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Abstract of Thesis (Ph. D.)--Cornell University, 1917. / "Reprinted from the Cornell civil engineer, vol. XXVI, pp. 184, 229, Feb., Mar., 1918."

Thesis (M.S.)--University of Wisconsin--Madison, 1937. / Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaf [73]).

A method of analysis of flexible arches under the action of axial deformation, support movements, and fabrication errors by the deflection theory method is presented in this thesis. The elastic moments and dimensionless magnification factors for parabolic hingeless arches with rise-span ratios of 1/8, 1/6, 1/4 and 1/3 are given. Although the data is given for parabolic hingeless arches with a constant EI and one prescribed variation of EI, it is shown, by numerical examples, that the tables may be used for other arches whose shapes do not differ greatly from a parabola and, by interpolation, to other variations of EI. It is also shown that these solutions for hingeless arches may be used to obtain the solution of one and two hinged arches. It is shown by theory and by numerical tests that the deflection theory moments are directly proportional to the magnitude of the axial deformation, support movement, or fabrication error. It is also shown that these moments, when determined separately, may be added to each other and to moments due to load to obtain the correct total moment. The solutions in the tables were calculated by a numerical procedure of successive approximations. The electronic computer, the ALWAC III E, at the University of British Columbia was used to perform the large amount of numerical work required. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

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. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

This thesis presents the investigation of the behaviour of continuous arches on elastic piers about which little is currently known. A series of studies were made to indicate the effects of pier dimensions on extreme fiber stresses at a number of critical sections of arches. Such results are of particular interest to the bridge designer. Six numerical examples of symmetrical arch systems have been solved, using an interesting variation of the force distribution method of the late Prof. Hardy Cross. The relative proportions of the system were based primarily upon aesthetic considerations. In the six structures which were investigated two systems of arches and piers, called I and II, were considered. Each system has five spans. The variable span lengths are the same in each system. The arches in each were selected from Whitney's paper and are linear arches for dead load only. In System I the arch ribs are lighter and flatter than in System II, and the piers are more flexible. In each system the variation of span lengths and arch rises are such so that there is no unbalanced dead load horizontal thrust on the piers. In both systems all piers are of single equal batter. Three different heights of pier 40', 60', and 80' were investigated in each system. Figs. 1, 2 and 3 clarify the foregoing while Tables 1 and 2 give the properties of the elements making up each system. In System I twenty influence lines for upper, lower, right, and left kern moments at springings, crowns, and pier tops were constructed. In System II "portions" of sixteen influence lines for upper and lower kern moment at springings and crowns sufficient to establish the trend of alteration of proportions were constructed. The large number of variables involved in the design of such indeterminate structures as continuous arch systems makes it inadvisable to draw too definite conclusions, but some results of studies obviously indicate that : (1) All controlling L.L. fiber stresses are greater than those in fixed ended arches and increase as the height of piers increases, but the maximum D.L. + L.L. fiber stresses do not exhibit this characteristic as might be expected but depend, of course, upon the ratio of dead load to live load as well as upon the proportions of the structure. (2) It would appear that the analysis may be confined to three spans for arches and two spans for piers, save in the case of a long centre span combined with very flexible piers. In such a case the complete structure must be involved in the analysis. (3) The effect of rotation of the pier tops on L.L. stresses at crowns is small and almost independent of pier height, whereas the effect on L.L. stresses at springings is somewhat greater and increases slowly as the height of piers increases. The effect of translation of the pier tops on L.L. stresses at springings and crowns is usually the greatest and increases rapidly as the height of piers increases. The results presented herein are, of course, true only for this particular type of system, but the author feels that the system chosen is more representative of a structure which might be constructed than are the typical three or four equal span systems of the text book variety with which some writers, more concerned with simplicity than reality, have dealt. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

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