WEAK ISOMETRIES OF HAMMING SPACES

In this thesis we study weak isometries of Hamming spaces. These are permutations of a Hamming space that preserve some but not necessarily all distances. We wish to find conditions under which a weak isometry is in fact an isometry. This type of problem was first posed by Beckman and Quarles for Rn. In chapter 2 we give definitions pertinent to our research. The 3rd chapter focuses on some known results in this area with special emphasis on papers by V. Krasin as well as S. De Winter and M. Korb who solved this problem for the Boolean cube, that is, the binary Hamming space. We attempted to generalize some of their methods to the non-boolean case. The 4th chapter has our new results and is split into two major contributions. Our first contribution shows if n=p or p < n2, then every weak isometry of Hnq that preserves distance p is an isometry. Our second contribution gives a possible method to check if a weak isometry is an isometry using linear algebra and graph theory.

Structural capacity of steel tubular cast-in-place piling

Steel tubular cast-in-place pilings are used throughout the country for many different project types. These piles are a closed-end pipe with varying wall thicknesses and outer diameters, that are driven to depth and then the core is filled with concrete. These piles are typically used for smaller bridges, or secondary structures. Mostly the piling is designed based on a resistance based method which is a function of the soil properties of which the pile is driven through, however there is a structural capacity of these members that is considered to be the upper bound on the loading of the member. This structural capacity is given by the AASHTO LRFD (2010), with two methods. These two methods are based on a composite or non-composite section. Many state agencies and corporations use the non-composite equation because it is requires much less computation and is known to be conservative. However with the trends of the time, more and more structural elements are being investigated to determine ways to better understand the mechanics of the members, which could lead to more efficient and safer designs. In this project, a set of these piling are investigated. The way the cross section reacts to several different loading conditions, along with a more detailed observation of the material properties is considered as part of this research. The evaluation consisted of testing stub sections of pile with varying sizes (10-¾”, 12-¾”), wall thicknesses (0.375”, 0.5”), and testing methods (whole compression, composite compression, push through, core sampling). These stub sections were chosen as they would represent a similar bracing length to many different soils. In addition, a finite element model was developed using ANSYS to predict the strains from the testing of the pile cross sections. This model was able to simulate the strains from most of the loading conditions and sizes that were tested. The bond between the steel shell and the concrete core, along with the concrete strength through the depth of the cross section were some of the material properties of these sections that were investigated.