Advanced Steel Construction

Vol. 11, No. 2, pp. 223-249 (2015)



C.K. Iu*,1 and M.A. Bradford2

1 School of Civil Engineering and Built Environment Queensland University of Technology QUT Brisbane, QLD, Australia

2 Centre for Infrastructure Engineering and Safety School of Civil and Environmental Engineering The University of New South Wales UNSW Sydney, NSW, Australia

*(Corresponding author: E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it. / This email address is being protected from spambots. You need JavaScript enabled to view it.)


Received: 4 May 2014; Revised: 16 June 2014; Accepted: 8 July 2014




In the finite element modelling of structural frames, external loads usually act along the elements rather than at the nodes only. Conventionally, when an element is subjected to these general transverse element loads, they are usually converted to nodal forces acting at the ends of the elements by either lumping or consistent load approaches. For a first- and second-order elastic analysis, the accurate displacement solutions of element load effect along an element can be simulated using neither lumping nor consistent load methods alone. It can be therefore regarded as a unique load method to account for the element load nonlinearly. In the second-order regime, the numerous prescribed stiffness matrices must indispensably be used for the plethora of specific transverse element loading patterns encountered. In order to circumvent this shortcoming, this paper shows that the principle of superposition can be applied to derive the generalized stiffness formulation for element load effect, so that the form of the stiffness matrix remains unchanged with respect to the specific loading patterns, but with only the magnitude of the loading (element load coefficients) being needed to be adjusted in the stiffness formulation, and subsequently the non-linear effect on element loadings can be commensurate by updating the magnitude of element load coefficients through the non-linear solution procedures. In principle, the element loading distribution is converted into a single loading magnitude at mid-span in order to provide the initial perturbation for triggering the member bowing effect due to its transverse element loads. This approach in turn sacrifices the effect of element loading distribution except at mid-span. Therefore, it can be foreseen that the load-deflection behaviour may not be as accurate as those at mid-span, but its discrepancy is still trivial as proved. This novelty allows for a very useful generalised stiffness formulation for a single higher-order element with arbitrary transverse loading patterns to be formulated. Moreover, another significance of this paper is placed on shifting the nodal solution (system analysis) to both nodal and element solution (sophisticated element formulation). For the conventional finite element method, such as cubic element, all accurate solutions can be only found at node. It means no accurate and reliable structural safety can be ensured within element, and as a result, it hinders the engineering applications.


Keywords: Elastic instability, Finite element, Transverse element load effect, Higher-order element formulation, Nodal solution, Element solution


Download PDF: