SHEAR BEHAVIOUR AND DESIGN OF COLD-FORMED CHANNEL SECTIONS WITH ELONGATED OPENINGS BASED ON DIRECT STRENGTH METHOD

This thesis presents a detailed study on shear buckling and shear strength of high strength cold-formed channel sections with elongated openings. The main purposes are to investigate experimentally and numerically the shear behaviour of perforated cold-formed steel plain C-lipped sections and to further develop a Direct Strength Method (DSM) of design to predict the shear capacity of such sections. Firstly, in order to achieve the primary aims as stated, an experimental program of thirty tests was performed to observe the shear behaviour of channel members with both non-elongated and elongated web holes. Based on the capability of minimising the bending moments at two ends of the shear span, a testing apparatus namely ‘Dual Actuator Test Rig’ previously developed at the University of Sydney was used throughout the test program to capture a state close to pure shear and to obtain the predominantly shear capacity of perforated members with an aspect ratio (shear span / web depth) up to 2.0. The experimental results were used to study the shear strength reduction due to enlarged web openings. Further, these test results were also used as the input to the current Direct Strength Method (DSM) equations for further comparisons, calibrations and validations. Secondly, in order to achieve more insights into the shear behaviour of cold-formed members with elongated web openings, numerical nonlinear simulations based on the Finite Element Method (FEM) using ABAQUS/Standard were developed to compare with and calibrate against the experimental results. Moreover, to study shear buckling behaviour and to generate shear buckling loads which is also a required input to the DSM in shear, a simplified method for shear buckling analysis using simplified finite element (FE) models including web holes were introduced in this study. The buckling results obtained from these models were calibrated against those of the full FE models which have the same configuration as the actual tests. On the basis of the accuracy of the finite element modelling, the FE models for both shear buckling and shear strength analyses were employed for parametric studies to extend the result database used for further verification of new proposals in this study. Finally, on the basis of the results from experimental and numerical investigations, a new DSM design for cold-formed channel sections subjected to shear with both non-elongated and elongated web openings was introduced. The new proposal is based on the use of the existing DSM design rules for shear together with introduced modifications of the shear yield loads as a result of the Vierendeel mechanism approach. As a consequence of the parametric study for shear buckling analyses, a dimensional transformation was also proposed to determine the equivalent hole dimensions in design

Inexact proximal methods for weakly convex functions

This paper proposes and develops inexact proximal methods for finding stationary points of the sum of a smooth function and a nonsmooth weakly convex one, where an error is present in the calculation of the proximal mapping of the nonsmooth term. A general framework for finding zeros of a continuous mapping is derived from our previous paper on this subject to establish convergence properties of the inexact proximal point method when the smooth term is vanished and of the inexact proximal gradient method when the smooth term satisfies a descent condition. The inexact proximal point method achieves global convergence with constructive convergence rates when the Moreau envelope of the objective function satisfies the Kurdyka-Lojasiewicz (KL) property. Meanwhile, when the smooth term is twice continuously differentiable with a Lipschitz continuous gradient and a differentiable approximation of the objective function satisfies the KL property, the inexact proximal gradient method achieves the global convergence of iterates with constructive convergence rates.Comment: 26 pages, 3 table

A Generalized Newton Method for Subgradient Systems

This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions defined by subgradients of extended-real-valued prox-regular functions. The proposed algorithm is formulated in terms of the second-order subdifferential of such functions that enjoys extensive calculus rules and can be efficiently computed for broad classes of extended-real-valued functions. Based on this and on metric regularity and subregularity properties of subgradient mappings, we establish verifiable conditions ensuring well-posedness of the proposed algorithm and its local superlinear convergence. The obtained results are also new for the class of equations defined by continuously differentiable functions with Lipschitzian derivatives ($\mathcal{C}^{1,1}$ functions), which is the underlying case of our consideration. The developed algorithm for prox-regular functions is formulated in terms of proximal mappings related to and reduces to Moreau envelopes. Besides numerous illustrative examples and comparison with known algorithms for $\mathcal{C}^{1,1}$ functions and generalized equations, the paper presents applications of the proposed algorithm to the practically important class of Lasso problems arising in statistics and machine learning.Comment: 35 page

A New Inexact Gradient Descent Method with Applications to Nonsmooth Convex Optimization

The paper proposes and develops a novel inexact gradient method (IGD) for minimizing C1-smooth functions with Lipschitzian gradients, i.e., for problems of C1,1 optimization. We show that the sequence of gradients generated by IGD converges to zero. The convergence of iterates to stationary points is guaranteed under the Kurdyka- Lojasiewicz (KL) property of the objective function with convergence rates depending on the KL exponent. The newly developed IGD is applied to designing two novel gradient-based methods of nonsmooth convex optimization such as the inexact proximal point methods (GIPPM) and the inexact augmented Lagrangian method (GIALM) for convex programs with linear equality constraints. These two methods inherit global convergence properties from IGD and are confirmed by numerical experiments to have practical advantages over some well-known algorithms of nonsmooth convex optimization.Comment: 23 pages, 8 figure

Coderivative-Based Newton Methods in Structured Nonconvex and Nonsmooth Optimization

This paper proposes and develops new Newton-type methods to solve structured nonconvex and nonsmooth optimization problems with justifying their fast local and global convergence by means of advanced tools of variational analysis and generalized differentiation. The objective functions belong to a broad class of prox-regular functions with specification to constrained optimization of nonconvex structured sums. We also develop a novel line search method, which is an extension of the proximal gradient algorithm while allowing us to globalize the proposed coderivative-based Newton methods by incorporating the machinery of forward-backward envelopes. Further applications and numerical experiments are conducted for the $\ell_0$-$\ell_2$ regularized least-square model appearing in statistics and machine learning