2016 Research Reports No R961-R964

Abstract
The formulation and implementation of beam-column finite elements for the general thin-walled open cross-sections within the OpenSees framework is presented. To account for the non-coincident location of shear centre and centroid as is the case in non-symmetric sections, a local cross-section transformation matrix is derived relating the axial force acting through the centroid and the shear centre. The stiffness relations are derived based on the Green-Lagrange strain for the displacement based beam-column element while the Wagner effect is incorporated in the torsional rigidity term for the elastic beam-column element. The beam-column elements are then implemented within the corotational framework of OpenSees. The performance of developed beam-column elements are demonstrated through the use of monosymmetric, doubly-symmetric and asymmetric sections in a series of numerical examples. The solutions obtained are verified with the results from the beam and shell element models in ABAQUS software.

Keywords
Centroid, shear centre, warping, monosymmetric, asymmetric, thin-walled open sections, postbuckling, co-rotation, nonlinear analysis

Abstract
A comprehensive experimental study on long-span cold-formed steel portal frames composed of back-to-back lipped channel sections is presented. The aim of the study is to determine appropriate design recommendations in order for engineers to safely and efficiently build larger spans. The specific frame system analyzed herein is a haunched frame with a knee brace connected between the column and rafter.

The objectives of the research were achieved through an extensive experimental study as well as numerical investigations. A comprehensive experimental program was completed to determine the strength and behavior of the frames. A total of nine full scale portal frame systems were tested, eight of which had unbraced columns, which was the main focus of this work, and one with braced columns. Variations to the frame layout including changes to the knee connection and the addition of sleeve stiffeners were tested, for both vertical loading and combined wind and vertical loading conditions. Column base rotational stiffness was quantified in the full scale experiments and in separate component tests, as well as the effects of different column base connection plates on the column base stiffness. Material and section properties were measured. All experimental work occurred in the Centre for Advanced Structural Engineering at the University of Sydney.

Keywords
Cold-Formed Steel, Portal Frames, Lateral-Torsional Buckling, Thin-Walled Sections, Column Base Stiffness

Abstract
The constrained finite strip method (cFSM) is an extension of the semi-analytical finite strip method (SAFSM) of structural analysis of thin-walled members, where consideration of the displacement fields utilised and of various mechanical criteria allows constraint matrices to be formed. The application of these constraint matrices to the linear buckling eigenvalue problem of the SAFSM results in deformation fields that satisfy the considered criteria and, therefore, isolate particular modes. Through careful selection of the mechanical criteria, the deformation fields obtained may be restricted to particular buckling modes. This is referred to as modal decomposition. While the cFSM has been applied to modal decomposition of thin-walled, prismatic members under the action of longitudinal normal stresses, it has yet to be applied to such members under the action of shear stresses. Recent work using the SAFSM to analyse the buckling behaviour of thin-walled, prismatic members under applied shear stresses, notably by Hancock and Pham, has shown that the issues of potentially indistinct minima or multiple minima in the signature curve can occur under this loading, as they did for compression and bending. This paper briefly presents the derivation of a SAFSM that permits coupling between longitudinal series terms of sines and cosines and also considers membrane instability due both to shear stresses and transverse normal stresses. It then presents the application of the cFSM to such a finite strip and results are produced for members under shear stresses. While the results are presented for members with unrestrained ends (equivalent to infinitely long members with simply-supported ends), simplification via removal of the degrees of freedom not present in typical FSM formulations would allow finite length members with simply-supported ends to be analysed.

Keywords
Constrained finite strip method, semi-analytical finite strip method, shear buckling analysis, membrane instability.

Abstract
A torsion differential equation previously used for analysing the elastic lateral buckling of simply supported doubly symmetric beams with distributed loads acting away from the centroidal axis omits an expected term and includes an unexpected term. A different equation is derived by two different methods, either by using the calculus of variations with the second variation of the total
potential, or by considering the equilibrium of the deflected and twisted beam.

Four different methods are used to find solutions for the elastic buckling of beams with uniformly distributed loads. Two of these solve the differential equations numerically, either by using a computer program based on the method of finite integrals, or by making hand calculations with a single term approximation of the buckled shape. These methods produce different solutions for the two torsion differential equations.

The two other methods used are based on the energy equation for lateral buckling. The first of these uses hand calculations and a limited series for the buckled shape, while the second uses a finite element computer program based on cubic deformation fields. Both of these produce solutions which agree closely with the finite integral and approximate solutions for the different differential equation derived in this paper, but are markedly different from the solutions for the previously used equation.

It is concluded that the previously used torsion differential equation is in error.

Keywords
Beam, buckling, differential equation, distributed load, elasticity, torsion