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UBC Theses and Dissertations

Ultimate load analysis of fixed arches Mill, Andrew John


The advent of Limit States Design has created the necessity for a better understanding of how structures behave when loaded beyond first local yielding and up to collapse. Because the problem of determining the ultimate load capacity of structures is complicated by geometric and material non-linearity, a closed form solution for anything but the simplest of structure is not practical. With this as motivation, the ultimate capacity of fixed arches is examined in this thesis. The results are presented in the form of dimensionless collapse curves. The form of these curves is analogous to column capacity curves in that an ultimate load parameter will be plotted as a function of slenderness. The ultimate capacity of a structure is often determined by Plastic Collapse analysis or Elastic Buckling. Plastic Collapse is attained when sufficient plastic hinges form in a structure to create a mechanism. This analysis has been proven valid for moment resisting frames subjected to large amounts of bending and whose second order effects are minimal. Elastic buckling is defined when a second order structure stiffness matrix becomes singular or negative definite. Pure elastic buckling correctly predicts the ultimate load if all components of the structure remain elastic. This may occur in slender structures loaded to produce large axial forces and small amounts of bending. Because arches are subject to a considerable amount of both axial and bending, it is clear that a reasonable ultimate load analysis must include both plastic hinge formation and second order effects in order to evaluate all ranges of arch slenderness. A computer program available at the University of British Columbia accomplishes the task of combining second order analysis with plastic hinge formation. This ultimate load analysis program, called "ULA", is interactive, allowing the user to monitor the behaviour of the structure as the load level is increased to ultimate. A second order analysis is continually performed on the structure. Whenever the load level is sufficient to cause the formation of a plastic hinge, the stiffness matrix and load vector are altered to reflect this hinge formation, and a new structure is created. Instability occurs when a sufficient loss of stiffness brought on by the formation of hinges causes the determinant of the stiffness matrix to become zero or negative. Two different load cases were considered in this work. These are a point load and a uniformly distributed load. Both load cases included a dead load distributed over the entire span of the arch. The load, either point load or uniform load, at which collapse occurs is a function of several independent parameters. It is convenient to use the Buckingham π Theorem to reduce the number of parameters which govern the behaviour of the system. For both load cases, it was necessary to numerically vary the location or pattern of the loading to produce a minimum dimensionless load. Because of the multitude of parameters governing arch action it was not possible to describe all arches. Instead, the dimensionless behaviour of a standard arch was examined and the sensitivity of this standard to various parameter variations was given. Being three times redundant, a fixed arch plastic collapse mechanism requires four hinges. This indeed was the case at low L/r. However, at intermediate and high values of slenderness, the loss of stiffness due to the formation of fewer hinges than required for a plastic mechanism was sufficient to cause instability. As well, it was determined that pure elastic buckling rarely, if ever, governs the design of fixed arches. Finally, the collapse curves were applied to three existing arch bridges; one aluminum arch, one concrete arch, and one steel arch. The ultimate capacity tended to be between three and five times the service level live loads.

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