Analytic solution of Timoshenko beam excited by real seismic support motions

김용우 국제구조공학회 2017 Structural Engineering and Mechanics, An Int'l Jou Vol.62 No.2

Beam-like structures such as bridge, high building and tower, pipes, flexible connecting rods and some robotic manipulators are often excited by support motions. These structures are important in machines and structures. So, this study proposes an analytic method to accurately predict the dynamic behaviors of the structures during support motions or an earthquake. Using Timoshenko beam theory which is valid even for non-slender beams and for high-frequency responses, the analytic responses of fixed-fixed beams subjected to a real seismic motions at supports are illustrated to show the principled approach to the proposed method. The responses of a slender beam obtained by using Timoshenko beam theory are compared with the solutions based on Euler-Bernoulli beam theory to validate the correctness of the proposed method. The dynamic analysis for the fixed-fixed beam subjected to support motions gives useful information to develop an understanding of the structural behavior of the beam. The bending moment and the shear force of a slender beam are governed by dynamic components while those of a stocky beam are governed by static components. Especially, the maximal magnitudes of the bending moment and the shear force of the thick beam are proportional to the difference of support displacements and they are influenced by the seismic wave velocity.

전단변형을 고려한 적층복합 I형 박벽보의 C유한요소

백성용,이승식 한국강구조학회 2006 韓國鋼構造學會 論文集 Vol.18 No.3

본 연구에서는 직교좌표계에 근거한 적층복합 I형 박벽보의 유한요소 해석을 위한 새로운 블록 강도행렬을 제안한다. 변위장은1차 전단변형을 고려한 보 이론을 사용하여 정의되었다. 축방향 변위는 Timoshenko 보이론과 수정된 Vlasov 박벽보 이론을 결합하여 투영단면의 면 변형과 면외 변형의 합으로 나타낸다. 유도된 강성행렬은 휨 전단변형과 ?? 비틂에 의한 영향을 고려한다. 본 유한요소 에서는 2절점, 3절점, 4절점의 세 가지 보요소를 제안하였다. 3절점과 4절점 보 요소는 적층복합 보의 휨 해석에 효과적이었다. 다른 연구자의 수치해석 결과와 비교 검토를 통하여 새로운 유한요소의 활용성과 정확성을 입증하였다. paper presents a new block stifnes matrix for the analyson an orthogonal Cartesian cordinate system. The displacement fields are defined using the first order shear deformable beam theory. The longitudinal displacement can be expresed as the sum of the projected plane deformation of the cros-section due to Timoshenko's beam theory and axial warping deformation due to modified Vlasov's thin-waled beam theory. The derived element takes into acount flexural shear d eformation and torsional warping deformation. Three diferent types of beam elements, namely, the two-noded, three-noded, and four-noded beam elements, are developed. The quadratic and cubic elements are found to be very efficient for the flexu accuracy of the new element are demonstrated by comparing the n umerical results available in the literature.

Large deflection analysis of edge cracked simple supported beams

Akbas, Seref Doguscan Techno-Press 2015 Structural Engineering and Mechanics, An Int'l Jou Vol.54 No.3

This paper focuses on large deflection static behavior of edge cracked simple supported beams subjected to a non-follower transversal point load at the midpoint of the beam by using the total Lagrangian Timoshenko beam element approximation. The cross section of the beam is circular. The cracked beam is modeled as an assembly of two sub-beams connected through a massless elastic rotational spring. It is known that large deflection problems are geometrically nonlinear problems. The considered highly nonlinear problem is solved considering full geometric non-linearity by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. There is no restriction on the magnitudes of deflections and rotations in contradistinction to von-Karman strain displacement relations of the beam. The beams considered in numerical examples are made of Aluminum. In the study, the effects of the location of crack and the depth of the crack on the non-linear static response of the beam are investigated in detail. The relationships between deflections, end rotational angles, end constraint forces, deflection configuration, Cauchy stresses of the edge-cracked beams and load rising are illustrated in detail in nonlinear case. Also, the difference between the geometrically linear and nonlinear analysis of edge-cracked beam is investigated in detail.

Timoshenko theory effect on the vibration of axially functionally graded cantilever beams carrying concentrated masses

Carlos A. Rossit,Diana V. Bambill,Gonzalo J. Gilardi 국제구조공학회 2018 Structural Engineering and Mechanics, An Int'l Jou Vol.66 No.6

In this paper is studied the effect of considering the theory of Timoshenko in the vibration of AFG beams that support ground masses. As it is known, Timoshenko theory takes into account the shear deformation and the rotational inertia, provides more accurate results in the general study of beams and is mandatory in the case of high frequencies or non-slender beams. The Rayleigh-Ritz Method is employed to obtain approximated solutions of the problem. The accuracy of the procedure is verified through results available in the literature that can be represented by the model under study. The incidence of the Timoshenko theory is analyzed for different cases of beam slenderness, variation of its cross section and compositions of its constituent material, as well as different amounts and positions of the attached masses.

A modified modal perturbation method for vibration characteristics of non-prismatic Timoshenko beams

Pan, Danguang,Chen, Genda,Lou, Menglin Techno-Press 2011 Structural Engineering and Mechanics, An Int'l Jou Vol.40 No.5

A new perturbation method is introduced to study the undamped free vibration of a non-prismatic Timoshenko beam for its natural frequencies and vibration modes. For simplicity, the natural modes of vibration of its corresponding prismatic Euler-Bernoulli beam with the same length and boundary conditions are used as Ritz base functions with necessary modifications to account for shear strain in the Timoshenko beam. The new method can transform two coupled partial differential equations governing the transverse vibration of the non-prismatic Timoshenko beam into a set of nonlinear algebraic equations. It significantly simplifies the solution process and is applicable to non-prismatic beams with various boundary conditions. Three examples indicated that the new method is more accurate than the previous perturbation methods. It successfully takes into account the effect of shear deformation of Timoshenko beams particularly at the free end of cantilever structures.

Timoshenko theory effect on the vibration of axially functionally graded cantilever beams carrying concentrated masses

Rossit, Carlos A.,Bambill, Diana V.,Gilardi, Gonzalo J. Techno-Press 2018 Structural Engineering and Mechanics, An Int'l Jou Vol.66 No.6

In this paper is studied the effect of considering the theory of Timoshenko in the vibration of AFG beams that support ground masses. As it is known, Timoshenko theory takes into account the shear deformation and the rotational inertia, provides more accurate results in the general study of beams and is mandatory in the case of high frequencies or non-slender beams. The Rayleigh-Ritz Method is employed to obtain approximated solutions of the problem. The accuracy of the procedure is verified through results available in the literature that can be represented by the model under study. The incidence of the Timoshenko theory is analyzed for different cases of beam slenderness, variation of its cross section and compositions of its constituent material, as well as different amounts and positions of the attached masses.

A modified modal perturbation method for vibration characteristics of non-prismatic Timoshenko beams

Danguang Pan,Genda Chen,Menglin Lou 국제구조공학회 2011 Structural Engineering and Mechanics, An Int'l Jou Vol.40 No.5

A new perturbation method is introduced to study the undamped free vibration of a nonprismatic Timoshenko beam for its natural frequencies and vibration modes. For simplicity, the natural modes of vibration of its corresponding prismatic Euler-Bernoulli beam with the same length and boundary conditions are used as Ritz base functions with necessary modifications to account for shear strain in the Timoshenko beam. The new method can transform two coupled partial differential equations governing the transverse vibration of the non-prismatic Timoshenko beam into a set of nonlinear algebraic equations. It significantly simplifies the solution process and is applicable to non-prismatic beams with various boundary conditions. Three examples indicated that the new method is more accurate than the previous perturbation methods. It successfully takes into account the effect of shear deformation of Timoshenko beams particularly at the free end of cantilever structures.

Deflections and rotations in rectangular beams with straight haunches under uniformly distributed load considering the shear deformations

José Daniel Barquero-Cabrero,Arnulfo Luévanos-Rojas,Sandra López-Chavarría,Manuel Medina-Elizondo,Francisco Velázquez-Santillán,Ricardo Sandoval-Rivas 국제구조공학회 2018 Smart Structures and Systems, An International Jou Vol.22 No.6

This paper presents a model of the elastic curve for rectangular beams with straight haunches under uniformly distributed load and moments in the ends considering the bending and shear deformations (Timoshenko Theory) to obtain the deflections and rotations on the beam, which is the main part of this research. The traditional model of the elastic curve for rectangular beams under uniformly distributed load considers only the bending deformations (Euler-Bernoulli Theory). Also, a comparison is made between the proposed and traditional model of simply supported beams with respect to the rotations in two supports and the maximum deflection of the beam. Also, another comparison is made for beams fixed at both ends with respect to the moments and reactions in the support A, and the maximum deflection of the beam. Results show that the proposed model is greater for simply supported beams in the maximum deflection and the traditional model is greater for beams fixed at both ends in the maximum deflection. Then, the proposed model is more appropriate and safe with respect the traditional model for structural analysis, because the shear forces and bending moments are present in any type of structure and the bending and shear deformations appear.

Finite element formulation and analysis of Timoshenko beam excited by transversely fl uctuating supports due to a real seismic wave

김용우,차승찬 한국원자력학회 2018 Nuclear Engineering and Technology Vol.50 No.6

Using the concept of quasi-static decomposition and using three-noded isoparametric locking-freeelement, this article presents a formulation of the finite element method for Timoshenko beam subjectedto spatially different time-dependent motions at supports. To verify the validity of the formulation,three fixed-hinged beams excited by the real seismic motions are examined; one is a slender beam,another is a stocky one, and the other is an intermediate one. The numerical results of time histories ofmotions of the three beams are compared with corresponding analytical solutions. The internal loadssuch as bending moment and shearing force at a specific time are also compared with analytic solutions. These comparisons show good agreements. The comparisons between static components of the internalloads and the corresponding total internal loads show that the static components predominate in thestocky beam, whereas the dynamic components predominate in the slender one. Thus, the total internalloads of the stocky beam, which is governed by static components, can be predicted simply by staticanalysis. Careful numerical experiments indicate that the fundamental frequency of a beam can be usedas a parameter identifying such a stocky beam.

Large deflection analysis of edge cracked simple supported beams

Şeref Doğuşcan Akbaş 국제구조공학회 2015 Structural Engineering and Mechanics, An Int'l Jou Vol.54 No.3

This paper focuses on large deflection static behavior of edge cracked simple supported beamssubjected to a non-follower transversal point load at the midpoint of the beam by using the total Lagrangian Timoshenko beam element approximation. The cross section of the beam is circular. The cracked beam is modeled as an assembly of two sub-beams connected through a massless elastic rotational spring. It is known that large deflection problems are geometrically nonlinear problems. The considered highly nonlinear problem is solved considering full geometric non-linearity by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. There is no restriction on the magnitudes of deflections and rotations in contradistinction to von-Karman strain displacement relations of the beam. The beams considered in numerical examples are made of Aluminum. In the study, the effects of the location of crack and the depth of the crack on the non-linear static response of the beam are investigated in detail. The relationships between deflections, end rotational angles, end constraint forces, deflection configuration, Cauchy stresses of the edge- cracked beams and load rising are illustrated in detail in nonlinear case. Also, the difference between the geometrically linear and nonlinear analysis of edge-cracked beam is investigated in detail.