[논문]Bar와 Beam 구조물의 기본적인 유한요소 모델의 수치해석

류용희; 주부석; 정우영

doi:10.11004/kosacs.2013.4.1.001

Bar와 Beam 구조물의 기본적인 유한요소 모델의 수치해석
Numerical Evaluation of Fundamental Finite Element Models in Bar and Beam Structures 원문보기

복합신소재구조학회 논문집 = Journal of the Korean Society for Advanced Composite Structures, v.4 no.1, 2013년, pp.1 - 8

류용희 (노스캐롤라이나주립대학교 토목공학과) , 주부석 (노스캐롤라이나주립대학교 토목공학과) , 정우영 (강릉원주대학교 토목공학과)

Abstract ▼ AI-Helper

The finite element analysis (FEA) is a numerical technique to find solutions of field problems. A field problem is approximated by differential equations or integral expressions. In a finite element, the field quantity is allowed to have a simple spatial variation in terms of linear or polynomial functions. This paper represents a review and an accuracy-study of the finite element method comparing the FEA results with the exact solution. The exact solutions were calculated by solid mechanics and FEA using matrix stiffness method. For this study, simple bar and cantilever models were considered to evaluate four types of basic elements - constant strain triangle (CST), linear strain triangle (LST), bi-linear-rectangle(Q4),and quadratic-rectangle(Q8). The bar model was subjected to uniaxial loading whereas in case of the cantilever model moment loading was used. In the uniaxial loading case, all basic element results of the displacement and stress in x-direction agreed well with the exact solutions. In the moment loading case, the displacement in y-direction using LST and Q8 elements were acceptable compared to the exact solution, but CST and Q4 elements had to be improved by the mesh refinement.

주제어

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가설 설정

This element has a node at each end, and each node has one degree of freedom (DOF) for translations. In this problem, the assumptions were made as: 1) the material properties are not affected by applied forces or moment loading 2) geometric nonlinearity is not considered, 3) material remains linearly elastic, isotropic, and homogeneous. The shape functions N for the bar elements can be defined by Eq.
1. Bernoulli Euler beam theory was used in the approach with the assumptions as: 1) the beam is initially straight, unstressed and symmetric, and long and slender, 2) plane sections remain plane and perpendicular to the neutral axis during bending, 3) Deflections are small, 3) the cross section needs to have a vertical axis of symmetry. 4) material is linearly elastic, isotropic, and homogeneous, 5) the beam has no twisting, 5) proportional limit is not exceeded.
Bernoulli Euler beam theory was used in the approach with the assumptions as: 1) the beam is initially straight, unstressed and symmetric, and long and slender, 2) plane sections remain plane and perpendicular to the neutral axis during bending, 3) Deflections are small, 3) the cross section needs to have a vertical axis of symmetry. 4) material is linearly elastic, isotropic, and homogeneous, 5) the beam has no twisting, 5) proportional limit is not exceeded. Having those assumptions, the solutions for the displacement and stress can be calculated by Equations (1) and (2):

제안 방법

In this paper, the principle concept of finite element analysis (FEA) was reviewed, and FEA using basic elements such as constant strain triangle (CST), linear strain triangle (LST), a four-node quadrilateral membrane element (Q4), and an eight-node membrane element (Q8) were evaluated by comparing the FEA results with the exact answer. For the application, practice models were subjected to uniaxial or bending moment loadings, and then the solid mechanics, matrix stiffness method, and FEM using basic elements were used to find the solutions for each case. In the uniaxial loading case, all results of displacements and stresses in x-direction agreed well compared to the exact solution, but the stress in y-direction were found even though stress in y-direction was not existed.
In this paper, the principle concept of finite element analysis (FEA) was reviewed, and FEA using basic elements such as constant strain triangle (CST), linear strain triangle (LST), a four-node quadrilateral membrane element (Q4), and an eight-node membrane element (Q8) were evaluated by comparing the FEA results with the exact answer. For the application, practice models were subjected to uniaxial or bending moment loadings, and then the solid mechanics, matrix stiffness method, and FEM using basic elements were used to find the solutions for each case.
The percentage error reduced significantly as finer mesh improved the model to have the quadratic behavior. In this paper, we examined the accuracy of FEA using commonly used basic elements - CST, LST, Q4, and Q8 by comparing FEA results with an exact solution referred to solid mechanics. FEM is a numerical technique to find approximate solutions.
This paper examines the accuracy of FEA using commonly used basic elements CST, LST, Q4, and Q8 by comparing FEA results with an exact solution referred to solid mechanics. Practice models were subjected to uniaxial or moment loadings, and solid mechanics, matrix stiffness method, and FEM using basice lements were used to find the solutions for each case. There view of basic concepts and shape functions for each approach were conducted to develop stiffness matrix for the matrix stiffness method and FEM.
As a result, the evaluation of FEA using basic elements is necessary for the accurate analysis. This paper examines the accuracy of FEA using commonly used basic elements CST, LST, Q4, and Q8 by comparing FEA results with an exact solution referred to solid mechanics. Practice models were subjected to uniaxial or moment loadings, and solid mechanics, matrix stiffness method, and FEM using basice lements were used to find the solutions for each case.

대상 데이터

10 shows the FE model using CST elements. Four plane triangle elements were used for this analysis. The field quantity is allowed to have linear displacements in X and Y directions.
Three conditions such as the compatibility, equilibrium, and boundary must be satisfied in this analysis. One of the basic models is the CST element which is a plane triangle element. In this model, a field quantity is allowed to have only a simple spatial variation described by linear terms such as x or y as shown in Fig.
1. This model has dimensions as h = 0.25m, L = 0.5m, and t =0.025m. The plane stress condition was considered because the thickness of the model is to be small compared to the other dimensions in two-dimensional analysis.

성능/효과

From the Eq. (3), the displacement at middle and end locations were calculated as 0.0025 m and 0.01 m, respectively. Solid Mechanics’ equation and the matrix analysis yield the same result, but there were significant errors using FEA with basic elements.

참고문헌 (10)

Cook, R. D. (2002). Concepts and applications of finite element analysis. 4th ed., Wiley, NewYork, USA.
Courant, R. L. (1943). "Vibrational Methods for the Solutionof Problems of Equilibrium and Vibration.", Bulletin of the American Mathematical Society, Vol. 49, pp. 1-23.
Iqbal, R. (2007). "Development of a finiteelement program incorporating advanced element types." M.S. Thesis, Osmania University Hyderabad, India
Kassimali, A. (2011). Matrix Analysis of Structures, Pacific Grove, Cengage Learning, CA, USA.
Kattan, P. I. (2007). MATLAB guide to finite elements: an interactive approach. 2nd ed. NewYork :Springer.
Melosh, R. J. (1985). "Self-qualification of finite element analysis results by polynomial extrapolation." Finite Elements in Analysis and Design, Vol. 1, pp. 49-60.

상세보기
Olson, M. D. and Bearden, T. W. (1979). A simple flat triangular shell element revisited, Int. J. Numer. Meth. Engng., Vol. 2, pp. 51-68.
Robinson, J. (1980). Four-node quadrilateral stress membrane element with rotational stiffness, Int. J. Numer. Meth. Engng., Vol. 3, pp. 1567-1569.
Sze, K. Y., Wanji, C., and Cheung, Y. K. (1992). "An efficient quadrilateral plane element with drilling degrees of freedom using orthogonal stress modes." Comp. Struct., Vol. 42, No. 5, pp. 695-705.

상세보기
Timoshenko, S. P. (1953). History of Strength of Materials, McGraw-Hill Book Co. New York.

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Bar와 Beam 구조물의 기본적인 유한요소 모델의 수치해석
Numerical Evaluation of Fundamental Finite Element Models in Bar and Beam Structures 원문보기

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Bar와 Beam 구조물의 기본적인 유한요소 모델의 수치해석 Numerical Evaluation of Fundamental Finite Element Models in Bar and Beam Structures 원문보기

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Bar와 Beam 구조물의 기본적인 유한요소 모델의 수치해석
Numerical Evaluation of Fundamental Finite Element Models in Bar and Beam Structures 원문보기

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