Six-dimensional Origin of N=4\mathcal{N}=4 SYM with Duality Defects

We study the topologically twisted compactification of the 6d $(2,0)$ M5-brane theory on an elliptically fibered K\"ahler three-fold preserving two supercharges. We show that upon reducing on the elliptic fiber, the 4d theory is $\mathcal{N}=4$ Super-Yang Mills, with varying complexified coupling $\tau$, in the presence of defects. For abelian gauge group this agrees with the so-called duality twisted theory, and we determine a non-abelian generalization to $U(N)$. When the elliptic fibration is singular, the 4d theory contains 3d walls (along the branch-cuts of $\tau$) and 2d surface defects, around which the 4d theory undergoes $SL(2,\mathbb{Z})$ duality transformations. Such duality defects carry chiral fields, which from the 6d point of view arise as modes of the two-form $B$ in the tensor multiplet. Each duality defect has a flavor symmetry associated to it, which is encoded in the structure of the singular elliptic fiber above the defect. Generically 2d surface defects will intersect in points in 4d, where there is an enhanced flavor symmetry. The 6d point of view provides a complete characterization of this 4d-3d-2d-0d `Matroshka'-defect configuration.Comment: 62 pages, 4 figure

M5-branes on S^2 x M_4: Nahm's Equations and 4d Topological Sigma-models

We study the 6d N=(0,2) superconformal field theory, which describes multiple M5-branes, on the product space S^2 x M_4, and suggest a correspondence between a 2d N=(0,2) half-twisted gauge theory on S^2 and a topological sigma-model on the four-manifold M_4. To set up this correspondence, we determine in this paper the dimensional reduction of the 6d N=(0,2) theory on a two-sphere and derive that the four-dimensional theory is a sigma-model into the moduli space of solutions to Nahm's equations, or equivalently the moduli space of k-centered SU(2) monopoles, where k is the number of M5-branes. We proceed in three steps: we reduce the 6d abelian theory to a 5d Super-Yang-Mills theory on I x M_4, with I an interval, then non-abelianize the 5d theory and finally reduce this to 4d. In the special case, when M_4 is a Hyper-Kahler manifold, we show that the dimensional reduction gives rise to a topological sigma-model based on tri-holomorphic maps. Deriving the theory on a general M_4 requires knowledge of the metric of the target space. For k=2 the target space is the Atiyah-Hitchin manifold and we twist the theory to obtain a topological sigma-model, which has both scalar fields and self-dual two-forms.Comment: 78 pages, 2 figure

Designing Cold-formed Steel Using the Direct Strength Method

The Direct Strength Method is an entirely new design method for cold-formed steel. Adopted in 2004 as Appendix 1 to the North American Specification for the Design of Cold-Formed Steel Structural Members, this paper introduces the Direct Strength Method and details some of the features of a new AISI Design Guide for this Method. The intent of this paper and the Guide is to provide engineers with practical guidance in the application of this new design method. The Direct Strength Method does not rely on effective width, nor require iteration for the determination of member design strength. Instead, the engineer must determine the elastic buckling load in local, distortional, and global buckling. This information along with the load that causes first yield are then employed in a series of simple equations to “directly ” provide the strength prediction. The primary complication with the method lies in determining the elastic local, distortional, and global buckling loads; once these values are determined application of the method is straightforward. Computational tools, such as the freely available open source program CUFSM, can provide the elastic buckling loads that the Direct Strength Method requires. This paper will highlight some of the features of the new Direct Strength Method Design Guide, including design examples, tutorial materials, beam and column charts, and discussion of the finer points and details that could trip up the conscientious engineer when first using the method in design

Stiffened Elements with Multiple Intermediate Stiffeners and Edge Stiffened Elements with Intermediate Stiffeners

Section B5 of the AISI Specification, covering stiffened elements with multiple intermediate stiffeners and edge stiffened elements with intermediate stiffeners, has been entirely replaced in the latest edition of the Cold-Formed Steel Specification (NAS 2001). The new design rules are based primarily on the work of Schafer and Pekoz (1998); however, subsequent to that work and prior to adoption by AISI, additional work was also completed. For elements with multiple intermediate stiffeners consideration of weblflange interaction was added. The resulting expressions are shown to agree well with both experimental and numerical data. New provisions for edge stiffened elements with intermediate stiffeners have also recently been adopted. The logic behind the development of these provisions is discussed herein. Compared with previously used procedures (AISI 1996) the new methods provide a more robust and reliable method for the design of these unique elements

Distortional Buckling Tests on Cold-formed Steel Beams

Laterally braced cold-formed steel beams generally fail due to local or distortional buckling. When the compression flange is not restrained by attachment to sheathing or paneling, such as in negative bending of continuous members (joist, purlins, etc.), members are prone to distortional failures. However, distortional buckling remains a largely unaddressed problem in the main body of the North American Specification (NAS 2001). Only a limited amount of experimental data on unrestricted distortional buckling in bending is available, therefore a new series of distortional buckling tests was completed. The test details are selected specifically to insure that distortional buckling is free to form, but lateral buckling is restricted. Several design methods, including those of the U.S., Canada, Australia, and Europe as well as the Direct Strength Method are compared with the test results. Combined with our previously conducted local buckling tests (Yu and Schafer 2003), we can now provide experimental upper and lower bounds for the capacity of laterally braced cold-formed steel beams in common use in North America. Further, the experimental results have been investigated and extended through the use of non-linear finite element analysis with ABAQUS. This paper covers the setup of the distortional buckling tests, test results, finite element analysis and discussion of current design methods

Laser Scanning to Develop Three-Dimensional Fields for the Precise Geometry of Cold-Formed Steel Members

Geometric imperfections play an important role in the performance and behavior of cold-formed steel members. The objective of this paper is to detail a newly developed imperfection measurement rig, where the full three-dimensional (3D) imperfect geometry of a cold-formed steel member can be measured and reconstructed. The measurement results in a dense three-dimensional point cloud that may be utilized to provide precise knowledge of the basic member dimensions (width, angle, radius including variation along the length), frequency content in the member (waviness, local dents, etc.), or directly as the exact geometry of the member. Practical applications of the data include basic quality control; however, the potential of the data is truly realized when applied to shell finite element models of cold-formed steel members to investigate imperfection sensitivity. The measurement rig set-up (Phase I) consists of three basic parts: a two-dimensional (2D) laser scanner with measurement range up to 304 mm [12 in.]; a linear drive system, allowing the laser to collect measurements of cross sections along the length of the target specimen; and a support beam. The raw point cloud data from the Phase I rig is input into MATLAB where custom postprocessing is employed to develop the full 3D reconstruction of the target specimen. The Phase II rig adds a rotary ring, providing a rotational stage for the laser so that the cross section of the target specimen may be profiled from any direction. This paper provides several examples of full-field imperfection measurements and compares against other methods in current use. The measured imperfections contribute to the database of realized imperfections appropriate for the generation of stochastic imperfections for use in simulation. Accurate knowledge of geometric imperfections is critical to the long-term success of analysis-based design paradigms for cold-formed steel