Elastic Foundation Model Research Articles

Surrogate articular contact models for computationally efficient multibody dynamic simulations

Contact occurs in a wide variety of multibody dynamic systems, including the human musculoskeletal system. However, sensitivity and optimization studies of such systems have been limited by the high computational cost of repeated contact analyses. This study presents a novel surrogate modeling approach for performing computationally efficient three-dimensional elastic contact analyses within multibody dynamic simulations. The approach fits a computationally cheap surrogate contact model to data points sampled from a computationally expensive elastic contact model (e.g., a finite element or elastic foundation model) and resolves several unique challenges involved in applying surrogate modeling techniques to elastic contact problems. As an example application, we performed multibody dynamic simulations of a Stanmore wear simulator machine using surrogate and elastic foundation (EF) contact models of a total knee replacement. Accuracy was assessed by performing eleven dynamic simulations with both types of contact models utilizing large variations in motion and load inputs to the machine. Wear volumes predicted with the surrogate contact models were within 1.5% of those predicted with the EF contact models. Computational speed was assessed by performing five Monte Carlo analyses (over 1000 dynamic simulations each) with surrogate contact models utilizing realistic variations in motion and load inputs. Computation time was reduced from an estimated 284 h per analysis with the EF contact models to 1.4 h with the surrogate contact models (i.e., 17 min vs. 5 s per simulation), with higher wear sensitivity observed for motion variations than for load variations. The proposed surrogate modeling approach can significantly improve the computational speed of multibody dynamic simulations incorporating three-dimensional elastic contact models with general surface geometry.

Nanomechanical Probes of Single Corneal Epithelial Cells: Shear Stress and Elastic Modulus

Living human corneal epithelial cells have been probed in vitro via atomic force microscopy, revealing the frictional characteristics of single cells. Under cell media, measured shear stresses of 0.40 kPa demonstrate the high lubricity of epithelial cell surfaces in contact with a microsphere probe. The mechanical properties of individual epithelial cells have been further probed through nanometer scale indentation measurements. A simple elastic foundation model, based on experimentally verifiable parameters, is used to fit the indentation data, producing an effective elastic modulus of 16.5 kPa and highlighting the highly compliant nature of the cell surface. The elastic foundation model is found to more accurately fit the experimental data, to avoid unverifiable assumptions, and to produce a modulus significantly higher than that of the widely used Hertz–Sneddon model.

A three-parameter elastic foundation model for interface stresses in curved beams externally strengthened by a thin FRP plate

Externally bonding of fiber reinforced polymer (FRP) plates or sheets has become a popular method for strengthening reinforced concrete structures. Stresses along the FRP–concrete interface are of great importance to the effectiveness of this type of strengthening because high stress concentration along the FRP–concrete interface can lead to the FRP debonding from the concrete beam. In this study, we develop an analytical solution of interface stresses in a curved structural beam bonded with a thin plate. A novel three-parameter elastic foundation model is used to describe the behavior of the adhesive layer. This adhesive layer model is an extension of the two-parameter elastic foundation commonly used in existing studies. It assumes that the shear stress in the adhesive layer is constant through the thickness, and the interface normal stresses along two concrete/adhesive and adhesive/FRP interfaces are different. Closed-form solutions are obtained for these two interfacial normal stresses, shear stress within the adhesive layer, and beam forces. The validation of these solutions is confirmed by finite element analysis.

The simulation of in-plane tyre modal behaviour: a broad modal range comparison between analytical and discretised modelling approaches

In-plane tyre modal behaviour determines the response of tyres to ride excitations and braking/traction manoeuvres. In many studies, the interest is limited to relatively low frequencies and a detailed investigation into the ability of models to accurately simulate higher-order responses is unnecessary. In cases where an in-plane model is to be used for the generation of the contact deformation and stresses, or where modal reduction methods are implemented, a detailed knowledge of the modal response is desirable. The present work forms a study on the ability of a number of frequently used modelling approaches to generate realistic modal data throughout a wide frequency range. The analytical ring on elastic foundation model is used as a benchmark throughout the paper. Its predictions are compared with those of two discretised models, namely a truss- and a beam-based model. The sensitivity of the ring’s response to a number of physical parameters is discussed. The results are used to inform the comparison between the analytical ring and the discretised models, providing explanations for any discrepancies observed. The limited applicability of the truss model is pointed out, while the accuracy of the beam-based model is enhanced by a circumferential inextensible string element. Both the ring and the enhanced beam models are further improved with the addition of a nonlinear string-based sidewall that accounts for the change in sidewall stiffness with inflation pressure. The findings may offer a reference when setting up in-plane models, including the stage of planning modal tests for parameter identification.

Analytical approach and field monitoring for mechanical behaviors of pipe roof reinforcement

Considering the delay effect of initial lining and revising the Winkler elastic foundation model, an analytical approach based on Pasternak elastic foundation beam theory for pipe roof reinforcement was put forward. With the example of a certain tunnel excavation, the comparison of the values of longitudinal strain of reinforcing pipe between field monitoring and analytical approach was made. The results indicate that Pasternak model, which considers a more realistic hypothesis in the elastic soil than Winkler model, gives more accurate calculation and agrees better with the result of field monitoring. The difference of calculation results between these two models is about 7%, and Pasternak model is proved to be a better way to study the reinforcement mechanism and improve design practice. The calculation results also reveal that the reinforcing pipes act as levers, which increases longitudinal load transfer to an unexcavated area, and consequently decreases deformation and increases face stability.

Elastic stability of Euler columns with a continuous elastic restraint using variational iteration method

In this paper, analysis of elastic stability for continuously restrained Euler columns is conducted by means of variational iteration method (VIM). A uniform homogeneous column is assumed to be restrained along its length. The restraint considered in this study is an elastic foundation model in engineering practice and it is of great interest to foundation engineers. Hence, the variation of the critical buckling loads with the stiffness of elastic restraint is investigated using VIM for the restrained columns with different end conditions. Obtaining analytical solutions for these types of problems is not a simple procedure since the equations of stability criteria are highly nonlinear. This study presents the application of VIM for obtaining exact solutions for continuously restrained Euler columns. The study proves that VIM is a very efficient and promising approach in the elastic stability analysis of specified problems.

Mechanical behaviour of a hyperelastic pad for an integrated cross-tie trackwork system

This paper presents the material characteristics and application of a hyperelastic pad used in an integrated cross-tie trackwork configuration, which is part of an advanced rapid transit (ART) system. The ART vehicle is powered by a linear induction motor (LIM); correspondingly, the steel cross-tie trackwork includes a LIM reaction rail as well as running rails to support the steel-wheeled vehicle. An elastomeric pad is placed under each end of the cross-tie to absorb the dynamic effects from the ART. The pad has been isolated from the integrated cross-tie system and tested to determine its material constants which are necessary for modeling of the cross-tie system. The pad is found to be nearly incompressible and demonstrates hyperelastic behaviour. The Mooney–Rivlin (three-term) model shows the best prediction of the pad behaviour among the five selected hyper-elasticity models and a simplified elastic foundation model also exhibits a reasonable engineering approximation. The contribution of variable pad stiffness values to the size of the required air gap between the LIM and the reaction rail, and to the fatigue life of welded connections in the cross-tie is studied.

Three-parameter, elastic foundation model for analysis of adhesively bonded joints

A novel three-parameter, elastic foundation model is proposed in this study to analyze interface stresses of adhesively bonded joints. The classical two-parameter, elastic foundation model of adhesive joints models the adhesive layer as a layer of normal and a layer of shear springs. This model does not satisfy the zero-shear-stress boundary conditions at the free edges of the adhesive layer due to the inherent flaw of the two-parameter, elastic foundation model, which violates the equilibrium condition of the adhesive layer. To eliminate this flaw, this study models the adhesive layer as two normal spring layers interconnected by a shear layer. This new three-parameter, elastic foundation model allows the peel stresses along the two adherend/adhesive interfaces of the joint to be different, and therefore, satisfies the equilibrium condition of the adhesive layer. This model regains the missing “degree of freedom” in the two-parameter, elastic foundation model of the adhesive layer by introducing the transverse displacement of the adhesive layer as a new independent parameter. Explicit closed-form expressions of interface stresses and beam forces are obtained. The new model not only satisfies all boundary conditions, but also predicts correctly which interface has the strongest stress concentration. The new model is verified by continuum models existing in the literature and finite element analysis. The new three-parameter, elastic foundation model provides an effective and efficient tool for analysis and design of general adhesive joints.

Analysis of gas foil bearings integrating FE top foil models

Gas foil bearings (GFBs) find widespread usage in oil-free turbo expanders, APUs, and micro gas turbines for distributed power due to their low drag friction and ability to tolerate high-level vibrations. The performance of GFBs depends largely on the support elastic structure, i.e. a smooth foil on top of bump strips. Conventional models include only the bumps as equivalent stiffnesses uniformly distributed around the bearing circumference. More complex finite element (FE) models couple the elastic deformations of the 2D shell or 1D beam-like top foil to the bump deflections as well as to the gas film hydrodynamics. Predictions of journal attitude angle and minimum film thickness for increasing static loads and two journal speeds are obtained for a GFB tested decades ago. For the GFB studied, 2D FE model predictions overestimate the minimum film thickness at the bearing centerline, while underestimating it at the bearing edges. Predictions from the 1D FE model compare best to the limited tests data, reproducing closely the experimental circumferential wavy-like film thickness profile. Predicted stiffness and damping coefficients versus excitation frequency show that the two FE models result in slightly lower direct stiffness and damping coefficients than those from the simple elastic foundation model.

Non-linear Oscillations of Orthotropic Plates on a Non-linear Elastic Foundation

In this paper, the non-linear vibration response of an orthotropic plate rests on a nonlinear foundation that subjected to non-uniform initial stress is investigated. The non-linear partial differential equations of motion for an orthotropic plate are derived by Hamilton's principle. By using these derived governing equations, the large amplitude vibration of an initially stressed plate on a non-linear elastic foundation model was studied. Galerkin's approximate method was applied to the governing partial differential equations to yield ordinary differential equations. The ordinary differential equations were solved by employing a Runge—Kutta method to obtain the ratio of nonlinear to linear frequencies. The initial stress is taken to be a combination of pure bending stresses plus extensional stresses in the plane of the plate. The softening non-linear elastic foundation model is used to describe the plate—foundation interaction. Numerical example of simply supported Mindlin plates subjected to the initial stress and resting on a Winkler non-linear foundation was solved. The frequency responses of non-linear vibration are sensitive of initial stress, amplitude of vibration, material properties, linear foundation stiffness and non-linear foundation stiffness.

Validation of a simplified numerical contact model

Surface roughness tends to have a significant effect on how loads are transmitted at the contact interface between solid bodies. Most numerical contact models for analyzing rough surface contacts are computational demanding and more computationally efficient contact models are required. Depending on the purpose of the simulation, simplified and less accurate models can be preferable to more accurate, but also more complex, models. This paper discusses a simplified contact model called the elastic foundation model and its applicability to rough surfaces. The advantage of the model is that it is fast to evaluate, but its disadvantage is that it only gives an approximate solution to the contact problem. It is studied how surface roughness influences the errors in the elastic foundation solution in terms of predicted pressure distribution, real contact area, and normal and tangential contact stiffness. The results can be used to estimate the extent of error in the elastic foundation model, depending on the degree of surface roughness. The conclusion is that the elastic foundation model is not accurate enough to give a correct prediction of the actual contact stresses and contact areas, but it might be good enough for simulations where contact stiffness are of interest.

Free vibration analysis of a plate on foundation with completely free boundary by finite integral transform method

The theoretical solutions of eigenfrequencies and vibration modes of a rectangular thin plate on an elastic foundation with completely free boundary are derived by using a double finite cosine integral transform method. In the analysis procedure, the elastic foundation is regarded as a Winkler elastic foundation model. Because the basic dynamic elasticity equations of the thin plate on elastic foundation are only used, it is not needed to select the deformation function arbitrarily. Therefore, the solution developed in the present paper is more reasonable and more accurate. To prove the correctness of the solutions, numerical results obtained using the present solutions are compared with those in the literatures.

Heavily Loaded Gas Foil Bearings: A Model Anchored to Test Data

Widespread usage of gas foil bearings (FBs) into microturbomachinery to midsize gas turbine engines requires accurate performance predictions anchored to reliable test data. This paper presents a simple yet accurate model predicting the static and dynamic force characteristics of gas FBs. The analysis couples the Reynolds equation for a thin gas film to a simple elastic foundation model for the top foil and bump strip layer. An exact flow advection model is adopted to solve the partial differential equations for the zeroth- and first-order pressure fields that render the FB load capacity and frequency-dependent force coefficients. As the static load imposed on the foil bearing increases, predictions show that the journal center displaces to eccentricities exceeding the bearing nominal clearance. A nearly constant FB static stiffness, independent of journal speed, is estimated for operation with large loads, and approaching closely the bearing structural stiffness derived from contact operation without rotor spinning. Predicted minimum film thickness and journal attitude angle demonstrate good agreement with archival test data for a first-generation gas FB. The bump-foil-strip structural loss factor, exemplifying a dry-friction dissipation mechanism, aids to largely enhance the bearing direct damping force coefficients. At high loads, the bump-foil structure influences most the stiffness and damping coefficients. The predictions demonstrate that FBs have greatly different static and dynamic force characteristics when operating at journal eccentricities in excess of the bearing clearance from those obtained for operation at low loads, i.e., small journal eccentricity.

Discussion of Pin-connected Joints Design Considering Contact

AbstractPin-connected joints are characteristic of contacted bearing loads. With no consideration of the contact, the disadvantage for pin-connected joints to bear loads will be overestimated. Based on the Winkler elastic foundation model, a new simplified model is proposed and a formula is derived to calculate the moment in the middle of span. It shows that the moment in the middle of span calculated according to the code JTJ025-86 should be reduced by coefficient β taking into account of the transferred load distribution along the pin axis. Through a practical example of pin-connected joint analysis, it is proved that the calculation result based on the derived formula and that based on the contact simulation model are well matched.

A modified elastic foundation contact model for application in 3D models of the prosthetic knee

Different models have been used in the literature for the simulation of surface contact in biomechanical knee models. However, there is a lack of systematic comparisons of these models applied to the simulation of a common case, which will provide relevant information about their accuracy and suitability for application in models of the implanted knee. In this work a comparison of the Hertz model (HM), the elastic foundation model (EFM) and the finite element model (FEM) for the simulation of the elastic contact in a 3D model of the prosthetic knee is presented. From the results of this comparison it is found that although the nature of the EFM offers advantages when compared with that of the HM for its application to realistic prosthetic surfaces, and when compared with the FEM in CPU time, its predictions can differ from FEM in some circumstances. These differences are considerable if the comparison is performed for prescribed displacements, although they are less important for prescribed loads. To solve these problems a new modified elastic foundation model (mEFM) is proposed that maintains basically the simplicity of the original model while producing much more accurate results. In this paper it is shown that this new mEFM calculates pressure distribution and contact area with accuracy and short computation times for toroidal contacting surfaces. Although further work is needed to confirm its validity for more complex geometries the mEFM is envisaged as a good option for application in 3D knee models to predict prosthetic knee performance.

Analysis of the sandwich DCB specimen for debond characterization

Analysis of the compliance and energy release rate of the sandwich double cantilever beam (DCB) specimen is presented. It is assumed that there is a starter crack at the upper face/core interface and that the crack remains at or near this interface during crack propagation. Beam, elastic foundation, and finite element analyses are presented and compared to experimentally measured compliance data, and compliance calibrated energy release rate over a range of crack lengths for foam cored sandwich DCB specimens. It is found that the beam analysis provides a conservative estimate on the compliance and energy release rate. The elastic foundation model is in agreement with finite element analysis and experimental compliance data. Recommendations for specimen design and an expression for an upper limiting crack length are provided.

Numerical Solution for Laterally Loaded Piles in a Two-Layer Soil Profile

Piles are often embedded in a layered soil profile, such as sand or clay layer underlain by rock. Several existing solutions are available for laterally loaded piles in a layered soil system. However, these solutions are only applicable to constant soil stiffness for each layer. In this paper, a variational approach is employed to numerically solve the problem of laterally loaded piles in layered soils using beam on an elastic foundation model. The soil stiffness can be either constant with depth or linearly varying with depth. The numerical solution is validated against an existing solution for linearly varying soil stiffness in a single soil layer system and an existing solution for a two-layer soil system with constant soil stiffness. Case studies using the proposed solution for field lateral load tests on full size drilled shafts embedded in weak rock with an overlying sand layer are presented. The simplicity and the relative ease of using the solution make it a good alternative approach for estimating the deflection and moment responses of a laterally loaded pile in a two-layer soil profile.

Quasi-non-linear fracture mechanics analysis of splitting failure in simply supported beams loaded perpendicular to grain by dowel joints

A quasi-non-linear fracture mechanics model based on beam on elastic foundation theory is applied for analysis of the splitting failure of dowel joints loaded perpendicular to grain. Simply supported beams symmetrically loaded by two dowels are considered, and the effects of edge distance, dowel spacing, and distance between dowels and supports are accounted for. The foundation modulus used in the beam on elastic foundation model is chosen so that the perpendicular-to-grain tensile strength and fracture energy properties of the wood are correctly represented. This ensures that a conventional stress analysis and failure criterion lead to the same solution as the compliance method of fracture mechanics. A semiempirical efficiency factor is proposed to account for the influence of the total beam depth, which does not enter the beam on elastic foundation model, but the effect of which is evident from tests. It is shown that the so-called Van der Put/Leijten model, which recently has been adopted in Eurocode 5, appears as a special case of the model presented. Tests on simply supported beams with a single dowel joint at midspan are compared with the theoretical predictions. Various edge distances, beam depths, and spans were tested.

Analysis of peel arm curvature for the determination of fracture toughness in metal-polymer laminates

Variable angle fixed arm peel and mandrel peel tests were performed on four metal-polymer laminate systems. In total, four polymeric adhesives and three grades of aluminium alloy (AA) substrates were used, enabling a wide range of material properties to be encompassed in the study. Mandrel peel tests provided a direct determination of the plastic bending energy (Gp) and adhesive fracture toughness (Ga). For the fixed arm tests, a global energy-balance analysis (ICPeel software) was used to determine Ga and Gp analytically. This was done via the calculation of the maximum curvature of the peel arm (1/R0) and the root rotation angle (θ0) from a beam on elastic foundation model. In order to investigate the accuracy of the analytical approach, an experimental method based on high resolution digital photography enabled 1/R0 and θ0 to be measured independently. It was then possible to compare these parameters by measurement and by analytical approach (ICPeel software). θ0 and R0 relate to the slope and curvature of the peel arm at the debonding front, respectively. In order to measure these parameters, the coordinates of the edge of the peel arm were extracted from each digital photograph, and the slope and curvature were calculated numerically from these curves. The crack tip was then defined as the point of maximum curvature 1/R0, in accordance with traditional beam theory. It was found that the smoothing in the calculation of first and second derivatives could generate significant errors in the value of θ0. On the other hand, R0 was found to be a more robust measurement, with little dependence on smoothing. Nevertheless, on most occasions, the measured values of θ0 and R0, as well as the resulting Ga were shown to be in good agreement with the analytical model. Since the peel fractures were generally cohesive, Ga was compared with the cohesive fracture toughness (Gc) obtained from Tapered Double Cantilever Beam (TDCB) tests with a fracture mechanics analysis. Good agreement was observed, confirming that Ga is likely to be a geometry-independent fracture parameter.

Face Sheet Buckling of Debonded Sandwich Panels Using a Two-Dimensional Elastic Foundation Approach

A two-dimensional elastic foundation model for analysis of the face sheet buckling behavior of sandwich panels with an embedded rectangular debond has been developed. The model is based on an energy method which assumes that the core behaves as an elastic foundation supporting the face sheets everywhere except over the debond. The face sheet buckling load is presented in a versatile form expressed into two buckling coefficients. Parametric studies are conducted to evaluate the influences of the foundation modulus, face sheet modulus, face sheet anisotropy, debond and panel dimensions, and aspect ratios. It is shown that the critical buckling load is very sensitive to the core modulus and flexural stiffness of the face and less sensitive to the debond size and face anisotropy. Model predictions are compared to experimental results obtained for sandwich panels consisting of glass/epoxy face sheets over a H100 PVC foam core with a range of square face/core debond sizes. Reasonable agreement with experimentally measured local buckling loads is observed for relatively small debonds. For large debonds the model tends to overpredict the buckling load.