Simple Beam Model Research Articles

An Investigation of Soil-Structure Interaction Effects Observed at the MIT Green Building

The soil-foundation impedance function of the MIT Green Building is identified from its response signals recorded during an earthquake. Estimation of foundation impedance functions from seismic response signals is a challenging task, because: (1) the foundation input motions (FIMs) are not directly measurable, (2) the as-built properties of the super-structure are only approximately known, and (3) the soil-foundation impedance functions are inherently frequency-dependent. In the present study, aforementioned difficulties are circumvented by using, in succession, a blind modal identification (BMID) method, a simplified Timoshenko beam model (TBM), and a parametric updating of transfer functions (TFs). First, the flexible-base modal properties of the building are identified from response signals using the BMID method. Then, a flexible-base TBM is updated using the identified modal data. Finally, the frequency-dependent soil-foundation impedance function is estimated by minimizing the discrepancy between TFs (of pairs instrumented floors) that are (1) obtained experimentally from earthquake data and (2) analytically from the updated TBM. Using the fully identified flexible-base TBM, the FIMs as well as building responses at locations without instruments can be predicted, as demonstrated in the present study.

Simplified analytical model for tapered sandwich beams using variable stiffness materials

A simplified layer-wise sandwich beam model to capture the effects of a combination of geometric taper and variable stiffness of the core on the static response of a sandwich beam is developed. In the present model, the face sheets are assumed to behave as Euler beams and the core is modelled with a first-order shear deformation theory. With geometrical compatibility enforced at both upper and lower skin/core interfaces, the beam’s field functions are reduced to only three, namely the extensional, transverse and rotational displacements at the mid-plane of the core. The minimum total potential energy method is used in combination with the Ritz technique to obtain an approximate solution. Geometrically nonlinear effects are considered in the present formulation by introducing von Kármán strains into the face sheets and core. Two types of sandwich beams, uniform and tapered, with different boundary conditions are studied. Results show that the proposed model provides accurate prediction of displacements and stresses, compared to three-dimensional finite element analysis. It is found that due to the axial stiffness variation in the core, displacements of beams and stresses of face sheets and core are significantly affected. The potential design space is shown to be expanded by utilizing variable stiffness materials in sandwich constructions.

A simplified model predicting the weight of the load carrying beam in a wind turbine blade

Based on a simplified beam model, the loads, stresses and deflections experienced by a wind turbine blade of a given length is estimated. Due to the simplicity of the model used, the model is well suited for work investigating scaling effects of wind turbine blades. Presently, the model is used to predict the weight of the load carrying beam when using glass fibre reinforced polymers, carbon fibre reinforced polymers or an aluminium alloy as the construction material. Thereby, it is found that the weight of a glass fibre wind turbine blade is increased from 0.5 to 33 tons when the blade length grows from 20 to 90 m. In addition, it can be seen that for a blade using glass fibre reinforced polymers, the design is controlled by the deflection and thereby the material stiffness in order to avoid the blade to hit the tower. On the other hand if using aluminium, the design will be controlled by the fatigue resistance in order to making the material survive the 100 to 500 million load cycles experience of the wind turbine blade throughout the lifetime. The aluminium blade is also found to be considerably heavier compared with the composite blades.

Field Measurement-Based System Identification and Dynamic Response Prediction of a Unique MIT Building

Tall buildings are ubiquitous in major cities and house the homes and workplaces of many individuals. However, relatively few studies have been carried out to study the dynamic characteristics of tall buildings based on field measurements. In this paper, the dynamic behavior of the Green Building, a unique 21-story tall structure located on the campus of the Massachusetts Institute of Technology (MIT, Cambridge, MA, USA), was characterized and modeled as a simplified lumped-mass beam model (SLMM), using data from a network of accelerometers. The accelerometer network was used to record structural responses due to ambient vibrations, blast loading, and the October 16th 2012 earthquake near Hollis Center (ME, USA). Spectral and signal coherence analysis of the collected data was used to identify natural frequencies, modes, foundation rocking behavior, and structural asymmetries. A relation between foundation rocking and structural natural frequencies was also found. Natural frequencies and structural acceleration from the field measurements were compared with those predicted by the SLMM which was updated by inverse solving based on advanced multiobjective optimization methods using the measured structural responses and found to have good agreement.

An improved static model for tool deflection in machining of Ti–6Al–4V acetabular shell produced by selective laser melting

Tool deflection during milling operation leads to dimensional error, decreasing surface quality and increasing rejection rate. In this study, tool deflection during the milling of the inner surfaces of Ti–6Al–4V prosthetic acetabular shell produced by selective laser melting (SLM) was modelled. The first purpose of this research is to provide a general static cutting tool deflection model for ball nose cutters where deviation of machine components and tool holder are so small as to be considered negligible. This is because the values of machine component and tool holder deflection were lower than standard tolerances (10 μm) and found to be lower than 1/15 of tool deflection. The second and third objectives of this work involve calculating contact surfaces by determining workpiece and tool geometry and choosing second moment of inertia using a novel cross section method (CSM). Static models for three quasi-analytical methods (QAM) that are simple cantilever beam model (SCBM), two-section model (TWSM) and our three section model (THSM) are presented. THSM showed high accuracy which was validated by 3D finite element method (FEM3D) and experimental measurements. The accuracy of tool deflection calculation using THSM by computing, shank, flute and ball head deflection and also utilizing CSM to determine second moment of inertia showed notable improvements.

A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method

A simple nonlocal beam model is proposed to study buckling response of protein microtubules. The size-effect for buckling model of microtubules is considered by using the nonlocal continuum theory. Finite element procedure is used for solution of nonlocal differential equation of microtubules for elastic stability. The influence of the small length scale on the buckling value is examined for different geometric parameters. The effect of elastic matrix surrounded of microtubules is also examined and some benchmark results are presented.

Suppression of bending waves in a beam using resonators with different separation lengths.

This work is concerned with the suppression of a bending wave in a beam using resonators. Particular focus is put on the separation length between resonators. It is demonstrated that, for a beam with identical resonators attached at equal intervals, the bending wave transmission efficiency varies with respect to the separation length. The phenomena and the underlying physics are investigated by resorting to a simple beam model having two resonators resting on it. The two resonators are coupled over the segment through various bending wave components, comprising both propagating waves and evanescent waves, generated at the resonator locations where the beam encounters impedance discontinuities. The separation length, specifying the phase change of the propagating waves and the amplitude decay of the evanescent waves travelling from one resonator to the other, is thereby the parameter determining the extent to which the resonators would be coupled and the degree of the power that is transmitted. Results show, qualitatively, the difference in the working mechanism of the resonators in different separation length regions, with criteria being defined to distinguish those regions. Particularly, in the intermediate separation region, the evanescent waves are shown to play an important role in the coupling and are responsible for transmitting power, comparable with that transmitted by propagating waves, to the far field.

Study of the mechanics of progressive collapse with simplified beam models

Methods for assessing structural robustness need to move away from the traditional norms of prescriptive rules and become more similar to those used in conventional structural design. They should therefore be based on a sound understanding of the mechanics of the problem and provide quantitative indication of its effects. Several Codes and Design Guides consider the sudden column removal approach as their principal method for progressive collapse assessment. The level of robustness is defined based on the capability of the remaining structure for sustaining the additional loading imposed by the column loss. Most likely, the beams adjacent to the lost column and their supporting connections form the principal load paths. The present paper presents a detailed study of the response of those components under the conditions experienced following column removal. Suitable analysis approaches that have been previously developed at Imperial College London are employed to investigate the basic features of the behaviour, while several simplifications are applied for exploring particular effects. The study concludes with the development of a simplified method for simulating the nonlinear dynamic response of axially restrained and unrestrained beams following column removal. The capability of the new simplified method to accurately describe performance is demonstrated through a set of suitable applications presented in a separate publication.

Robustness assessment of frame structures using simplified beam and grillage models

Simplified analysis methods derived in previous studies are employed for studying the progressive collapse behaviour of steel and composite buildings. A regular frame building is considered and various scenarios of sudden column removal, each affecting different floor areas in terms of geometry and boundary conditions, are applied. Descriptions of the pseudo-static responses of the various constitutive beams are obtained based on both detailed representations of the nonlinear static responses and by applying a new simplified approach proposed in a separate publication. Comparisons between the results of the two methods confirm that the simplified approach is capable of describing behaviour with reasonable accuracy. By employing a simplified multi-level assessment approach that has been previously derived at Imperial College, grillage-type approximations are obtained and used to examine the floor dynamic behaviour for the various column removal cases. It is found that, although the structural response varies depending on the location of the initial damage, substantial connection strength is required in all cases in order to provide resistance to progressive collapse. In addition, for average levels of connection ductility, failure most likely occurs prior to the development of tensile catenary action in the beams, which indicates that the provision of tying resistance may not be effective in enhancing robustness. Therefore, the combined action of flexure and compressive arching in the beams is likely to form the principal collapse resisting mechanism in common practical applications, which confirms similar conclusions made in previous studies at Imperial. The provision of adequate levels of connection moment capacity – in combination with sufficient ductility supply – is, therefore, the most effective way of securing structural robustness.

Non-prismatic beams: A simple and effective Timoshenko-like model

The present paper discusses simple compatibility, equilibrium, and constitutive equations for a non-prismatic planar beam. Specifically, the proposed model is based on standard Timoshenko kinematics (i.e., planar cross-section remain planar in consequence of a deformation, but can rotate with respect to the beam center-line). An initial discussion of a 2D elastic problem highlights that the boundary equilibrium deeply influences the cross-section stress distribution and all unknown fields are represented with respect to global Cartesian coordinates. A simple beam model (i.e. a set of Ordinary Differential Equations (ODEs)) is derived, describing accurately the effects of non-prismatic geometry on the beam behavior and motivating equation’s terms with both physical and mathematical arguments. Finally, several analytical and numerical solutions are compared with results existing in literature. The main conclusions can be summarized as follows. (i) The stress distribution within the cross-section is not trivial as in prismatic beams, in particular the shear stress distribution depends on all generalized stresses and on the beam geometry. (ii) The derivation of simplified constitutive relations highlights a strong dependence of each generalized deformation on all the generalized stresses. (iii) Axial and shear-bending problems are strictly coupled. (iv) The beam model is naturally expressed as an explicit system of six first order ODEs. (v) The ODEs solution can be obtained through the iterative integration of the right hand side term of each equation. (vi) The proposed simple model predicts the real behavior of non-prismatic beams with a good accuracy, reasonable for the most of practical applications.

Van der Waals force-induced crack healing in dry rough interfaces

Spontaneous crack healing due to van der Waals forces is an important phenomenon in diverse areas such as precision assembly, locomotion, soft robotics, and micro- and nanomachines. For rough surfaces that can be described as a collection of asperities, parallel plate models are used to gain insight into the adhesion values. A single adhesion value is then found for a given surface description. However, experiments reveal a range of values. Here, implementing a simple beam model to gain physical insight, we show that an important contribution to the range can be due to the placement of asperities relative to the crack tip. For example, tall asperities far from the crack tip resist crack healing if they contact the substrate, but promote healing if not in contact with it. Due to this effect, the beam model predicts a range of values that is significant compared with the observed experiment variation. Furthermore, as the crack approaches mechanical equilibrium, the resisting action tends to dominate over the healing action, and the beam model predicts a lower adhesion value than the parallel plate model. These effects will be greatest in the case where the elasticity (Tabor) parameter is small.

Correlated spin-down rates and radio emission in PSR B1859+07

We study the spin-down changes of PSR B1859$+$07 over a period of more than 28 years of radio observation. We identify that the time derivative of the rotational frequency ($\nu$) varies quasi-periodically with a period of $\sim$350 days, switching mainly between two spin-down states. The profile shape of the pulsar is correlated with the $\dot \nu$ variation, producing two slightly different profile shapes corresponding to high- and low-$\dot \nu$ states. In addition to these two normal emission states, we confirm the occasional flare-state of the pulsar, in which the emission appears early in spin phase compared to that of the common normal emission. The profile of the flare-state is significantly different from that of the two normal emission states. The correlation analysis further shows that the flare-state is not directly linked with the $\dot \nu$ changes. With a simple emission beam model, we estimate the emission altitude of the normal emission to be 240~km, and explain the origin of the flare-state as an emission height variation from the leading edge of the beam. We also argue that the emission of these states can be explained with a partially active beam model. In this scenario, the trailing portion of the radio beam is usually active and the normal emission is produced. The flare-state occurs when the leading edge of the beam becomes active while the trailing part is being blocked. This model estimates a fixed emission altitude of 360~km. However, the cause of the flare-state (i.e. the emission height variation, or the time-dependent activity across the radio beam) is not easily explained.

중형 상용차용 프레스 성형 차축빔의 경량화 설계

This paper represents the structural design of the light weight axle beam for medium duty commercial vehicle using hot press. To reduce the weight of the axle, axle beam of solid type was replaced by hollow type which was made by hot press. According to the change of axle beam structure and manufacturing method, we have to investigate the structural strength and fatigue performance. To verify the axle beam performance, the structural analysis was carried out by simplified axle beam model and various design parameters that are axle beam height, thickness and width. From the analysis results, the light weight axle beam structure was founded and applied the full model analysis. This study will be used as a guidance in development of the light weight axle for medium duty commercial vehicle.

Compliant mechanism synthesis by using elastic similitude

Compliant mechanisms have several advantages, especially smaller number of elements and therefore less movable joints. The flexural members furthermore allow an integration of special functions like balancing or locking. Especially fiber reinforced materials exhibit a wide range of function integration considering their compliance in passive as well active applications. To take advantage of compliant elements in applications a robust synthesis tool is needed. The synthesis based on topology optimization method or the pseudo rigid body approach leads to complex structures. Considering the use of fiber reinforced material a synthesis approach which leads to less complex structures is more suitable. For building up simple structures, with only one cantilever beam as compliant element(B) a graphical approach using the elastic similitude is the most efficient method. A step-by-step synthesis procedure is presented to synthesize compliant mechanisms with rotatory joints(R) and prismatic joints(P) to develop RRB/PRB- and RPB-linkages. Using the elastic similitude to implement these results into a graphical synthesis algorithm is the innovation part of this paper. It can be shown that this approach leads to a comfortable handling of beam elements during the synthesis, where the two free parameters can be directly coupled to scale and fix the orientation of the beam element. This advantage inherently shortens the development process. In giving an example the focus lies of the experimental approach, which also shows that the simple BERNOULLI beam model is valid and so the synthesis by using the elastic similitude. The method is presented and discussed by using an application for a cup holder mechanism made of fiber reinforced material.

Sensitivities of biomechanical assessment methods for fracture healing of long bones

There is a controversy as to whether the biomechanical methods are feasible to assess fracture healing of long bones. This paper investigated the sensitivities of two biomechanical methods, torsion and bending, for assessing fracture healing of long bones; both a simplified beam model and finite element model of an artificial femur were employed. The results demonstrated that, in the initial healing stage, the whole-bone stiffness of the fractured bone is extremely sensitive to the variation of the callus stiffness at the fracture site; when the shear (or Young's) modulus of the callus reaches 15% that of the intact bone, the whole-bone stiffness rises up to 90% that of the intact bone. After that, the whole-bone torsional (or bending) stiffness increases slowly; it becomes less sensitive to the variation of the callus stiffness. These results imply that the whole-bone stiffness is of limited reliability to assess the healing quality particular at late stages of the healing process. The simplified model in this paper provided a theoretical framework to explain why the whole-bone stiffness is insensitive to the healing process of fractured long bones in the late stage of healing. The conclusions obtained from the simplified model were verified with the finite element simulations of the artificial femur.

Evaluating the fidelity and robustness of calibrated numerical model predictions

Purpose – Numerical models are being increasingly relied upon to evaluate wind turbine performance by simulating phenomena that are infeasible to measure experimentally. These numerical models, however, require a large number of input parameters that often need to be calibrated against available experiments. Owing to the unavoidable scarcity of experiments and inherent uncertainties in measurements, this calibration process may yield non-unique solutions, i.e. multiple sets of parameters may reproduce the available experiments with similar fidelity. The purpose of this paper is to study the trade-off between fidelity to measurements and the robustness of this fidelity to uncertainty in calibrated input parameters. Design/methodology/approach – Here, fidelity is defined as the ability of the model to reproduce measurements and robustness is defined as the allowable variation in the input parameters with which the model maintains a predefined level of threshold fidelity. These two vital attributes of model predictiveness are evaluated in the development of a simplified finite element beam model of the CX-100 wind turbine blade. Findings – Findings of this study show that calibrating the input parameters of a numerical model with the sole objective of improving fidelity to available measurements degrades the robustness of model predictions at both tested and untested settings. A more optimal model may be obtained by calibration methods considering both fidelity and robustness. Multi-criteria Decision Making further confirms the conclusion that the optimal model performance is achieved by maintaining a balance between fidelity and robustness during calibration. Originality/value – Current methods for model calibration focus solely on fidelity while the authors focus on the trade-off between fidelity and robustness.

Optimal control of a viscoelastic rack feeder based on the method of integrodifferential relations

In this paper, control-oriented models are derived for an experimental setup representing the structure of a typical high bay rack feeder. To develop a real-time applicable control algorithm, a frequency analysis is performed for the original viscoelastic double-beam structure. This leads to a simplified Bernoulli beam model with specific boundary conditions. On the basis of the proposed model, a feedforward control strategy is designed. The control objective under consideration is to move the flexible structure to a desired position in a given time interval and to minimize the relative mean energy stored in the beams during the process. A modification of the method of integrodifferential relations, which is based on a projection approach and a suitable finite element technique, is employed to optimize the controlled motions. Results of numerical simulations are presented and compared with experimentally measured data from the experimental setup.

Simulating Building Motions Using Ratios of the Building's Natural Frequencies and a Timoshenko Beam Model

A simple prismatic Timoshenko beam model with soil-structure interaction (SSI) is developed to approximate the dynamic linear elastic behavior of buildings. A closed-form solution with complete vibration modes is derived. It is demonstrated that building properties, including mode shapes, can be derived from knowledge of the natural frequencies of the first two translational modes in a particular direction and from the building dimensions. In many cases, the natural frequencies of a building's first two vibrational modes can be determined from data recorded by a single seismometer. The total building's vibration response can then be simulated by the appropriate modal summation. Preliminary analysis is performed on the Caltech Millikan Library, which has significant bending deformation because it is much stiffer in shear.

Natural frequency of a composite girder with corrugated steel web

This paper presents the natural frequency of a composite girder with corrugated steel web (CGCSW). A corrugated steel web has negligible in-plane axial stiffness, due to the unique characteristic of corrugated steel webs, which is called the accordion effect. Thus, the corrugated steel web only resists shear force. Further, the shear buckling resistance and out-of-plane stiffness of the web can be enhanced by using a corrugated steel web, since the inclined panels serve as transverse stiffeners. To take these advantages, the corrugated steel web has been used as an alternative to the conventional pre-stressed concrete girder. However, studies about the dynamic characteristics, such as the natural frequency of a CGCSW, have not been sufficiently reported, and it is expected that the natural frequency of a CGCSW is different from that of a composite girder with flat web due to the unique characteristic of the corrugated steel web. In this study, the natural frequency of a CGCSW was investigated through a series of experimental studies and finite element analysis. An experimental study was conducted to evaluate the natural frequency of CGCSW, and the results were compared with those from finite element analysis for verification purpose. A parametric study was then performed to investigate the effect of the geometric characteristics of the corrugated steel web on the natural frequency of the CGCSW. Finally, a simplified beam model to predict the natural frequency of a CGCSW was suggested.

Numerical Simulation for the Quasi-static Behavior of Superelastic Nitinol Shape Memory Alloys (SMAs)

초탄성 형상기억합금은 상온에서 소성 범위를 초월하여 상당량의 변위를 가하더라도 하중을 제거 후에 별도의 열처리를 가하지 않더라도 원상태로 복원이 가능한 특수한 금속이다. 자동치유가 가능한 형상기억합금의 특유한 재료적인 성질로 인하여 구조물에서 변위가 집중되는 부분에 기존에 주로 사용되는 강재를 대체하여 이러한 특수 합금 재료가 널리 활용되기 시작하였다. 하지만 형상기억합금을 활용한 구조물의 기본적인 설계와 성능 검증을 하기 위해 고등적인 구조해석에 필요한 재료적인 모델의 개발과 연구의 노력이 부족하기 때문에 본 재료를 현장에서 적용하기에는 여전히 많은 제약을 받고 있다. 따라서 본 연구에서는 초탄성 형상기억합금의 거동을 수치해석적인 방법으로 재현이 가능한 구성적인 재료 모델의 소개와 프로그램 코딩에 대하여 다루고자 한다. 또한 본 연구에서 제시된 재료 모델의 타당성을 입증하기 위하여 수치해석적으로 재현된 물리적인 거동을 실험에서 얻어진 데이터에 비교 및 보정 작업도 수행하였다. 아울러 이러한 재료 모델로 구현된 초탄성 형상기억합금의 물리적인 물성치를 구조 해석에 적용하고 정확성을 검증하여 현장 적용의 타당성을 입증하였다. Superelastic shape memory alloys (SMAs) are metallic materials that can automatically recover to their original condition without heat treatment only after the removal of the applied load. These smart materials have been wildly applied instead of steel materials to the place where large deformation is likely to concentrate. In spite of many advantages, superelastic SMA materials have been limited to use in the construction filed because there is lack of effort and research involved with the development of the material model, which is required to reproduce the behavior of superelastic SMA materials. Therefore, constitutive material models as well as algorithm codes are mainly treated in this study for the purpose of simulating their hysteretic behavior through numerical analyses. The simulated curves are compared and calibrated to the experimental test results with an aim to verify the adequacy of material modeling. Furthermore, structural analyses incorporating the material property of the superelastic SMAs are conducted on simple and cantilever beam models. It can be shown that constitutive material models presented herein are adequate to reliably predict the behavior of superelastic SMA materials under cyclic loadings.