This study of the bending of a spherical shell of constant thickness consists of five chapters. In the first chapter, general differential equations of the problem are derived and presented with the variables in dimensionless form. The second chapter deals with the solution of the two second order differential equations in terms of Bessel Functions. The solution is applicable for shells with a colatitude of about 25 degrees or less. Expressions are given for the internal stresses and displacements in terms of these values at the boundary. The third chapter gives a numerical solution of the problem. Solutions are found for three cases of radius to thickness ratio, namely 30, 100 and 500, and for colatitudes from 8 to 90 degrees. In the preparation of tables the Alwac III-E Electronic Digital Computer was used. The flow diagram and programme for the Alwac III-E are given. The fourth chapter compares the numerical method with one of Timoshenko's approximate solutions. The wave lengths, deflections, rotations and moments are calculated by two methods and compared. As the conclusion to these comparisons a graph is given, which shows the approximate error in the damped harmonic solution. In the last chapter, a numerical example is solved using tables from the digital computer. The ratio of radius to thickness is 500; radius being 125 feet and the colatitude 30 degrees. The edges are considered as being on rollers. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

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Call number: LD2668 .T4 1984 R36 / Master of Science

This thesis investigates the effects of elastic restraint at the supports on the behaviour of shallow concrete arches, and the time-dependent effects of shrinkage and creep of concrete on the nonlinear behaviour of shallow arches. The nonlinear behaviour of shallow circular arches with elastic rotational restraints at each support and subjected to a uniformly distributed radial load is firstly investigated. A virtual work formulation is used to establish both the nonlinear equilibrium equations and the buckling equilibrium equation for shallow arches. The analytical results show that the effects of the stiffness of the rotational restraints on the prebuckling and buckling response are significant. An analytical model is developed for the in-plane elastic stability analysis of shallow parabolic arches with horizontal spring supports subjected to a uniformly distributed vertical load. A parametric study is undertaken using the proposed analytical model. It is found that the effects of the stiffness of the horizontal springs on the prebuckling response, buckling load and buckling behaviour of arches are significant. An analytical model is developed to simulate the time-dependent behaviour of shallow concrete parabolic arches with horizontal spring supports subjected to a sustained loading, and in particular to investigate creep buckling. The time-dependent buckling load and the critical time (or age) of the arches are calculated by using an iterative process based on the proposed model. A systematic parametric study is undertaken, and the results show that the various parameters have a profound effect on the time-dependent buckling load and the prebuckling life of arches. Both short-term and long-term experimental investigations of shallow parabolic tied concrete arches are described and used to validate the analytical models. For the short-term tests, three concrete arches were subjected to a uniformly distributed vertical load and were loaded to failure. For the long-term tests, seven concrete arches were subjected to sustained service loads. The instantaneous and time-dependent deflections were recorded throughout the period of loading, together with the distribution of the horizontal thrust at the supports. Comparisons between the experimental results and the analytical predictions using the analytical models are made to verify the accuracy of the theoretical models.

This thesis investigates the effects of elastic restraint at the supports on the behaviour of shallow concrete arches, and the time-dependent effects of shrinkage and creep of concrete on the nonlinear behaviour of shallow arches. The nonlinear behaviour of shallow circular arches with elastic rotational restraints at each support and subjected to a uniformly distributed radial load is firstly investigated. A virtual work formulation is used to establish both the nonlinear equilibrium equations and the buckling equilibrium equation for shallow arches. The analytical results show that the effects of the stiffness of the rotational restraints on the prebuckling and buckling response are significant. An analytical model is developed for the in-plane elastic stability analysis of shallow parabolic arches with horizontal spring supports subjected to a uniformly distributed vertical load. A parametric study is undertaken using the proposed analytical model. It is found that the effects of the stiffness of the horizontal springs on the prebuckling response, buckling load and buckling behaviour of arches are significant. An analytical model is developed to simulate the time-dependent behaviour of shallow concrete parabolic arches with horizontal spring supports subjected to a sustained loading, and in particular to investigate creep buckling. The time-dependent buckling load and the critical time (or age) of the arches are calculated by using an iterative process based on the proposed model. A systematic parametric study is undertaken, and the results show that the various parameters have a profound effect on the time-dependent buckling load and the prebuckling life of arches. Both short-term and long-term experimental investigations of shallow parabolic tied concrete arches are described and used to validate the analytical models. For the short-term tests, three concrete arches were subjected to a uniformly distributed vertical load and were loaded to failure. For the long-term tests, seven concrete arches were subjected to sustained service loads. The instantaneous and time-dependent deflections were recorded throughout the period of loading, together with the distribution of the horizontal thrust at the supports. Comparisons between the experimental results and the analytical predictions using the analytical models are made to verify the accuracy of the theoretical models.

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