2013 Research Reports No R931-R942

Abstract
Buckling of thin-walled sections in pure shear has been recently investigated using the Semi-Analytical Finite Strip Method (SAFSM) to develop the “signature curve” for sections in shear. The method assumes that the buckle is part of an infinitely long section unrestrained against distortion at its ends. For sections restrained at finite lengths by transverse stiffeners or other similar constraints, the Spline Finite Strip Method (SFSM) has been used to determine the elastic buckling loads in pure shear. These loads are higher than those from the SAFSM due to the constraints.

The SFSM requires considerable computation to achieve the buckling loads due to the large numbers of degrees of freedom of the system. In the 1980’s, Anderson and Williams developed a shear buckling analysis for sections in shear where the ends are simply supported based on the exact finite strip method. The current report further develops the SAFSM buckling theory of YK Cheung for sections in pure shear accounting for simply supported ends using the methodology of Anderson and Williams. The theory is applied to the buckling of plates of increasing length and channel sections in pure shear also for increasing length. The method requires increasing numbers of series terms as the sections become longer. Convergence studies with strip subdivision and number of series terms is provided in the report.

Keywords
Cold-formed channel sections; Simply supported ends; Shear buckling analysis; Finite strip method; Semi-analytical finite strip method; Complex mathematics.

Abstract
The finite strip method is computationally efficient for the static, stability, post-buckling and vibration analyses of thin-walled structures. The finite strip employs simple polynomial functions to describe the transverse variations of the displacements and continuous harmonic series functions or discontinuous spline functions to describe the longitudinal variation of the strip displacements. While the Semi-Analytical Finite Strip Method (SAFSM) generally uses the longitudinal harmonic series to satisfy the boundary conditions at the longitudinal ends and to give compatibility between strips, the Spline Finite Strip Method (SFSM) employs local spline functions in the longitudinal direction to account for different boundary conditions.

The Semi-Analytical Finite Strip Method (SAFSM) has been widely used in computer software (THIN-WALL, CUFSM) to develop the signature curves of the buckling stress versus buckling half-wavelength for a thinwalled section under compression or bending to allow identification of buckling modes. Recently, a complex mathematical technique has been applied in the SAFSM theory to allow for the case of shear. The shear buckling modes produced include local, distortional and overall with phase shifts along the member. This report provides the analysis and comparison between the new SAFSM development for shear and the SFSM for whole plain channel sections including flanges and lips where the sections are loaded in pure shear parallel to the web. The main variables are the flange widths and lip sizes. For the longitudinal direction, the SAFSM determines the shear signature curves versus buckling half-wavelength and the SFSM determines the elastic shear buckling stresses versus the member lengths of the whole channel section. The SAFSM is limited to a single half-wavelength whereas the SFSM can include multiple buckles as seen in the wellknown garland curve. The report demonstrates the potential for coupling between multiple short halfwavelength modes in shear and longer single half-wavelength as may occur in distortional buckling.

Keywords
Cold-formed channel sections; Semi-Analytical Finite Strip; Spline Finite Strip; Shear buckling analysis; Signature curve; Coupling buckling modes.

Abstract
The buckling behaviour of T-section columns is discussed in detail followed by a numerical study using geometric and material (GMNIA) analysis to produce column strengths for a wide range of geometries of Tsections and column lengths. The T-sections are assumed to be hot-rolled and include residual stresses and geometric imperfections typical of hot-rolled sections. Based on the numerical strengths thus produced and available test strengths for T-section columns, the design provisions of the Australian, European and American Specifications for hot-rolled steel structures are evaluated. It is shown that while the Australian Standard provides fairly consistent and accurate predictions of strength, the design provisions for T sections of current European and American Specifications are conservative and associated with large variability, particularly for T-sections with slender elements. The paper recommends modifications to the European and American specifications which improve the design strength predictions of these specifications for T-section columns.

Keywords
Columns; Buckling; Design; Finite Element Method; T-sections; Steel structures; Numerical models.

Abstract
Steel structural members are not perfectly straight due to manufacturing and erection tolerances and they contain internal residual stresses to various degrees because of non-uniform cooling of rolled sections or welding of fabricated sections. The ultimate strength of a steel structure is sensitive to geometrical imperfections (initial out-of- straightness and initial out-of-plumb) and material imperfections (residual stress). Consequently, these imperfections need to be modelled accurately when determining the load carrying capacity of a steel frame by advanced structural analysis.

In general, the shape and magnitude of geometrical imperfections may have a significant influence on the response of a structure. Most conveniently, geometric imperfections can be introduced in structural models as scaled eigenmodes obtained a priori from an elastic buckling analysis. However, it remains unanswered how many eigenmodes need to be incorporated and how to choose the scaling factors of each mode.

This report presents a study of how the strength of steel frames varies with the number and magnitudes of eigenmodes. Frames with random geometric imperfections are produced using the statistics of measurements of out-of-plumb and member imperfection, and analysed using advanced geometric and material nonlinear analysis. The imperfections are then resolved into eigenmodes and a second set of advanced analysis is carried out using a finite number of modes to represent the geometric imperfections. Conclusions are drawn about the appropriate number and magnitudes of eigenmodes to use in advanced structural analyses of steel frames.

Additionally, a new perspective is proposed for the modeling of residual stress as a random variable in advanced analysis of steel structures. Randomness in residual stress has a significant impact on the strength and reliability of steel frames by increasing lateral deflections and thereby second-order effects. Thus, when investigating the behaviour of steel structures numerically, it is important to have proper statistical characteristics of such stresses. The paper presents an extensive survey of literature on a large number of experimental measurements of residual stress in hot-rolled I-sections. An error minimization is then performed to find an appropriate scale factor to apply to common residual stress patterns to obtain the best agreement with available experimental data. Frames with deterministic and random residual stresses are then analysed by advanced geometric and material nonlinear analysis using proposed scale factors to find the influence of different residual stress patterns on the ultimate strength of steel frames.

Keywords
Initial geometric imperfection; Residual stress; Advanced analysis; Steel structures; Inelastic analysis; Structural engineering; Buckling analysis.

Abstract
A formwork for determining a system resistance factor in LRFD is presented. A truss composed of HSS members based on current design specifications is chosen for analysis.

To reflect real structures, the truss system is analyzed by advanced second order analysis, which takes into account the variation in physical properties and directly models geometric and material nonlinear behavior. Measured data on the variation of physical properties reported in literature is located including member imperfections, residual stresses, member thickness, Young’s modulus and yield stress. Distributions of these values are obtained from data for recreation in the finite element models. Member imperfection profiles are generated and residual stress patterns through the thickness and around the cross-section are formulated. The variations of physical properties are represented in the finite element simulations using Latin Hypercube sampling of the random variables. Random loads are also modeled.

A connection modeling technique is devised, and finite element models of the structure are created and compared to benchmark tests to assess their validity. A 2D model of a single truss and a 3D model of a system of trusses are created. Simulations are completed to obtain strength distributions of each system, followed by a reliability analysis to determine resistance factors for each system. The results of the system analysis and resulting reliability index and system resistance factors are compared to that of component based design. It is found that the system resistance factor for the specific system analyzed herein is lower than that of the resistance factor for individual components.

Keywords
Hollow structural sections, Cold-formed steel, Residual stresses, Geometric imperfections, System reliability, Finite element modeling

Abstract
This paper examines the effects of torsional post-buckling on the strengths of tee section steel columns which fail in local, torsional or flexural modes. Flexural-torsional buckling is not considered, but the post-buckling movement of the effective centroid and the ensuing bending effects are.

It is shown that the elastic torsional buckling loads of tee section columns are greater than or equal to their local buckling loads, while their torsional and local post-buckling behaviours are very similar. Design codes which require separate account to be taken of torsional and local buckling duplicate their allowances for these effects and ignore the torsional post-buckling strength.

It is found that the effective section method used in most codes to allow for local buckling and post-buckling is probably conservative, and that allowances for the movement of the centroid are over-conservative.

It is suggested that close design approximations for the strengths of tee section columns can be obtained by ignoring the effects of movement of the centroid and torsional buckling, and instead using the lower of the axial compression section resistance determined for the effective section and the member resistance calculated by ignoring local buckling.

Keywords
Buckling, Columns, Design, Flexure, Post-buckling, Steel, Tee sections, Torsion, Yield.

Abstract
This paper describes an efficient finite element method of analysing the elastic in-plane bending and out-ofplane buckling of indeterminate beam structures whose members may be tapered and of mono-symmetric I cross-section. The structure’s loading includes concentrated moments and concentrated or uniformly distributed off-axis transverse and longitudinal forces, and its deformations may be prevented or resisted by concentrated or continuous rigid or elastic off-axis restraints.

Tapered finite element formulations are developed by numerical integration instead of the closed forms often used for uniform elements. Difficulties in specifying the load positions for tapered mono-symmetric members caused by the variations of the centroid and shear centre axes are avoided by using an arbitrary axis system based on the web mid-line. Account is taken of additional Wagner torque terms arising from the inclination of the shear centre axis.

A computer program based on this method is used to analyse a number of examples of the elastic in-plane bending of tapered cantilevers and built-in beams, and very close agreement is found between its predictions and closed form solutions.

The program’s predictions of the elastic out-of-plane flexural-torsional buckling of a large number of uniform and tapered doubly and mono-symmetric beams and cantilevers under various loading and restraint conditions are generally in close agreement with existing predictions and test results. The common approximation in which tapered elements are replaced by uniform elements is shown to converge slowly, and to lead to incorrect predictions for tapered mono-symmetric beams.

Keywords
Beam-columns, Bending, Buckling, Mono-symmetry, Steel, Structures, Taper, Torsion