The linear elastic stiffness matrix model analysis of pre twisted Euler Bernoulli beam

Ying Huang,Haoran Zou,Chang Hong Chen,Songlin Bai,Yao Yao,Leon M. Keer 국제구조공학회 2019 Structural Engineering and Mechanics, An Int'l Jou Vol.72 No.5

Based on the finite element method of traditional straight Euler-Bernoulli beams and the coupled relations between linear displacement and angular displacement of a pre-twisted Euler-Bernoulli beam, the shape functions and stiffness matrix are deduced. Firstly, the stiffness of pre-twisted Euler-Bernoulli beam is developed based on the traditional straight Euler-Bernoulli beam. Then, a new finite element model is proposed based on the displacement general solution of a pre-twisted Euler-Bernoulli beam. Finally, comparison analyses are made among the proposed Euler-Bernoulli model, the new numerical model based on displacement general solution and the ANSYS solution by Beam188 element based on infinite approach. The results show that developed numerical models are available for the pre-twisted Euler-Bernoulli beam, and which provide more accurate finite element model for the numerical analysis. The effects of pre-twisted angle and flexural stiffness ratio on the mechanical property are investigated.

On the static and dynamic stability of beams with an axial piezoelectric actuation

C. Zehetner,H. Irschik 국제구조공학회 2008 Smart Structures and Systems, An International Jou Vol.4 No.1

The present contribution is concerned with the static and dynamic stability of a piezo-laminated Bernoulli-Euler beam subjected to an axial compressive force. Recently, an inconsistent derivation of the equations of motions of such a smart structural system has been presented in the literature, where it has been claimed, that an axial piezoelectric actuation can be used to control its stability. The main scope of the present paper is to show that this unfortunately is impossible. We present a consistent theory for composite beams in plane bending. Using an exact description of the kinematics of the beam axis, together with the Bernoulli-Euler assumptions, we obtain a single-layer theory capable of taking into account the effects of piezoelectric actuation and buckling. The assumption of an inextensible beam axis, which is frequently used in the literature, is discussed afterwards. We show that the cited inconsistent beam model is due to inadmissible mixing of the assumptions of an inextensible beam axis and a vanishing axial displacement, leading to the erroneous result that the stability might be enhanced by an axial piezoelectric actuation. Our analytical formulations for simply supported Bernoulli-Euler type beams are verified by means of three-dimensional finite element computations performed with ABAQUS.

On the static and dynamic stability of beams with an axial piezoelectric actuation

Zehetner, C.,Irschik, H. Techno-Press 2008 Smart Structures and Systems, An International Jou Vol.4 No.1

The present contribution is concerned with the static and dynamic stability of a piezo-laminated Bernoulli-Euler beam subjected to an axial compressive force. Recently, an inconsistent derivation of the equations of motions of such a smart structural system has been presented in the literature, where it has been claimed, that an axial piezoelectric actuation can be used to control its stability. The main scope of the present paper is to show that this unfortunately is impossible. We present a consistent theory for composite beams in plane bending. Using an exact description of the kinematics of the beam axis, together with the Bernoulli-Euler assumptions, we obtain a single-layer theory capable of taking into account the effects of piezoelectric actuation and buckling. The assumption of an inextensible beam axis, which is frequently used in the literature, is discussed afterwards. We show that the cited inconsistent beam model is due to inadmissible mixing of the assumptions of an inextensible beam axis and a vanishing axial displacement, leading to the erroneous result that the stability might be enhanced by an axial piezoelectric actuation. Our analytical formulations for simply supported Bernoulli-Euler type beams are verified by means of three-dimensional finite element computations performed with ABAQUS.

Dynamic modeling and analysis of a spinning Rayleigh beam under deployment

Zhu, K.,Chung, J. Pergamon Press ; Elsevier Science Ltd 2016 International journal of mechanical sciences Vol.115 No.-

In this paper we propose a new model for a spinning beam under deployment and present analyses of the beam's dynamic responses and characteristics. The proposed model is established in an inertial reference frame by using the Rayleigh beam theory to consider the rotary inertia effect. This model is an advanced version of a model that we previously reported; that model was based on the Euler-Bernoulli beam theory, described in a rotating reference frame. We compare the dynamic responses and natural frequencies of a spinning beam under deployment between the proposed Rayleigh beam model and the previous Euler-Bernoulli beam model. In addition, we analyze the beat phenomena of a spinning Rayleigh beam under deployment in an inertial reference frame. Furthermore, we investigate the effects of the choice between the two reference frames (inertial and rotating reference frames) upon the analytical results for dynamic responses and natural frequencies. We show that the proposed model yields more accurate and reliable results for dynamic responses than the previous model, for the case in which a spinning beam is deployed.

Tailoring the second mode of Euler-Bernoulli beams: an analytical approach

Korak Sarkar,Ranjan Ganguli 국제구조공학회 2014 Structural Engineering and Mechanics, An Int'l Jou Vol.51 No.5

In this paper, we study the inverse mode shape problem for an Euler-Bernoulli beam, using ananalytical approach. The mass and stiffness variations are determined for a beam, having various boundary conditions, which has a prescribed polynomial second mode shape with an internal node. It is found that physically feasible rectangular cross-section beams which satisfy the inverse problem exist for a variety of boundary conditions. The effect of the location of the internal node on the mass and stiffness variations and on the deflection of the beam is studied. The derived functions are used to verify the p-version finite element code, for the cantilever boundary condition. The paper also presents the bounds on the location of the internal node, for a valid mass and stiffness variation, for any given boundary condition. The derived property variations, corresponding to a given mode shape and boundary condition, also provides a simple closed-form solution for a class of non-uniform Euler-Bernoulli beams. These closed-form solutions canalso be used to check optimization algorithms proposed for modal tailoring.

Natural Frequencies of Euler-Bernoulli Beam with Open Cracks on Elastic Foundations

Shin Young-Jae,Yun Jong-Hak,Seong Kyeong-Youn,Kim Jae-Ho,Kang Sung-Hwang The Korean Society of Mechanical Engineers 2006 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.20 No.4

A study of the natural vibrations of beam resting on elastic foundation with finite number of transverse open cracks is presented. Frequency equations are derived for beams with different end restraints. Euler-Bernoulli beam on Pasternak foundation and Euler-Bernoulli beam on Pasternak foundation are investigated. The cracks are modeled by massless substitute spring. The effects of the crack location, size and its number and the foundation constants, on the natural frequencies of the beam, are investigated.

Tailoring the second mode of Euler-Bernoulli beams: an analytical approach

Sarkar, Korak,Ganguli, Ranjan Techno-Press 2014 Structural Engineering and Mechanics, An Int'l Jou Vol.51 No.5

In this paper, we study the inverse mode shape problem for an Euler-Bernoulli beam, using an analytical approach. The mass and stiffness variations are determined for a beam, having various boundary conditions, which has a prescribed polynomial second mode shape with an internal node. It is found that physically feasible rectangular cross-section beams which satisfy the inverse problem exist for a variety of boundary conditions. The effect of the location of the internal node on the mass and stiffness variations and on the deflection of the beam is studied. The derived functions are used to verify the p-version finite element code, for the cantilever boundary condition. The paper also presents the bounds on the location of the internal node, for a valid mass and stiffness variation, for any given boundary condition. The derived property variations, corresponding to a given mode shape and boundary condition, also provides a simple closed-form solution for a class of non-uniform Euler-Bernoulli beams. These closed-form solutions can also be used to check optimization algorithms proposed for modal tailoring.

Natural Frequencies of Euler-Bernoulli Beam with Open Cracks on Elastic Foundations

Youngjae Shin,Jonghak Yun,Kyeongyoun Seong,Jaeho Kim,Sunghwang Kang 대한기계학회 2006 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.20 No.4

A study of the natural vibrations of beam resting on elastic foundation with finite number of transverse open cracks is presented. Frequency equations are derived for beams with different end restraints. Euler-Bernoulli beam on Winkler foundation and Euler-Bernoulli beam on Pasternak foundation are investigated. The cracks are modeled by massless substitute spring. The effects of the crack location, size and its number and the foundation constants, on the natural frequencies of the beam, are investigated.

Free vibration analysis of beams with non-ideal clamped boundary conditions

이진희 대한기계학회 2013 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.27 No.2

A non-ideal boundary condition is modeled as a linear combination of the ideal simply supported and the ideal clamped boundary conditions with the weighting factors k and 1-k, respectively. The proposed non-ideal boundary model is applied to the free vibration analyses of Euler-Bernoulli beam and Timoshenko beam. The free vibration analysis of the Euler-Bernoulli beam is carried out analyti-cally, and the pseudospectral method is employed to accommodate the non-ideal boundary conditions in the analysis of the free vibration of Timoshenko beam. For the free vibration with the non-ideal boundary condition at one end and the free boundary condition at the other end, the natural frequencies of the beam decrease as k increases. The free vibration where both the ends of a beam are restrained by the non-ideal boundary conditions is also considered. It is found that when the non-ideal boundary conditions are close to the ideal clamped boundary conditions the natural frequencies are reduced noticeably as k increases. When the non-ideal boundary conditions are close to the ideal simply supported boundary conditions, however, the natural frequencies hardly change as k varies, which indicate that the proposed boundary condition model is more suitable to the non-ideal boundary condition close to the ideal clamped boundary condition.

Bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams using integral form of Eringen's non-local elasticity theory

Nejad, Mohammad Zamani,Hadi, Amin,Omidvari, Arash,Rastgoo, Abbas Techno-Press 2018 Structural Engineering and Mechanics, An Int'l Jou Vol.67 No.4

The main aim of this paper is to investigate the bending of Euler-Bernouilli nano-beams made of bi-directional functionally graded materials (BDFGMs) using Eringen's non-local elasticity theory in the integral form with compare the differential form. To the best of the researchers' knowledge, in the literature, there is no study carried out into integral form of Eringen's non-local elasticity theory for bending analysis of BDFGM Euler-Bernoulli nano-beams with arbitrary functions. Material properties of nano-beam are assumed to change along the thickness and length directions according to arbitrary function. The approximate analytical solutions to the bending analysis of the BDFG nano-beam are derived by using the Rayleigh-Ritz method. The differential form of Eringen's non-local elasticity theory reveals with increasing size effect parameter, the flexibility of the nano-beam decreases, that this is unreasonable. This problem has been resolved in the integral form of the Eringen's model. For all boundary conditions, it is clearly seen that the integral form of Eringen's model predicts the softening effect of the non-local parameter as expected. Finally, the effects of changes of some important parameters such as material length scale, BDFG index on the values of deflection of nano-beam are studied.