Many researchers (see e.g. Bolotin [1], Kounadis and Katsikadelis [2], Venkateswara and Kanaka [3], Lee, et al. [4], Lee and Yang [5]) have analyzed the non-conservative instability of beams resting on an elastic foundation. Takahashi [6,7] studied the vibration and stability of a cracked Timoshenko beam subjected to a follower force. Takahashi [8,9] proposed the identification method for the axial force (or critical force) and boundary conditions of a beam using the neural networks.

The problem of experimental design or the design of experiments (DOE) is encountered in many fields. A common situation for using the DOE is when the designer does not know the exact underlying relationship between responses and design variables. The empirical model is called a response surface model or curve fit. The basic idea of the response surface methodology is to create explicit approximation functions to the objective and constraints, and then use these when performing the optimization. The approximation functions are typically in the form of low-order polynomials fit by least squares regression analysis. In order to construct the approximation function , it is necessary to have some results for a minimum number of points in the design space. The proper selection of points could drastically improve the quality of a response surface model. The response at the most suitable points, which are selected by the design of experiments (DOE) could have been obtained either by some analysis program or through physical experiments.

In this paper the possibility of using a response surface methodology, which consists of a design of experiments, for estimating the critical flutter load of the beam is studied. An analysis is presented for the vibration and stability of a tapered beam simultaneously subjected to a follower force with an axial force by the use of the transfer matrix approach. Once the matrix has been determined by the numerical integration of equations, the eigenvalues of vibration and the critical flutter load are obtained. The method is applied to beams with linearly varying cross-sections, and the natural frequencies and flutter loads are calculated numerically, to provide information about the effect on them of varying cross-section, the slenderness, the axial force and the span and stiffness of intermediate supports.

Some numerical examples were presented to demonstrate the possibility of the response surface approximation. From the results of the numerical examples we can draw the following conclusions. First, the critical flutter load can be predicted by using the response surface approximation with three-level orthogonal Latin squares. Second, the generalization capability of the response surface with three-level orthogonal Latin squares L₂₇(3¹³) is sufficient for estimating the critical flutter loads.

1

Bolotin,V.V., "Nonconservative Problems of Theory of Elastic Stability", Pergamon Press, Oxford, 1963.

2

Kounadis, A.N. and Katsikadelis, J.K., "Shear and rotatory inertia effects on Beck's column", J. Sound Vibr. 49, pp.171-178, 1976. doi:10.1016/0022-460X(76)90494-6

3

Venkateswara Rao, G. and Kanaka Raju, K., "Stability of tapered cantilever columns with an elastic foundation subjected to a concentrated follower force at the free end", J. Sound Vibr. 81, pp.147-151, 1982. doi:10.1016/0022-460X(82)90186-9

4

Lee, S.Y., Kuo, Y.H. and Lin, F.Y., "Stability of a Timoshenko beam resting on a Winkler elastic foundation", J. Sound Vibr. 153, pp.193-202, 1992. doi:10.1016/S0022-460X(05)80001-X

5

Lee, S.Y. and Yang, C.C., "Non-conservative instability of non-uniform beams resting on an elastic foundation", J. Sound Vibr. 169, pp.433-444, 1994. doi:10.1006/jsvi.1994.1027

6

Takahashi, I., "Vibration and stability of a non-uniform cracked Timoshenko beam subjected to follower force", Computers Struct. 71, pp.585-591, 1999. doi:10.1016/S0045-7949(98)00233-8

7

Takahashi, I., "Vibration and stability of a cracked shaft simultaneously subjected to follower force with an axial forces", Int. J. Solid Struct. 35, pp.3071-3080, 1998. doi:10.1016/S0020-7683(97)00364-8

8

Takahashi, I., "Identification for axial force and boundary conditions of a beam using neural networks", Proceedings of the 1997 International Conference on Engineering Applications of Neural Networks, Neural Networks in Engineering Systems, pp.253-256, 1997.

9

Takahashi, I., "Identification for critical force and boundary conditions of a beam using neural networks", J. Sound Vibr, 228, pp.857-870, 1999. doi:10.1006/jsvi.1999.2451

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £135 +P&P)