Effect of higher order terms of Maclaurin expansion in nonlinear analysis of the Bernoulli beam by single finite element

- Journal title : Structural Engineering and Mechanics
- Volume 58, Issue 6, 2016, pp.949-966
- Publisher : Techno-Press
- DOI : 10.12989/sem.2016.58.6.949

Title & Authors

Effect of higher order terms of Maclaurin expansion in nonlinear analysis of the Bernoulli beam by single finite element

Zahrai, Seyed Mehdi; Mortezagholi, Mohamad Hosein; Mirsalehi, Maryam;

Zahrai, Seyed Mehdi; Mortezagholi, Mohamad Hosein; Mirsalehi, Maryam;

Abstract

The second order analysis taking place due to non-linear behavior of the structures under the mechanical and geometric factors through implementing exact and approximate methods is an indispensible issue in the analysis of such structures. Among the exact methods is the slope-deflection method that due to its simplicity and efficiency of its relationships has always been in consideration. By solving the differential equations of the modified slope-deflection method in which the effect of axial compressive force is considered, the stiffness matrix including trigonometric entries would be obtained. The complexity of computations with trigonometric functions causes replacement with their Maclaurin expansion. In most cases only the first two terms of this expansion are used but to obtain more accurate results, more elements are needed. In this paper, the effect of utilizing higher order terms of Maclaurin expansion on reducing the number of required elements and attaining more rapid convergence with less error is investigated for the Bernoulli beam with various boundary conditions. The results indicate that when using only one element along the beam length, utilizing higher order terms in Maclaurin expansion would reduce the relative error in determining the critical buckling load and kinematic parameters in the second order analysis.

Keywords

non-linear behavior;slope-deflection method;axial compressive force;Maclaurin expansion;critical load;kinematic parameters;

Language

English

Cited by

References

1.

Al-Bermani, F.G. and Kitipornchai, S. (1990), "Nonlinear analysis of thin-walled structures using least element/member", J. Struct. Eng., 116(1), 215-234.

2.

Aristizabal-Ochoa, J.D. (1997), "First- and second-order stiffness matrices and load vector of beam-columns with semi-rigid connections", J. Struct. Eng., ASCE, 123(5), 669-78.

3.

Aristizabal-Ochoa, J.D. (2008), "Slope-deflection equations for stability and second-order analysis of Timoshenko beam-column structures with semi-rigid connections", Eng. Struct., 30(9), 2517-2527.

4.

Aristizabal-Ochoa, J.D. (2012), "Matrix method for stability and second-order analysis of Timoshenko beam-column structures with semi-rigid connections", Eng. Struct., 34, 289-302.

5.

Balling, R.J. and Lyon, J.W. (2010), "Second-order analysis of plane frames with one element per member", J. Struct. Eng., 137(11), 1350-1358.

6.

Bryant, R.H. and Baile, O.C. (1977), "Slope deflection analysis including transverse shear", J. Struct. Div., 103(2), 443-6.

7.

Chan, S.L. and Zhou, Z.H. (1994), "Point wise equilibrating polynomial element for nonlinear analysis of frames", J. Struct. Eng., 120(6), 1703-1717.

8.

Chan, S.L. and Zhou, Z.H. (1995), "Second-order elastic analysis of frames using single imperfect element per member", J. Struct. Eng., 121(6), 939-945.

9.

Eltaher, M.A., Khater, M.E., Park, S., Abdel-Rahman, E. and Yavuz, M. (2016) "On the static stability of nonlocal nanobeams using Higher-order beam theories", Adv. Nano. Res., 4(1), 51-64.

10.

Ermopoulos, J C. (1988), "Slope-deflection method and bending of tapered bars under stepped loads", J. Constr. Steel Res., 11(2), 121-141.

11.

Ibearugbulem, O.M., Ettu, L.O. and Ezeh, J.C. (2013), "A new stiffness matrix for analysis of flexural line continuum", Int. J. Eng. Sci., 2(2), 57-62.

12.

Iu, C.K. and Bradford, M.A. (2010), "Second-order elastic finite element analysis of steel structures using a single element per member", Eng. Struct., 32(9), 2606-2616.

13.

Nguyen, N.H. and Lee, D.Y. (2015), "Bending analysis of a single leaf flexure using higher-order beam theory", Struct. Eng. Mech., 53(4), 781-790.

14.

Saffari, H., Rahgozar, R. and Jahanshahi, R. (2008), "An efficient method for computation of effective length factor of columns in a steel gabled frame with tapered members", J. Constr. Steel Res., 64(4), 400-406.

15.

Samuelsson, A. and Zienkiewicz, O.C. (2006), "History of the stiffness method", Int. J. Numer. Meth. Eng., 67, 149-157.

16.

Senjanovic, I., Vladimir, N. and Cho, D. S. (2012), "A simplified geometric stiffness in stability analysis of thin-walled structures by the finite element method", Int. J. Nav. Arch. Ocean., 4(3), 313-321.

17.

Bathe, K.J. (1996), Finite Element Procedures, Prentice Hall.

18.

Kassimali, A. (1998), Structural Analysis, 2nd Edition, Thomson-Engineering.

19.

Lui, E.M. and Chen, W.F. (1987), Structural Stability: Theory and Implementation, Elsevier.

20.

Norris, C.H. and Wilbur, J.B. (1960), Elementary Structural Analysis, McGraw-Hill Book Co.

21.

Salmon, C.G. and Johnson, J.E. (1996), Steel structures: design and behavior, 4th Edition, Harper Collins College Publisher, Chapter 14.

22.

Timoshenko, S.P. and Gere, J.M (1961), Theory of Elastic Stability, McGraw-Hill New York.

23.

Wilson, W.M. and Maney, G.A. (1915), Slope-deflection Method, University of Illinois Engineering Experiment Station, Bulletin.

24.

Zienkiewicz, O.C. (1971), The Finite Element Method In Engineering Science, London, McGraw-Hill.