Structural behaviour and design of cold-formed steel beams at elevated temperatures

Cold-formed steel members are extensively used in the building construction industry, especially in residential, commercial and industrial buildings. In recent times, fire safety has become important in structural design due to increased fire damage to properties and loss of lives. However, past research into the fire performance of cold-formed steel members has been limited, and was confined to compression members. Therefore a research project was undertaken to investigate the structural behaviour of compact cold-formed steel lipped channel beams subject to inelastic local buckling and yielding, and lateral-torsional buckling effects under simulated fire conditions and associated section and member moment capacities. In the first phase of this research, an experimental study based on tensile coupon tests was undertaken to obtain the mechanical properties of elastic modulus and yield strength and the stress-strain relationship of cold-formed steels at uniform ambient and elevated temperatures up to 700oC. The mechanical properties deteriorated with increasing temperature and are likely to reduce the strength of cold-formed beams under fire conditions. Predictive equations were developed for yield strength and elastic modulus reduction factors while a modification was proposed for the stressstrain model at elevated temperatures. These results were used in the numerical modelling phases investigating the section and member moment capacities. The second phase of this research involved the development and validation of two finite element models to simulate the behaviour of compact cold-formed steel lipped channel beams subject to local buckling and yielding, and lateral-torsional buckling effects. Both models were first validated for elastic buckling. Lateral-torsional buckling tests of compact lipped channel beams were conducted at ambient temperature in order to validate the finite element model in predicting the non-linear ultimate strength behaviour. The results from this experimental study did not agree well with those from the developed experimental finite element model due to some unavoidable problems with testing. However, it highlighted the importance of magnitude and direction of initial geometric imperfection as well as the failure direction, and thus led to further enhancement of the finite element model. The finite element model for lateral-torsional buckling was then validated using the available experimental and numerical ultimate moment capacity results from past research. The third phase based on the validated finite element models included detailed parametric studies of section and member moment capacities of compact lipped channel beams at ambient temperature, and provided the basis for similar studies at elevated temperatures. The results showed the existence of inelastic reserve capacity for compact cold-formed steel beams at ambient temperature. However, full plastic capacity was not achieved by the mono-symmetric cold-formed steel beams. Suitable recommendations were made in relation to the accuracy and suitability of current design rules for section moment capacity. Comparison of member capacity results from finite element analyses with current design rules showed that they do not give accurate predictions of lateral-torsional buckling capacities at ambient temperature and hence new design rules were developed. The fourth phase of this research investigated the section and member moment capacities of compact lipped channel beams at uniform elevated temperatures based on detailed parametric studies using the validated finite element models. The results showed the existence of inelastic reserve capacity at elevated temperatures. Suitable recommendations were made in relation to the accuracy and suitability of current design rules for section moment capacity in fire design codes, ambient temperature design codes as well as those proposed by other researchers. The results showed that lateral-torsional buckling capacities are dependent on the ratio of yield strength and elasticity modulus reduction factors and the level of non-linearity in the stress-strain curves at elevated temperatures in addition to the temperature. Current design rules do not include the effects of non-linear stress-strain relationship and therefore their predictions were found to be inaccurate. Therefore a new design rule that uses a nonlinearity factor, which is defined as the ratio of the limit of proportionality to the yield stress at a given temperature, was developed for cold-formed steel beams subject to lateral-torsional buckling at elevated temperatures. This thesis presents the details and results of the experimental and numerical studies conducted in this research including a comparison of results with predictions using available design rules. It also presents the recommendations made regarding the accuracy of current design rules as well as the new developed design rules for coldformed steel beams both at ambient and elevated temperatures.

Creation of a validated 3D finite element model of an ovine tibia

Introduction Ovine models are widely used in orthopaedic research. To better understand the impact of orthopaedic procedures computer simulations are necessary. 3D finite element (FE) models of bones allow implant designs to be investigated mechanically, thereby reducing mechanical testing. Hypothesis We present the development and validation of an ovine tibia FE model for use in the analysis of tibia fracture fixation plates. Material & Methods Mechanical testing of the tibia consisted of an offset 3-pt bend test with three repetitions of loading to 350N and return to 50N. Tri-axial stacked strain gauges were applied to the anterior and posterior surfaces of the bone and two rigid bodies – consisting of eight infrared active markers, were attached to the ends of the tibia. Positional measurements were taken with a FARO arm 3D digitiser. The FE model was constructed with both geometry and material properties derived from CT images of the bone. The elasticity-density relationship used for material property determination was validated separately using mechanical testing. This model was then transformed to the same coordinate system as the in vitro mechanical test and loads applied. Results Comparison between the mechanical testing and the FE model showed good correlation in surface strains (difference: anterior 2.3%, posterior 3.2%). Discussion & Conclusion This method of model creation provides a simple method for generating subject specific FE models from CT scans. The use of the CT data set for both the geometry and the material properties ensures a more accurate representation of the specific bone. This is reflected in the similarity of the surface strain results.

The effect of friction on indenter force and pile-up in numerical simulations of bone nanoindentation

Nanoindentation is a useful technique for probing the mechanical properties of bone, and finite element (FE) modeling of the indentation allows inverse determination of elasto-plastic constitutive properties. However, FE simulations to date have assumed frictionless contact between indenter and bone. The aim of this study was to explore the effect of friction in simulations of bone nanoindentation. Two dimensional axisymmetric FE simulations were performed using a spheroconical indenter of tip radius 0.6m and angle 90°. The coefficient of friction between indenter and bone was varied between 0.0 (frictionless) and 0.3. Isotropic linear elasticity was used in all simulations, with bone elastic modulus E=13.56GPa and Poisson’s ratio =0.3. Plasticity was incorporated using both Drucker-Prager and von Mises yield surfaces. Friction had a modest effect on the predicted force-indentation curve for both von Mises and Drucker-Prager plasticity, reducing maximum indenter displacement by 10% and 20% respectively as friction coefficient was increased from zero to 0.3 (at a maximum indenter force of 5mN). However, friction has a much greater effect on predicted pile-up after indentation, reducing predicted pile-up from 0.27m to 0.11m with a von Mises model, and from 0.09m to 0.02m with Drucker-Prager plasticity. We conclude that it is important to include friction in nanoindentation simulations of bone.

Mechanical properties of thin sol-gel films

This thesis presents a study of the mechanical properties of thin films. The main aim was to determine the properties of sol-gel derived coatings. These films are used in a range of different applications and are known to be quite porous. Very little work has been carried out in this area and in order to study the mechanical properties of sol-gel films, some of the work was carried out on magnetron sputtered metal coatings in order to validate the techniques developed in this work. The main part of the work has concentrated on the development of various bending techniques to study the elastic modulus of the thin films, including both a small scale three-point bending, as well as a novel bi-axial bending technique based on a disk resting on three supporting balls. The bending techniques involve a load being applied to the sample being tested and the bending response to this force being recorded. These experiments were carried out using an ultra micro indentation system with very sensitive force and depth recording capabilities. By analysing the result of these forces and deflections using existing theories of elasticity, the elastic modulus may be determined. In addition to the bi-axial bending study, a finite element analysis of the stress distribution in a disk during bending was carried out. The results from the bi-axial bending tests of the magnetron sputtered films was confirmed by ultra micro indentation tests, giving information of the hardness and elastic modulus of the films. It was found that while the three point bending method gave acceptable results for uncoated steel substrates, it was very susceptible to slight deformations of the substrate. Improvements were made by more careful preparation of the substrates in order to avoid deformation. However the technique still failed to give reasonable results for coated specimens. In contrast, biaxial bending gave very reliable results even for very thin films and this technique was also found to be useful for determination of the properties of sol-gel coatings. In addition, an ultra micro indentation study of the hardness and elastic modulus of sol-gel films was conducted. This study included conventionally fired films as well as films ion implanted in a range of doses. The indentation tests showed that for implantation of H⁺ ions at doses exceeding 3x10¹⁶ ions/cm², the mechanical properties closely resembled those of films that were conventionally fired to 450°C.

Mind change complexity of learning logic programs

The present paper motivates the study of mind change complexity for learning minimal models of length-bounded logic programs. It establishes ordinal mind change complexity bounds for learnability of these classes both from positive facts and from positive and negative facts. Building on Angluin’s notion of finite thickness and Wright’s work on finite elasticity, Shinohara defined the property of bounded finite thickness to give a sufficient condition for learnability of indexed families of computable languages from positive data. This paper shows that an effective version of Shinohara’s notion of bounded finite thickness gives sufficient conditions for learnability with ordinal mind change bound, both in the context of learnability from positive data and for learnability from complete (both positive and negative) data. Let Omega be a notation for the first limit ordinal. Then, it is shown that if a language defining framework yields a uniformly decidable family of languages and has effective bounded finite thickness, then for each natural number m >0, the class of languages defined by formal systems of length <= m: • is identifiable in the limit from positive data with a mind change bound of Omega (power)m; • is identifiable in the limit from both positive and negative data with an ordinal mind change bound of Omega × m. The above sufficient conditions are employed to give an ordinal mind change bound for learnability of minimal models of various classes of length-bounded Prolog programs, including Shapiro’s linear programs, Arimura and Shinohara’s depth-bounded linearly covering programs, and Krishna Rao’s depth-bounded linearly moded programs. It is also noted that the bound for learning from positive data is tight for the example classes considered.

An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields

In this paper, an enriched radial point interpolation method (e-RPIM) is developed the for the determination of crack tip fields. In e-RPIM, the conventional RBF interpolation is novelly augmented by the suitable trigonometric basis functions to reflect the properties of stresses for the crack tip fields. The performance of the enriched RBF meshfree shape functions is firstly investigated to fit different surfaces. The surface fitting results have proven that, comparing with the conventional RBF shape function, the enriched RBF shape function has: (1) a similar accuracy to fit a polynomial surface; (2) a much better accuracy to fit a trigonometric surface; and (3) a similar interpolation stability without increase of the condition number of the RBF interpolation matrix. Therefore, it has proven that the enriched RBF shape function will not only possess all advantages of the conventional RBF shape function, but also can accurately reflect the properties of stresses for the crack tip fields. The system of equations for the crack analysis is then derived based on the enriched RBF meshfree shape function and the meshfree weak-form. Several problems of linear fracture mechanics are simulated using this newlydeveloped e-RPIM method. It has demonstrated that the present e-RPIM is very accurate and stable, and it has a good potential to develop a practical simulation tool for fracture mechanics problems.

Lie group symmetry analysis for granular media stress equations

The Airy stress function, although frequently employed in classical linear elasticity, does not receive similar usage for granular media problems. For plane strain quasi-static deformations of a cohesionless Coulomb–Mohr granular solid, a single nonlinear partial differential equation is formulated for the Airy stress function by combining the equilibrium equations with the yield condition. This has certain advantages from the usual approach, in which two stress invariants and a stress angle are introduced, and a system of two partial differential equations is needed to describe the flow. In the present study, the symmetry analysis of differential equations is utilised for our single partial differential equation, and by computing an optimal system of one-dimensional Lie algebras, a complete set of group-invariant solutions is derived. By this it is meant that any group-invariant solution of the governing partial differential equation (provided it can be derived via the classical symmetries method) may be obtained as a member of this set by a suitable group transformation. For general values of the parameters (angle of internal friction and gravity g) it is found there are three distinct classes of solutions which correspond to granular flows considered previously in the literature. For the two limiting cases of high angle of internal friction and zero gravity, the governing partial differential equation admit larger families of Lie point symmetries, and from these symmetries, further solutions are derived, many of which are new. Furthermore, the majority of these solutions are exact, which is rare for granular flow, especially in the case of gravity driven flows.

Estimating the latent demand for rail transit : a case study in Perth, Western Australia

This paper presents a study on estimating the latent demand for rail transit in Australian context. Based on travel mode-choice modelling, a two-stage analysis approach is proposed, namely market population identification and mode share estimation. A case study is conducted on Midland-Fremantle rail transit corridor in Perth, Western Australia. The required data mainly include journey-to-work trip data from Australian Bureau of Statistics Census 2006 and work-purpose mode-choice model in Perth Strategic Transport Evaluation Model. The market profile is analysed, such as catchment areas, market population, mode shares, mode specific trip distributions and average trip distances. A numerical simulation is performed to test the sensitivity of the transit ridership to the change of fuel price. A corridor-level transit demand function of fuel price is thus obtained and its characteristics of elasticity are discussed. This study explores a viable approach to developing a decision-support tool for the assessment of short-term impacts of policy and operational adjustments on corridor-level demand for rail transit.

The effect of friction on indenter force and pile-up in numerical simulations of bone nanoindentation

Nanoindentation is a useful technique for probing the mechanical properties of bone, and finite element (FE) modeling of the indentation allows inverse determination of elasto-plastic constitutive properties. However, all but one FE study to date have assumed frictionless contact between indenter and bone. The aim of this study was to explore the effect of friction in simulations of bone nanoindentation. Two dimensional axisymmetric FE simulations were performed using a spheroconical indenter of tip radius 0.6 m and angle 90°. The coefficient of friction between indenter and bone was varied between 0.0 (frictionless) and 0.3. Isotropic linear elasticity was used in all simulations, with bone elastic modulus E=13.56GPa and Poisson‟s ratio f 0.3. Plasticity was incorporated using both Drucker-Prager and von Mises yield surfaces. Friction had a modest effect on the predicted force-indentation curve for both von Mises and Drucker-Prager plasticity, reducing maximum indenter displacement by 10% and 20% respectively as friction coefficient was increased from zero to 0.3 (at a maximum indenter force of 5mN). However, friction has a much greater effect on predicted pile-up after indentation, reducing predicted pile-up from 0.27 to 0.11 m with a von Mises model, and from 0.09 to 0.02 m with Drucker-Prager plasticity. We conclude that it is potentially important to include friction in nanoindentation simulations of bone if pile-up is used to compare simulation results with experiment.

Mechanical properties of cold-formed steels at elevated temperatures

Mechanical properties have an important role in the fire safety design of cold-formed steel structures due to the rapid reduction in mechanical properties such as yield strength and elastic modulus under fire conditions and associated reduction to the load carrying capacities. Hence there is a need to fully understand the deterioration characteristics of yield strength and elastic modulus of cold-formed steels at elevated temperatures. Although past research has produced useful experimental data on the mechanical properties of cold-formed steels at elevated temperatures, such data do not yet cover different cold-formed steel grades and thicknesses. Therefore, an experimental study was undertaken to investigate the elevated temperature mechanical properties of two low and high strength steels with two thicknesses that are commonly used in Australia. Tensile coupon tests were undertaken using a steady state test method for temperatures in the range 20–700 °C. Test results were compared with the currently available reduction factors for yield strength and elastic modulus, and stress–strain curves, based on which further improvements were made. For this purpose, test results of many other cold-formed steels were also used based on other similar studies undertaken at the Queensland University of Technology. Improved equations were developed to predict the yield strength and elastic modulus reduction factors and stress–strain curves of a range of cold-formed steel grades and thicknesses used in Australia. This paper presents the results of this experimental study, comparisons with the results of past research and steel design standards, and the new predictive equations.

Theoretical and numerical investigation of bending properties of Cu nanowires

Based on the AFM-bending experiments, a molecular dynamics (MD) bending simulation model is established which could accurately account for the full spectrum of the mechanical properties of NWs in a double clamped beam configuration, ranging from elasticity to plasticity and failure. It is found that, loading rate exerts significant influence to the mechanical behaviours of nanowires (NWs). Specifically, a loading rate lower than 10 m/s is found reasonable for a homogonous bending deformation. Both loading rate and potential between the tip and the NW are found to play an important role in the adhesive phenomenon. The force versus displacement (F-d) curve from MD simulation is highly consistent in shapes with that from experiments. Symmetrical F-d curves during loading and unloading processes are observed, which reveal the linear-elastic and non-elastic bending deformation of NWs. The typical bending induced tensile-compressive features are observed. Meanwhile, the simulation results are excellently fitted by the classical Euler-Bernoulli beam theory with axial effect. It is concluded that, axial tensile force becomes crucial in bending deformation when the beam size is down to nanoscale for double clamped NWs. In addition, we find shorter NWs will have an earlier yielding and a larger yielding force. Mechanical properties (Young’s modulus & yield strength) obtained from both bending and tensile deformations are found comparable with each other. Specifically, the modulus is essentially similar under these two loading methods, while the yield strength during bending is observed larger than that during tension.

Application of high voltage, high frequency pulsed electromagnetic field on cortical bone tissue

Over the last few decades, electric and electromagnetic fields have achieved important role as stimulator and therapeutic facility in biology and medicine. In particular, low magnitude, low frequency, pulsed electromagnetic field has shown significant positive effect on bone fracture healing and some bone diseases treatment. Nevertheless, to date, little attention has been paid to investigate the possible effect of high frequency, high magnitude pulsed electromagnetic field (pulse power) on functional behaviour and biomechanical properties of bone tissue. Bone is a dynamic, complex organ, which is made of bone materials (consisting of organic components, inorganic mineral and water) known as extracellular matrix, and bone cells (live part). The cells give the bone the capability of self-repairing by adapting itself to its mechanical environment. The specific bone material composite comprising of collagen matrix reinforced with mineral apatite provides the bone with particular biomechanical properties in an anisotropic, inhomogeneous structure. This project hypothesized to investigate the possible effect of pulse power signals on cortical bone characteristics through evaluating the fundamental mechanical properties of bone material. A positive buck-boost converter was applied to generate adjustable high voltage, high frequency pulses up to 500 V and 10 kHz. Bone shows distinctive characteristics in different loading mode. Thus, functional behaviour of bone in response to pulse power excitation were elucidated by using three different conventional mechanical tests applying three-point bending load in elastic region, tensile and compressive loading until failure. Flexural stiffness, tensile and compressive strength, hysteresis and total fracture energy were determined as measure of main bone characteristics. To assess bone structure variation due to pulse power excitation in deeper aspect, a supplementary fractographic study was also conducted using scanning electron micrograph from tensile fracture surfaces. Furthermore, a non-destructive ultrasonic technique was applied for determination and comparison of bone elasticity before and after pulse power stimulation. This method provided the ability to evaluate the stiffness of millimetre-sized bone samples in three orthogonal directions. According to the results of non-destructive bending test, the flexural elasticity of cortical bone samples appeared to remain unchanged due to pulse power excitation. Similar results were observed in the bone stiffness for all three orthogonal directions obtained from ultrasonic technique and in the bone stiffness from the compression test. From tensile tests, no significant changes were found in tensile strength and total strain energy absorption of the bone samples exposed to pulse power compared with those of the control samples. Also, the apparent microstructure of the fracture surfaces of PP-exposed samples (including porosity and microcracks diffusion) showed no significant variation due to pulse power stimulation. Nevertheless, the compressive strength and toughness of millimetre-sized samples appeared to increase when the samples were exposed to 66 hours high power pulsed electromagnetic field through screws with small contact cross-section (increasing the pulsed electric field intensity) compare to the control samples. This can show the different load-bearing characteristics of cortical bone tissue in response to pulse power excitation and effectiveness of this type of stimulation on smaller-sized samples. These overall results may address that although, the pulse power stimulation can influence the arrangement or the quality of the collagen network causing the bone strength and toughness augmentation, it apparently did not affect the mineral phase of the cortical bone material. The results also confirmed that the indirect application of high power pulsed electromagnetic field at 500 V and 10 kHz through capacitive coupling method, was athermal and did not damage the bone tissue construction.