Practice in the field of limit design has tended to place certain restrictions on structural loading patterns in order to simplify the calculations involved in the limit design procedure. The loads considered in this simplified approach are assumed to either remain constant and fixed, or if they vary then this is to be in such a manner that their magnitudes stand in a constant relationship one to the other. Actual structural loadings seldom satisfy these restrictive conditions and the question naturally arises as to whether or not this simplified limit design procedure is valid for general use in practical design problems in which external loads may be wholly independent in their individual actions. This question is investigated in the present paper through the examination of several practical forms of structure which portray the more adverse conditions of independent and variable loading to be met in practice. These structures are, respectively, single and double bay gable bents of lightweight construction, and two forms of multispan bridge girders. The study indicates that all of these structures are able to support the ultimate loads predicted by the simplified limit design method; the actual ultimate loads exceeding the predicted values by up to twenty percent. It is concluded that structural failure in practice can always be expected to occur within acceptable limits of the ultimate load capacity as predicted by the simplified method. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

This thesis presents the results of a theoretical and experimental study of the lateral buckling of two- and three-hinged arches of rectangular cross-section with laterally restrained top edges. The structure is analysed with and without a linear torsional restraint along the top edge. The problem is formulated using the stiffness method. A stiffness matrix including the effects of lateral bending and torsion is used. The buckling load is defined as the smallest load at which the structure stiffness matrix becomes singular. The method of solution of the theoretical lateral buckling is iteration (eigen value problem) and determinant plot. This theoretical approach is verified by model tests with two- and three-hinged parabolic glulam arches in the laboratory. The method of solution for model test is the Southwell plot. The results of the tests are presented and are shown to be satisfactory. A set of numerical results are given for a range of arches with torsional restraint at the top edge and for various load distributions. A sample of calculations of a practical arch shows that, although the arch is safe according to the existing code, it is only safe considering lateral buckling including a torsional restraint at the top edge. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

The lateral buckling of two hinged glulam parabolic arches of rectangular cross-section with laterally restrained top edges is investigated. The problem is formulated using the stiffness method. The structure is idealised as a series of straight segments. A stiffness matrix including the effects of lateral bending and torsion is derived and it is shown how the structure stiffness matrix is generated. The buckling load is defined as the smallest load at which the structure stiffness matrix becomes singular. Three methods of solution are given; iteration, determinant plot and Southwell plot. Experimental tests were carried out to check the validity of the theory. The results of the tests are presented and are shown to be satisfactory. A set of numerical results are given for a range of arches and load distributions and a set of design parameters are proposed. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

The analysis of structures which contain catenary cables is made difficult by the non-linear force-deformation relationships of the cables. For all but the smallest deflections it is not possible to linearize these relationships without causing significant inaccuracies. Newton's Method solves non-linear equations by solving a succession of linearized problems, the answer converging to the solution of the non-linear problem. Newton's Method so used to analyze cable-containing structures results in a succession of linear stiffness analysis problems. As a result, conventional stiffness analysis computer programs may be modified without great difficulty to solve cable structures by Newton's Method. The use of Newton's Method to solve cable structures forms the body of this thesis. The two basic innovations necessary, which are the provision of methods for calculating the end-forces of a cable in an arbitrary position, and for evaluating the stiffness matrix of a cable, are presented. Also discussed are the co-ordinate transformations necessary to describe the cable stiffness matrix and cable end forces in a Global Co-ordinate System. The virtues of the method are demonstrated in two example problems, and the theoretical basis for Newton's Method is examined. Finally, the value of the method presented is briefly discussed. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

published_or_final_version / Civil Engineering / Master / Master of Philosophy

A finite element formulation for the shear-deformable analysis of beams, plates and shells, based on a strain energy expression defined in terms of total and flexural displacement components, is presented. The effects of transverse shear deformation are considered while the normal strain is neglected. The finite element representation requires independent descriptions of total and flexural displacement components. The flexural strain energy term involves second derivatives of flexural displacement component and thereby necessitates slope-compatible shape functions. This requirement is relaxed by adopting the 'discrete Kirchhoff' hypothesis for the flexural displacement component. An element of triangular shape is formulated for the analysis of laminated composite plates and shallow shells. Numerically exact integration is employed in the calculation of element stiffness matrix and corresponding load vectors. The resulting finite element possesses twelve degrees of freedom at each corner node of the triangle. Numerical results obtained for an extensive range of thickness and planform aspect ratios, laminate configurations, mesh sizes, edge conditions, types of loading and geometry of the structure demonstrate the efficacy of the finite element formulation. The element is applicable to a full range of thicknesss ratios. The present formulation is employed for dynamic and stability analysis of beams, as a precursor to the inclusion of these effects in the analysis of plates and shells.

The research work presented here deals with problems associated with finite element analysis of laminated composite thin-shell structures. The specific objective was to develop a thin shell finite element to model the linear elastic behavior of these shells, which would be efficient and simple to use by the practicing engineer. A detailed discussion of the issues associated with the development of thin shell finite element has been presented. It has been pointed out that the problems encountered with formulation of these elements stem from the need for satisfaction of the interelement normal slope continuity and the rigid body displacement condition by the assumed displacement functions. These difficulties have been surmounted by recourse to the discrete Kirchhoff theory approach and an isoparametric representation of the shell middle surface. A detailed derivation of the strain energy density in a thin laminated composite shell, based on a linear shear deformation theory formulated in a general curvilinear coordinate system, has been presented. The strain-displacement relations are initially derived in terms of the displacement and rotation vectors of the shell middle surface, and are subsequently expressed in terms of the cartesian components of these vectors to enable an isoparametric representation of the shell geometry. A three-node curved triangular element with the tangent and normal displacement components and their first-order derivatives as the final nodal degrees of freedom has been developed. The element formulation, however, starts with the independent interpolation of cartesian components of the displacement and rotation vectors using complete cubic and quadratic polynomials, respectively. The rigid-body displacement condition is satisifed by isoparametric interpolation of the shell geometry within an element. A convergence to the thin shell solution is achieved by enforcement of the Kirchhoff hypothesis at a discrete number of points in the element. A detailed numerical evaluation through a number of standard problems has been carried out. Results of application of the "patch test solutions" to spherical shells demonstrate a satisfactory performance of the element under limiting states of deformation. It is concluded that the DKT approach in conjunction with isoparametric representation results in a simple and efficient thin shell element.

No description available.

Research report submitted to the Faculty of Engineering and the Built Environment, University of Witwatersrand, Johannesburg, in partial fulfilment of the requirements for the degree of Master of Science in Engineering Johannesburg 2017 / This report investigates the seismic response of catenary vaults. Through a series of tests, the inherent seismic resilience of catenary vaults was assessed and a number of reinforcement strategies were investigated to improve this. An analytical model, based on the virtual work method, was developed by Ochsendorf (2002) for the assessment of circular voussoir arches. This model was adapted for catenary vaults. This model is used to calculate the minimum lateral acceleration required to cause the collapse of a catenary vault (λmin) for any catenary profile. The model indicates that there is a linear relationship between cross sectional depth of the arch and λmin until the depth to ratio passes approximately 0.3, where the change in λmin becomes exponential. Using the model, it is also predicted that λmin decreases exponentially with an increase in the height to width ratio up to a value of approximately 1.6. After this point λmin linearly decreases with increased height to width ratios and approaches zero. The first series of tests involved subjecting unreinforced catenary vaults to seismic loading. In these tests the frequency of vibration was varied and the stroke was kept constant. From the results of the tests, it was found that there was no frequency at which the vaults underwent excessive vibration due to resonance. It was observed that during seismic loading, hinges form at locations where pre-existing cracks occur despite the higher computed λmin values for these positions. The tests also indicate that the vaults’ behaviour changes drastically with each hinge that forms. In the next series of tests the frequency was set and the stroke was increased. The vaults were subjected to seismic loading at 2 Hz and 6 Hz, representative of low and high frequencies respectively. The tests indicated that the collapse acceleration of arches subjected to vibration at 2 Hz was lower than that of the vaults subjected to vibrations at 6 Hz. Despite this, the stroke, representing ground movement, required to cause collapse at 2 Hz was substantially higher than that of the 6 Hz tests. This indicates that the duration of load cycles has an effect on the collapse acceleration. In comparing the computed collapse acceleration, λmin, with the actual collapse accelerations, it was found that the computed values are highly conservative. Yet this is expected as the model is based on an infinite duration of lateral loading. It was found that the analytical model was more accurate for low frequency tests as compared to high frequency tests in terms of the predicted hinge locations. Finally, three reinforcement strategies were investigated using basalt fibre geogrid. This was found to be an economical and viable reinforcement material. The first strategy consisted of laying the geogrid over the arch and securing it at the arch base. The second was the same as the first with the addition of anchors which held the geogrid down. The final strategy involved prestressing the arch using the geogrid. The latter 2 methods were found to be the most effective, with observed collapse accelerations being over 60% higher than that of the same unreinforced arch. The anchorage solution was found to be the most viable due to the substantially higher technical input required for the prestressing solution. / MT2017

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