The beam-column method which corresponds to the analytical resolution of the differential equilibrium equation gives the well-known stability functions. However, by using this method, no information is obtained about the equivalent nodal loads, for an arbitrarily transverse load, which need the appropriate displacement's shape functions. Unfortunately, these shape functions are not yet derived from this way of resolution.

The second approach, namely the finite element method gives the stiffness matrix coefficients and the equivalent nodal loads. In this approach, Higher order terms in the strain-displacement relationship are included to account for the effect of the geometric non-linearity [1,2].

In the present paper, new displacement shape functions, expressed with respect of nodal generalised coordinates are derived analytically from the differential equilibrium equation. These shape functions are used in the context of the finite element method to formulate the stiffness matrix and the equivalent load vector.

This combined approach, allows to link up the accuracy of the exact stability methods to the simplicity of the finite element approach. These shape functions offer a significant practical advantages to overcome limitations arising from the cubic Hermite element and other formulations [3].

The accuracy of this formulation makes the convergence rate better than by the conventional cubic element or other used elements. Euler's critical load for bi-articulated beam is computed readily by a single element per member.

By using these exact shape functions in computer software, the second-order analysis and the non-linear integrated design become reliable and easy to use for practical purposes.

1

E.M. Lui, W.-F. Chen, "Analysis and Behaviour of flexibly-jointed frames, Engineering structures", 8, 107-118, 1986. doi:10.1016/0141-0296(86)90026-X

2

S.L. Chan, Z.H. Zou, W.F. Chen, J.L. Peng, A.D. Pan "Stability analysis of semi-rigid steel scaffolding", Engineering Structures, 17(8), 568-574, 1995. doi:10.1016/0141-0296(95)00011-U

3

R. Adman "Analyse des ossatures en phase élasto-plastique second ordre par rigidité sécantes", Master thesis, U.S.T.H.B., Algiers, Algeria, 1988.

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)