The use of the photoelastic coating method in determining the stresses in the torispherical head of a pressure vessel was investigated. It was found that the method is valuable to obtain the distribution, direction, and magnitude of stresses on the surface of any structure. The results obtained with the method showed close agreement with the theoretical investigations. The maximum stresses in a torispherical head of a pressure vessel occur in the torus. The same conclusion was drawn from the results obtained with the method. It also revealed, that these stresses were compressive on the outer surface. The mobility of the instruments, the relatively simple way of coating the surface of the structure are other features of the method. / Applied Science, Faculty of / Mechanical Engineering, Department of / Graduate

The effect of the mean stress on the endurance limit is a matter of considerable academic and practical importance. So many variable and uncertain factors are involved that it is not surprising that many different formulas and theories have been proposed since the start of investigations into this subject in about 1858. In this thesis, factual data obtained in the course of tests planned primarily for the purpose of determining the effect of the mean stress are presented and discussed. A standard fatigue testing machine vas used for all the tests. Various mean stresses, both for axial-load and torsional tests, were employed. A critical examination of the proposed theories of fatigue failure has been made. The theory which considered that the inception of fatigue resulted from alternating shearing stress and that the resistance to fatigue was influenced by the magnitude and sign of the steady normal stress was found to explain best the damage done to the material structure. It has been the opinion of a number of investigators that the fatigue strength under axial-load is decreased by an increasing tensile mean stress and in torsion the fatigue strength is unaffected by the mean stress. The experiments done here clearly indicate that this conclusion is true. / Applied Science, Faculty of / Mechanical Engineering, Department of / Graduate

An influence line method for the analysis of flexible hingeless arch by the deflection theory is presented in this thesis. To facilitate the work, tables of dimensionless magni fication factors are provided. Prom these tables, influence lines taking into account the flexibility of the arch may be readily drawn and used very much in the ordinary way. The flexibility of the arch was conveniently measured by a dimensionless ratio, [ equation omitted ], and called the stiffness factor of the arch. The tables are for parabolic hingeless arches having rise ratios of 1/8,1/6,1/4,1/3, with constant EI or a prescribed variable EI. Values are given for β = 3 and 5 with some for β = 7. Also the tables contain magnification factors for maximum moments at eleven points in the arch, when the arch is loaded with a uniform load. Although the given tables are good only for parabolic hingeless arches with constant EI or a prescribed variation in EI, the tables may be reasonably extended to other hingeless arches whose shapes are not too different from a parabola and to a wide variety of variation in moment of inertia, provided these variations are not unrealistic. The possibility of using superposition in the deflection theory is based on the fact that calculations showed the horizontal thrust acting on the arch was approximately the same either by the deflection theory or the elastic theory. Because of this, the horizontal thrust becomes independent of deflection and the differential equation for bending of an arch is linear. Thus superposition may be used. The differential equation was hot convenient for calculation. Instead, the solutions in the tables were calculated by a numerical procedure of successive approximations, using the conjugate beam concept. This procedure was conveniently programmed for an electronic computer, the ALWAC III E, at the University of British Columbia. In the first cycle of approximation, the programme assumed the horizontal and vertical deflections were zero. This represented the elastic theory analysis. In subsequent cycles, the deflected shape of the arch from previous analysis was assumed. Successive approximation as such led to a solution based on the deflection theory. Three numerical examples shown in this thesis indicated that the error introduced by the linearized deflection theory was small, and the influence line method may be used for analysis of flexible arches. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

This thesis presents a simple method for the determination of bending moments in flexible symmetrical hingeless arches. The deflection theory bending moments are obtained by multiplying elastic theory bending moments by predetermined magnification factors. The problem was to provide magnification factors for all cases of loading on a wide range of flexible symmetrical hingeless arches. By studying the differential equation a modified method of superposition was developed. Therefore, it was only necessary to determine magnification factors for a concentrated load at a number of positions along the arch axis. A convenient set of coordinates for the magnification factor was determined by dimensional analysis. Finally, an electronic computer was used to calculate the required magnification factors by a numerical method. Tables of magnification factors are presented for symmetrical parabolic hingeless arches. Magnification factors are given for rise to span ratios of 1/8, 1/6, 1/4, and 1/3 ; for constant and variable moment of inertia; and for values of [formula omitted]from 0 to 7, where H = the total horizontal thrust, L = the span, EIa = the average flexural rigidity. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

Drained behaviour of sands under simultaneous increase in stress ratio and principal stress rotation is investigated. The hollow cylinder torsional (HCT) device which permits independent control of four stress parameters: mean normal stress σ'[subscript]m, stress ratio R, intermediate principal stress parameter b and the inclination α[subscript]σ of σ'₁ to the vertical, is adopted as the testing device. In order to conduct complex stress path testing in the HCT device, a new automatic stress path control system is developed. The stress non-uniformities due to the curvature of the HCT specimen is assessed using an incremental elastic representation of sand behaviour, in order to delineate the domain of stress space that could reliably be explored using the HCT device. It is shown that previous assessments of stress non-uniformities assuming linear elastic soil grossly overestimate the stress non-uniformities in a HCT sand specimen. A much larger domain of stress space with acceptable levels of non-uniformities is apparent from the results of incremental elastic analysis. New domain of stress space for reliable exploration using the HCT device is delineated and the testing program is developed so that all stress paths lie within these acceptable limits. Tests are carried out on pluviated sand under saturated drained conditions. The deformations under increasing R and α[subscript]σ is shown to be path independent, if the final stress state is within the approximate bounds of R ≤ 2 and α[subscript]σ ≤ 45°, regardless of the b or relative density D[subscript]r, levels. With increasing stress ratio R and/or principal stress rotation α[subscript]σ, the deformations gradually become path dependent. Once loaded to a stress state within the domain R ≤ 2 and α[subscript]σ ≤ 45°, the strain response under subsequent principal stress rotation is shown to be independent of the previous loading history. It is demonstrated that the strain response under any general increasing R - α path in the domain of R ≤ 2 and α[subscript]σ ≤ 45° can be predicted using the results of a limited number of tests characterizing that domain. It is shown that these concepts can be extended to loading paths which involve simultaneous increase of three stress parameters. Strain increment direction α[subscript]Δε is shown to be approximately coincident with and totally governed by the stress increment direction α[subscript]Δσ when the stress increment direction α[subscript]Δσ is more inclined towards the vertical deposition direction. When the stress increment direction is inclined closer to the bedding plane, the strain increment direction depends in addition, on other parameters such as R, α[subscript]σ and R[subscript]r etc. Under any stress path involving principal stress rotation, the deformations decrease with increasing density and therefore the principal stress rotation is more crucial in loose sands. Deformations increase with the level of stress ratio R. Level of b parameter does not affect deformations under principal stress rotation, if the rotations are small. However, with increasing α[subscript]σ deformations due to principal stress rotation tend to increase with decreasing b value. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

The deformation characteristics of continuous tungsten fibre-reinforced zinc composites have been investigated. Composites with a single crystal matrix containing up to 4.5 volume per cent of tungsten fibres were studied. The stress-strain curves of W-Zn composites showed positive deviations from the "rule of mixture" predictions. Theoretical work attributes the positive deviations to matrix hardening due to either one of the phenomena: (a) the difference in the lateral contractions of the fibre and the matrix; (b) the pile up of dislocations in the matrix at the matrix-fibre interface. In the present work the positive deviations in the elastic-plastic region of the stress-strain curves of the composites have been attributed to both (a) and (b). The positive deviations in the ultimate tensile strengths of the composites have been attributed to (b). Composites containing up to 0.08 volume per cent of the tungsten wires deformed even after the fracture of the fibres. Dissolution of the matrix of these deformed composites showed that multiple necking had occurred in the fibres fractured to 1-5 mms length. Composites containing greater than 0.08 volume per cent of tungsten fibres fractured by cleaving through the basal plane of the matrix. No fibre fracture inside the matrix was seen except at the fracture end of the composite. Multiple necking of the fibres near the fractured end has been seen only in those composites which have deformed more than the free fibres tested individually. / Applied Science, Faculty of / Materials Engineering, Department of / Graduate

Analysis of a general shell of revolution with arbitrary loading and boundary conditions using the Finite Element approach, well-suited for use with the electronic computer, is presented. The shell is approximated by an assemblage of flat, equilateral trapezoids and isosceles triangles connected to each other at the corners. The assumptions involved in transforming a piece of plate into a finite element are defined. Uncoupled plane stress and flexure stiffness matrices for the above-mentioned shapes of the finite elements are derived from considerations of (i) statics, and (ii) virtual work (energy). Statics matrices are asymmetric with the exception of the triangle plane stress stiffness matrix. However, it is important to note that irrespective of the size of the trapezoid element, in conditions of uniform stress the nodal forces satisfy Betti's reciprocal theorem. When a trapezoid reduces to a rectangle, the asymmetry of plane stress and flexure stiffness matrices disappears. Asymmetry of the Statics matrix is removed by averaging the matrix and its transpose. This process corresponds to introducing self-equilibrating nodal forces which disappear in conditions of uniform stress. Suitable direction cosine matrices are derived to transform the displacements and forces from the element coordinate system to the shell coordinate system. The accuracy of the formulation is demonstrated in several examples by comparing the finite element solution with the elasticity solution. The comparison suggests convergence of the results to the correct solution on reduction of the element size. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

The strongest column problem is defined, for this thesis, as the determination of the column shape which gives the maximum Euler buckling load for a given length, volume of material, type of cross-section and type of tapering. This thesis presents a new method - the matrix method - for solving some strongest column problems. In the matrix method a member is approximated by a number of uniform sub-members. A structure stiffness matrix, with the effect of axial force on deflections included, is generated from the sub-members. By setting the determinant of this matrix equal to zero the critical buckling load and the buckled shape of the member in the first mode are found. The section properties of the sub-members are then altered, according to the constant stress criterion, so that the extreme fibre bending stress, determined from the first mode, is the same in each sub-member. The process is repeated until the stresses are sufficiently close to being equal so that no further alterations are required. The optimum shape is taken from the last iteration. The constant stress criterion is based on the fact that, when certain conditions are satisfied, the extreme fibre bending stress at any section is constant along the length of the strongest column when it is buckled in the first mode. The matrix method gives results in very close agreement with those found by previous authors for cases where the constant stress criterion is valid. For the one example presented where the constant stress criterion was not valid, the column buckling under its own weight, the matrix method gave poor results. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

Plane stress stiffness matrices are derived explicitly for square Isotropic elements under different assumptions on the stress distribution. An explicit (8 x 8) matrix is obtained under the assumption of uniform σx, σy, linear τxy and thus it is shown that the Gallagher matrix belongs to the class of parametric matrices. Two (10 x 10) matrices are obtained under the assumption of linear σx, σy, τxy using interior nodal translations and corner edge rotations respectively as additional generalized displacements. These two matrices do not appear suitable for general usage but will perform as well as the Turner matrix under the same nodal loads. A (12 x 12) matrix is derived under the assumption of hyperbolic σx, σy, and parabolic τxy, again exemplifying the use of corner edge rotations as additional generalized displacements. This matrix behaves unexpectedly with varying Boisson's ratio. A method of evaluating stiffness matrices, which reduces the necessity of comparing finite element solutions with analytical ones, is formulated. In this method a comparison is made of the strain energy of deformation produced within a finite element by the different matrices under the same nodal loads. It is shown that such comparisons require the study of special matrices i.e. the stiffness difference matrix and the inverse difference matrix which are obtained from the matrices under comparison. It is proved that the results of the element matrix comparisons apply to the structure. It is shown that the strain energy of a finite element under normalised loads is bounded between the maximum and minimum eigenvalues of the inverse matrix. The strain energy comparison criterion is used in the study of parametric matrices. An explicit parametric inverse is obtained. Explicit parametric eigenvalues are obtained for the inverse difference matrix and the stiffness difference matrix, and it is verified that they give identical results for the matrix comparisons. It is proved that the parametric matrices produce the exact strain energy under uniform nodal loads. It is shown that the stiffness matrix parameter and the inverse matrix parameter represent a measure of the strain energy under non-uniform nodal loads so that the strain energy can always be bounded by varying the parameter. It is proved that if strain energy curves are drawn with respect to structure sub-division then no two curves will intersect. It is proved that all parametric strain energy curves will converge towards the true solution with progressive structure subdivision. A strain energy ordering is obtained for the parametric matrices and the following conclusions are drawn. The Pian matrix is the best displacement matrix. The Gallagher matrix is inferior to the Turner, Pian, and Argyris-Melosh matrices. Constant stress tri-nodal triangles are generally inferior to the use of square elements. Matrices satisfying microscopic equilibrium or capable of representing uniform stresses will not necessarily yield good results. A method is proposed for obtaining upper bounds on the strain energy of a region under plane stress by replacing the continuum with a psuedo-truss system, the bar forces of which provide the equilibrium and self-straining solutions. Two examples of its application are presented, and an indication is obtained that upper bounding by varying the matrix parameter will give better results for the same structure subdivision. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

Many field problems in soils approximate plane strain conditions. Conventional laboratory practice, however, consistently uses triaxial compression testing for evaluating strength and deformation properties of soils. Possible differences between triaxial and plane strain behaviour of natural soils have so far received little attention. New plane strain and K₀-triaxial apparatuses have been designed in which it is now possible to shear samples under various stress paths. A series of triaxial and plane strain tests were performed on identically K₀-consolidated samples of an undisturbed, sensitive, marine clay. Drained and undrained compression and extension tests under both increasing and decreasing stresses were carried out. Similar series of tests were also made on heavily overconsolidated specimens. It is demonstrated that the use of triaxial test results where plane strain conditions prevail invariably leads to an overestimate of deformations and an underestimate of strength. Methods to estimate strains in undrained plane strain shear from the results of triaxial tests under identical stress paths have been developed. Similarly, strains during drained plane strain shear of normally consolidated clay could be predicted from drained and undrained triaxial results, using an extended form of Rendulic's hypothesis. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate