scholarly journals Improving the Accuracy of Analytical Relationships for Mechanical Properties of Permeable Metamaterials

2021 ◽  
Vol 11 (3) ◽  
pp. 1332 ◽  
Author(s):  
Reza Hedayati ◽  
Naeim Ghavidelnia ◽  
Mojtaba Sadighi ◽  
Mahdi Bodaghi
Keyword(s):  
Timoshenko Beam ◽  
Beam Theory ◽  
Cell Types ◽  
Timoshenko Beam Theory ◽  
Unit Cell ◽  
Order Of Magnitude ◽  
Analytical Formulas ◽  
The Difference ◽  
Micro Structures ◽  
Euler Bernoulli Beam

Permeable porous implants must satisfy several physical and biological requirements in order to be promising materials for orthopaedic application: they should have the proper levels of stiffness, permeability, and fatigue resistance approximately matching the corresponding levels in bone tissues. This can be achieved using designer materials, which exhibit exotic properties, commonly known as metamaterials. In recent years, several experimental, numerical, and analytical studies have been carried out on the influence of unit cell micro-architecture on the mechanical and physical properties of metamaterials. Even though experimental and numerical approaches can study and predict the behaviour of different micro-structures effectively, they lack the ease and quickness provided by analytical relationships in predicting the answer. Although it is well known that Timoshenko beam theory is much more accurate in predicting the deformation of a beam (and as a result lattice structures), many of the already-existing relationships in the literature have been derived based on Euler–Bernoulli beam theory. The question that arises here is whether or not there exists a convenient way to convert the already-existing analytical relationships based on Euler–Bernoulli theory to relationships based on Timoshenko beam theory without the need to rewrite all the derivations from the start point. In this paper, this question is addressed and answered, and a handy and easy-to-use approach is presented. This technique is applied to six unit cell types (body-centred cubic (BCC), hexagonal packing, rhombicuboctahedron, diamond, truncated cube, and truncated octahedron) for which Euler–Bernoulli analytical relationships already exist in the literature while Timoshenko theory-based relationships could not be found. The results of this study demonstrated that converting analytical relationships based on Euler–Bernoulli to equivalent Timoshenko ones can decrease the difference between the analytical and numerical values for one order of magnitude, which is a significant improvement in accuracy of the analytical formulas. The methodology presented in this study is not only beneficial for improving the already-existing analytical relationships, but it also facilitates derivation of accurate analytical relationships for other, yet unexplored, unit cell types.

scholarly journals Improving the accuracy of analytical relationships for mechanical properties of additively manufactured lattice structures

2020 ◽  
Author(s):  
Reza Hedayati ◽  
Naeim Ghavidelnia ◽  
Mojtaba Sadighi ◽  
Bodaghi
Keyword(s):  
Timoshenko Beam ◽  
Beam Theory ◽  
Cell Types ◽  
Timoshenko Beam Theory ◽  
Unit Cell ◽  
Lattice Structures ◽  
Starting Point ◽  
Unit Cells ◽  
Order Of Magnitude ◽  
The Difference

Porous implants must satisfy several physical and biological requirements in order to be promising materials for orthopedic application: they should have the proper levels of stiffness, permeability, and fatigue resistance and in proximity to how much they are in bone tissues. In recent years, several experimental, numerical, and analytical studies have been carried out on the influence of unit cell geometry on such properties. Even though experimental and numerical techniques can effectively study and predict the behavior of different micro-structure, they lack the ease the analytical relationships provide for such predictions. Even though it is well-known that Timoshenko beam theory gives much better accuracy in predicting the deformation of a beam (and as a result lattice structures), many of the already-existing relationships in the literature have been derived based on Euler-Bernoulli beam theory. The question that arises here is that can there be a convenient way to convert the already-existing relationships based on Euler-Bernoulli to relationships based on Timoshenko beam theory without the need to rewrite all the derivations from the starting point. In this paper, this question is addressed and answered, and a simple approach is presented. This technique is applied to six unit cells for which Euler-Bernoulli analytical relationships could be found in the literature, but Timoshenko theories could not be found: BCC, hexagonal packing, rhombicuboctahedron, diamond, truncated cube, and truncated octahedron. The results of this study demonstrated that converting analytical relationships based on Euler-Bernoulli to equivalent Timoshenko ones can decrease the difference between the analytical and numerical values for one order of magnitude which is a significant improvement in accuracy of the analytical formulas. The methodology presented in this study is not only beneficial to the already-existing analytical relationships but also facilitates derivation of accurate analytical relationships for other, yet unexplored, unit cell types.

Medium Frequency Vibration Analysis of Beam Structures Modeled by the Timoshenko Beam Theory

2020 ◽  
Author(s):  
Yichi Zhang ◽  
Bingen Yang
Keyword(s):  
Shear Deformation ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Bernoulli Beam ◽  
Beam Structure ◽  
Medium Frequency ◽  
Beam Structures ◽  
Euler Bernoulli Beam

Abstract Vibration analysis of complex structures at medium frequencies plays an important role in automotive engineering. Flexible beam structures modeled by the classical Euler-Bernoulli beam theory have been widely used in many engineering problems. A kinematic hypothesis in the Euler-Bernoulli beam theory is that plane sections of a beam normal to its neutral axis remain normal when the beam experiences bending deformation, which neglects the shear deformation of the beam. However, as observed by researchers, the shear deformation of a beam component becomes noticeable in high-frequency vibrations. In this sense, the Timoshenko beam theory, which describes both bending deformation and shear deformation, may be more suitable for medium-frequency vibration analysis of beam structures. This paper presents an analytical method for medium-frequency vibration analysis of beam structures, with components modeled by the Timoshenko beam theory. The proposed method is developed based on the augmented Distributed Transfer Function Method (DTFM), which has been shown to be useful in various vibration problems. The proposed method models a Timoshenko beam structure by a spatial state-space formulation in the s-domain, without any discretization. With the state-space formulation, the frequency response of a beam structure, in any frequency region (from low to very high frequencies), can be obtained in an exact and analytical form. One advantage of the proposed method is that the local information of a beam structure, such as displacements, bending moment and shear force at any location, can be directly obtained from the space-state formulation, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated with the FEA in numerical examples, where the efficiency and accuracy of the proposed method is present. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are illustrated through comparison of the Timoshenko beam theory and the Euler-Bernoulli beam theory.

scholarly journals Extension Effects in Compliant Joints and Pseudo-Rigid-Body Models

2016 ◽  
Vol 138 (9) ◽  
Author(s):  
Venkatasubramanian Kalpathy Venkiteswaran ◽  
Hai-Jun Su
Keyword(s):  
Aspect Ratio ◽  
Rigid Body ◽  
Timoshenko Beam ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Element Analysis ◽  
Thin Beam ◽  
New Approach ◽  
Compliant Joints ◽  
Euler Bernoulli Beam

Compliant members come in a variety of shapes and sizes. While thin beam flexures are commonly used in this field, they can be replaced by soft members with lower aspect ratio. This paper looks to study the behavior of such elements by analyzing them from the view of beam theory for 2D cases. A modified version of the Timoshenko beam theory is presented which incorporates extension and Poisson's effects. The utility and validity of the new approach are demonstrated by comparing against Euler–Bernoulli beam theory, Timoshenko beam theory, and finite-element analysis (FEA). The results from this are then used to study the performance of pseudo-rigid-body models (PRBMs) for the analysis of low aspect ratio soft compliant joints for 2D quasi-static applications. A parallel-guiding mechanism comprised of similar compliant elements is analyzed using the new results to validate the contribution of this work.

An Analytical Model for Beam Flexure Modules Based on the Timoshenko Beam Theory

2017 ◽  
Author(s):  
M. H. Kahrobaiyan ◽  
M. Zanaty ◽  
S. Henein
Keyword(s):  
Timoshenko Beam ◽  
Axial Load ◽  
Compliant Mechanisms ◽  
Slenderness Ratio ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Bernoulli Beam ◽  
Model Based ◽  
Bernoulli Beam Theory ◽  
Euler Bernoulli Beam

Short beams are the key building blocks in many compliant mechanisms. Hence, deriving a simple yet accurate model of their elastokinematics is an important issue. Since the Euler-Bernoulli beam theory fails to accurately model these beams, we use the Timoshenko beam theory to derive our new analytical framework in order to model the elastokinematics of short beams under axial loads. We provide exact closed-form solutions for the governing equations of a cantilever beam under axial load modeled by the Timoshenko beam theory. We apply the Taylor series expansions to our exact solutions in order to capture the first and second order effects of axial load on stiffness and axial shortening. We show that our model for beam flexures approaches the model based on the Euler-Bernoulli beam theory when the slenderness ratio of the beams increases. We employ our model to derive the stiffness matrix and axial shortening of a beam with an intermediate rigid part, a common element in the compliant mechanisms with localized compliance. We derive the lateral and axial stiffness of a parallelogram flexure mechanism with localized compliance and compare them to those derived by the Euler-Bernoulli beam theory. Our results show that the Euler-Bernoulli beam theory predicts higher stiffness. In addition, we show that decrease in slenderness ratio of beams leads to more deviation from the model based on the Euler-Bernoulli beam theory.

Estimation of damping in riveted short cantilever beams

2020 ◽  
Vol 26 (23-24) ◽  
pp. 2163-2173
Author(s):  
Yemineni Siva Sankara Rao ◽  
Kutchibotla Mallikarjuna Rao ◽  
V V Subba Rao
Keyword(s):  
Timoshenko Beam ◽  
Damping Ratio ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Analytical Models ◽  
Cantilever Beams ◽  
Bernoulli Beam ◽  
Bernoulli Beam Theory ◽  
Half Power ◽  
Euler Bernoulli Beam

In layered and riveted structures, vibration damping happens because of a micro slip that occurs because of a relative motion at the common interfaces of the respective jointed layers. Other parameters that influence the damping mechanism in layered and riveted beams are the amplitude of initial excitation, overall length of the beam, rivet diameter, overall beam thickness, and many layers. In this investigation, using the analytical models such as the Euler–Bernoulli beam theory and Timoshenko beam theory and half-power bandwidth method, the free transverse vibration analysis of layered and riveted short cantilever beams is carried out for observing the damping mechanism by estimating the damping ratio, and the obtained results from the Euler–Bernoulli beam theory and Timoshenko beam theory analytical models are validated by the half-power bandwidth method. Although the Euler–Bernoulli beam model overestimates the damping ratio value by a very less fraction, both the models can be used to evaluate damping for short riveted cantilever beams along with the half-power bandwidth method.

Dynamic Mechanical Properties of Spirally Wound Paper Tubes

1989 ◽  
Vol 111 (4) ◽  
pp. 489-490 ◽  
Author(s):  
L. C. Bank ◽  
T. D. Gerhardt ◽  
J. H. Gordis
Keyword(s):  
Mechanical Properties ◽  
Timoshenko Beam ◽  
Beam Theory ◽  
Dynamic Mechanical Properties ◽  
Timoshenko Beam Theory ◽  
Dynamic Mechanical ◽  
Stiffness Properties ◽  
Free Mode ◽  
Euler Bernoulli Beam ◽  
Beam Theories

The use of experimental modal analysis to obtain the dynamic mechanical properties of spirally wound paper tubes is investigated. Based on experimentally measured natural frequencies in the free-free mode of transverse vibration, tube flexural stiffness properties are predicted using three beam theories: Euler-Bernoulli beam theory, Timoshenko beam theory for isotropic materials, and Timoshenko beam theory for anisotropic materials.

ANALYSIS OF AUXETIC BEAMS AS RESONANT FREQUENCY BIOSENSORS

2012 ◽  
Vol 12 (05) ◽  
pp. 1240027 ◽  
Author(s):  
TEIK-CHENG LIM
Keyword(s):  
Shear Deformation ◽  
Cross Sections ◽  
Timoshenko Beam ◽  
Poisson’S Ratio ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Poisson's Ratio ◽  
Bernoulli Beam ◽  
Bernoulli Beam Theory ◽  
Euler Bernoulli Beam

The mechanics of beam vibration is of fundamental importance in understanding the shift of resonant frequency of microcantilever and nanocantilever sensors. Unlike the simpler Euler–Bernoulli beam theory, the Timoshenko beam theory takes into consideration rotational inertia and shear deformation. For the case of microcantilevers and nanocantilevers, the minute size, and hence low mass, means that the topmost deviation from the Euler–Bernoulli beam theory to be expected is shear deformation. This paper considers the extent of shear deformation for varying Poisson's ratio of the beam material, with special emphasis on solids with negative Poisson's ratio, which are also known as auxetic materials. Here, it is shown that the Timoshenko beam theory approaches the Euler–Bernoulli beam theory if the beams are of solid cross-sections and the beam material possess high auxeticity. However, the Timoshenko beam theory is significantly different from the Euler–Bernoulli beam theory for beams in the form of thin-walled tubes regardless of the beam material's Poisson's ratio. It is herein proposed that calculations on beam vibration can be greatly simplified for highly auxetic beams with solid cross-sections due to the small shear correction term in the Timoshenko beam deflection equation.

Horizontal Dynamic Response of a Combined Loaded Large-Diameter Pipe Pile Simulated by the Timoshenko Beam Theory

2019 ◽  
Vol 20 (02) ◽  
pp. 2071003 ◽  
Author(s):  
Changjie Zheng ◽  
Lubao Luan ◽  
Hongyu Qin ◽  
Hang Zhou
Keyword(s):  
Timoshenko Beam ◽  
Dynamic Performance ◽  
Large Diameter ◽  
Beam Theory ◽  
Analytical Framework ◽  
Timoshenko Beam Theory ◽  
Pipe Pile ◽  
Large Diameter Pipe ◽  
Diameter Pipe ◽  
Euler Bernoulli Beam

This paper presents an analytical framework for the horizontal dynamic analysis of a large-diameter pipe pile subjected to combined loadings, in which the pipe pile is simulated by the Timoshenko beam theory. The derived solution allows us to evaluate the effects of both the shear deformations and vertical loads on the horizontal dynamic performance of the pipe pile. The proposed solution provides appropriate estimates of complex impedances of large-diameter pipe piles, unlike the earlier solutions based on the Euler–Bernoulli beam theory for describing the pile behavior, which ignores the shear deformation of the pile. The results indicate that the Euler–Bernoulli theory overestimates the pipe pile’s horizontal impedance, while overestimating the effect of vertical loads on its horizontal performance.

A Time-Dependent Finite Element Formulation for Thick Shape Memory Polymer Beams Considering Shear Effects

2018 ◽  
Vol 10 (04) ◽  
pp. 1850043 ◽  
Author(s):  
Amir H. Eskandari ◽  
Mostafa Baghani ◽  
Saeed Sohrabpour
Keyword(s):  
Finite Element ◽  
Shape Memory ◽  
Timoshenko Beam ◽  
Beam Theory ◽  
Beam Element ◽  
Timoshenko Beam Theory ◽  
Bernoulli Beam ◽  
3D Finite Element ◽  
Bernoulli Beam Theory ◽  
Euler Bernoulli Beam

In this paper, employing a thermomechanical small strain constitutive model for shape memory polymers (SMP), a beam element made of SMPs is presented based on the kinematic assumptions of Timoshenko beam theory. Considering the low stiffness of SMPs, the necessity for developing a Timoshenko beam element becomes more prominent. This is due to the fact that relatively thicker beams are required in the design procedure of smart structures. Furthermore, in the design and optimization process of these structures which involves a large number of simulations, we cannot rely only on the time consuming 3D finite element analyses. In order to properly validate the developed formulations, the numeric results of the present work are compared with those of 3D finite element results of the authors, previously available in the literature. The parametric study on the material parameters, e.g., hard segment volume fracture, viscosity coefficient of different phases, and the external force applied on the structure (during the recovery stage) are conducted on the thermomechanical response of a short I-shape SMP beam. For instance, the maximum beam deflection error in one of the studied examples for the Euler–Bernoulli beam theory is 7.3%, while for the Timoshenko beam theory, is 1.5% with respect to the 3D FE solution. It is noted that for thicker or shorter beams, the error of the Euler–Bernoulli beam theory even more increases. The proposed beam element in this work could be a fast and reliable alternative tool for modeling 3D computationally expensive simulations.

Exact Dynamic Analysis of Space Structures Using Timoshenko Beam Theory

AIAA Journal ◽  
2004 ◽  
Vol 42 (4) ◽  
pp. 833-839 ◽  
Author(s):  
Jen-Fang Yu ◽  
Hsin-Chung Lien ◽  
B. P. Wang
Keyword(s):  
Dynamic Analysis ◽  
Timoshenko Beam ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Space Structures
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