method (FEM). This metastructure isolator was fabricated by a 3D printing equipment and the vibration isolation test was carried out. Vibration transmissibility characteristics is obtained through the analytical method, FEM and experiment show that the local resonance near the natural frequency of the oscillators can effectively suppress the vibration transmission. The specific performance shows that the vibration transmissibility is as low as 21 % at the center frequency of the bandgap, while the mass of the isolator is only increased by about 2.3 % compared with the ordinary structure which has no oscillators. With the increasing of the mass of vibration-isolated objects (650 g), the transmissibility can be as low as 4 %. In addition, the contribution of different frame bodies and different oscillators of the isolator on the isolation effect and a way of widening the isolation frequency ranges are clarified. The proposed fish-skeleton-like metastructure in the present work provides a new design approach for vibration isolation in engineering applications.

Coupled Eulerian-Lagrangian Debris Flow Model with Flexible Barrier

Natural hazards such as large debris flow events can have catastrophic effects on the environment and critical infrastructure, posing a significant threat to human life. Debris flows often exhibit high velocity, high-pressure discharges due to their bulk volume, and the capacity to transport considerable volumes of large rocks, boulders, and woody debris. Although debris flow run-out simulations are commonly performed using hydraulic modelling software, these environments are seldom capable of assessing the interaction between the debris fluid, transported material, and protective structures. In this research, large deformation numerical models are calibrated using input parameters from hydraulic modelling software. Due to the computational cost of the large deformation models involving fluid-solid-structure simulation with flexible net barriers, an equivalent stiffness method is implemented to provide comparable performance through a membrane structure. The Coupled Eulerian-Lagrangian Finite Element method is used to model the impact forces of rocky boulders on the membrane, exhibiting damage characteristics consistent with flexible ring-net protective structures. The Coupled Eulerian-Lagrangian model results highlight the performance of the simplified membrane, as shown through a benchmark simulation of debris flow with boulders.

Optimization Design of Short Shell Structure of Rocket Fuel Tank Based on Kriging Surrogate Model

Abstract Aiming at the weight reduction optimal design of short shell structure of rocket fuel tank, this paper deals with the structural optimization under the action of axial load. First, the accuracy of equivalent stiffness method is verified by post-buckling analysis of short shell in ABAQUS. Then, according to the parametric model of short shell, based on Kriging surrogate model and EGO algorithm, the optimal structure without and with fillet are obtained under the given design load. Finally, a parametric study is carried out on the influence of fillet on the bearing capacity per unit mass of short shell. The results show that when the width of rib or the spacing of transverse rib is small, and the height of rib, the thickness of skin or the spacing of longitudinal rib is large, processing large fillet for load-bearing purpose is less beneficial or even disadvantageous.

Micromechanical constitutive modeling of tensile and cyclic behaviors of nano-clay reinforced metal matrix nanocomposites

Metal matrix nanocomposites are emerging rapidly among both research and industry communities. Nonetheless, the studies for their use in Low-Cycle Fatigue (LCF) applications are rarely documented. Prior to tailoring the microstructure for the desired LCF performance, it is a fundamental task to understand the structure-property correlations of the nanocomposites. In the present paper, a continuum-based micromechanical model starting from the nanostructure is proposed to fulfill such a task. The proposed model is applied to a piston aluminum alloy reinforced with 1 wt% nano-clay subjected to fully-reversed strain-controlled LCF loading at room temperature. The model, formulated within Eshelby's inclusion framework, considers the active clay/matrix interactions, the intercalation/exfoliation effect via the equivalent stiffness method, and the size effect through the introduction of the interphase thickness as a characteristic length scale. To render the model able to reproduce the cyclic hysteretic behavior of the nanocomposites, the isotropic/kinematic hardening rule is considered to describe the aluminum alloy plasticity. The model results are favorably compared to the experimental observations in terms of monotonic and cyclic elastoplastic stress-strain responses of the aluminum/clay nanocomposite. A parametric study is then carried out in order to understand the separate and synergic effects of clay weight fraction, clay geometrical parameters (e.g., shape and size), and clay structural parameters (e.g., number of silicate layers and interlayer spacing) on the overall monotonic and cyclic responses of the present nanocomposite. The reinforcing effect of each parameter is discussed concerning the micromechanical model.

Modal analysis of the stator system of a permanent magnet synchronous motor with integer slot multi‐pole pair for electric vehicles

The 0-order natural frequency resonance is the main cause of electromagnetic vibration noise in integer slot multi-pole pair permanent magnet synchronous motors (PMSMs) for electric vehicles. To study this phenomenon, this study proposes a finite element (FE) model of the stator system based on a 48-slot 8-pole permanent magnet synchronous motor for electric vehicles. Based on this model, the effects of winding, frame end cover and temperature on the natural frequencies of a stator system are analysed by using the equivalent stiffness method (ESM). The simulation results show that the 0-order natural frequency of the stator core system decreases after considering the winding varnished, and the 0-order natural frequency is almost unaffected after considering the frame and end cover. However, the other natural frequencies increase in different degrees. In addition, the natural frequencies of the stator core decrease with the increase of temperature. The accuracy of the finite element model is verified by the modal experiment of the hammer method (HM).

An Equivalent Radial Stiffness Method of Laboratory SEPT on Anchorage Performance Prediction of Rockbolts under Different Field Geoconditions

The short encapsulation pull-out test (SEPT) is extensively used in rockbolting research or engineering. The field SEPT is time-consuming and labor-intensive, and its result is only applicable to the tested in situ. The laboratory SEPT is usually employed in theoretical rockbolting research due to its easily controlled variables. However, the design of laboratory SEPT is quite different, as there is no standard testing method, resulting in the applicability and limitations of each study not being clear. Accordingly, the aim of this paper is to bridge the gap between laboratory SEPT research and field application. On the basis of thick-walled cylinder theory, a mechanical model of a rock bolt subjected to axial load was established under consideration of the deformational behavior of confining materials around the bolt. Plane stress analysis was introduced to derive the analytical relationship between the axial force of the bolt and the deformation of the confining materials. A new approach of laboratory SEPT sample design was established, namely, equivalent radial stiffness theory, to simulate anchorage performance in a specific in-situ geocondition. Consequently, the field SETP could be replaced by laboratory testing using properly designed bolting samples with a certain level of accuracy. In addition, the application scope of previous laboratory SEPT research could be accurately defined. Laboratory SEPT was carried out to study the anchoring performance of right spiral rebar bolts under different confining materials. Poly Vinyl Chloride (PVC) tubes with a thickness of 31 mm, #60 aluminum (Al) tubes with a thickness of 5.8 mm, and #20 steel tubes with a thickness of 5.5, 7.0 mm were used in sample preparation to simulate soft, medium, and hard surrounding rocks in the field. The anchorage performance of the bolt under different geoconditions was systematically proposed, which provides a technical approach for similar research using different anchoring materials. A negative exponential expression formulating the axial load capacity of the right spiral bolts for the full spectrum of the surrounding rocks’ strength was derived on the basis of theoretical analysis and data regression. It can be used for preliminary reinforcement design, as well as the accurate key parameter setting in the numerical calculation of roadway deformation using right spiral bolts. The theoretical prediction is highly consistent with the testing results in the literature, which confirms the validity and reliability of this research. This study contributes to the establishment of a laboratory SEPT standard in rock mechanics.

A New Cam-follower Safety Joint Mechanism Design based on Variable-length Four-bar Linkage for Robot Safety

Abstract Humans come into physical contacts with various machines such as robots in daily life. This leads to the underlying issue of guaranteeing safety during such human-robot interactions. Thus, many devices and methods have been studied for impact damage reduction. A safety joint mechanism (SJM) using four-bar linkages has been highlighted as an impact cutoff device owing to its capabilities of nonlinear load transfer. This paper focuses on a new design and testing for a kinematic element of an SJM based on four-bar linkages to improve the impact cutoff performances. In the present work, a set of variable-length floating link designs is proposed, and the mechanism is implemented by mechanical contact surface profile shaping between the cams and followers. The performance of the cam-follower mechanism is evaluated depending on the variable length of the floating link, by using equivalent stiffness method, which successfully quantifies the performance of the proposed mechanism. Based on this design and analysis, two SJMs having symmetrical arrangements for four numbers of cam-follower mechanisms are fabricated: one SJM has fixed-length floating links and the other has variable-length floating links. The effect of the new kinematic elements on the performance improvement is verified by comparing the absorbed impact rates of the two SJMs by impact hammer-like drop tests. Consequently, it is confirmed that the rapid length change of the floating link is the core element for improving the performance of the safety mechanism.

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.

An Optimization Approach for Stiffener Layout of Composite Stiffened Panels Based on Moving Morphable Components (MMCs)

An explicit topology optimization method for the stiffener layout of composite stiffened panels is proposed based on moving morphable components (MMCs). The skin and stiffeners are considered as panels with different bending stiffnesses, with the use of equivalent stiffness method. Then the location and geometric properties of composite stiffeners are determined by several MMCs to perform topology optimization, which can greatly simplify the finite element model. With the objective of maximizing structural stiffness, several typical cases with various loading and boundary conditions are selected as numerical examples to demonstrate the proposed method. The numerical examples illustrate that the proposed method can provide clear stiffener layout and explicit geometry information, which is not limited within the framework of parameter and size optimization. The mechanical properties of composite stiffened panels can be fully enhanced.

Research on Application of Buckling Restrained Braces in Strengthening of Concrete Frame Structures

This paper examines the application of BRB in strengthening of reinforced concrete frame structures to meet seismic requirements according to Chinese seismic design code. Elastic response spectrum analysis and nonlinear time history analysis are performed by taking a real engineering example that suffers weak first floor irregularity due to added loads and addition of one floor. With the method of equivalent stiffness and displacement-based design method, buckling restrained brace parameters are deduced and are used to model BRB in ETABS using plastic wen model. Three configurations of buckling restrained braces are studied together with ordinary braces. Under elastic state, the relationship between the required cross section area of BRB and ordinary braces is deduced from the formula of calculating elastic bearing capacity where it is shown that the area of the ordinary braces must be 1.25 times that of BRB for ensuring the same performance. The results show that Inverted V brace configuration demonstrated better performance over single brace and V brace configurations and X brace configuration, although not recommended by Chinese code, is simulated and used in this paper and has demonstrated better performance over other configurations, and the further research on the practical use of this brace is recommended. Also, under action of strong earthquakes, by nonlinear time history analysis, buckling restrained braces demonstrated better performance of strengthening the structure and make it meet the requirement of code. Under this same condition, ordinary braces losses their bearing capacity due to excessive buckling.

A Model Predictive Controller with Longitudinal Speed Compensation for Autonomous Vehicle Path Tracking

Autonomous vehicle path tracking accuracy faces challenges in being accomplished due to the assumption that the longitudinal speed is constant in the prediction horizon in a model predictive control (MPC) control frame. A model predictive control path tracking controller with longitudinal speed compensation in the prediction horizon is proposed in this paper, which reduces the lateral deviation, course deviation, and maintains vehicle stability. The vehicle model, tire model, and path tracking model are described and linearized using the small angle approximation method and an equivalent cornering stiffness method. The mechanism of action of longitudinal speed changed with state vector variation, and the stability of the path tracking closed-loop control system in the prediction horizon is analyzed in this paper. Then the longitudinal speed compensation strategy is proposed to reduce tracking error. The controller designed was tested through simulation on the CarSim-Simulink platform, and it showed improved performance in tracking accuracy and satisfied vehicle stability constrains.