The focus of this work is the buckling response of symmetrically laminated composite plates having a planform area in the shape of an isosceles trapezoid. The loading is assumed to be inplane and applied perpendicular to the parallel ends of the plate. The tapered edges of the plate are assumed to have simply supported boundary conditions, while the parallel ends are assumed to have either simply supported or clamped boundary conditions. Plates with one end being up to 3 times narrower than the other end, and the plate being up to 3 time longer than the width of the wide end are considered. A semi-analytic closed-form solution based on energy principles and the Trefftz stability criterion is derived and solutions are obtained using the Rayleigh-Ritz method. Intrinsic in this solution is a simplified prebuckling analysis which approximates the inplane force resultant distributions by the forms N_x=P/W(x) and N_y=N_xy=0, where P is the applied load and W(x) is the plate width which, for the trapezoidal planform, varies linearly with the lengthwise coordinate x. The out-of plane displacement is approximated by a double trigonometric series. This analysis is posed in terms of four nondimensional parameters representing orthotropic and anisotropic material properties, and two nondimensional parameters representing geometric properties. With nondimensionalization, the analysis is well suited for parametric studies. The analysis uses standard eigenvalue extraction routines and converges using 5 terms in the out-of-plane displacement series. It appears that this analysis captures the buckling response of plates having tapered planform and should be a useful design tool. For comparison purposes, a number of specific plate geometry, ply orientation, and stacking sequence combinations are investigated using the general purpose finite element code ABAQUS. Comparison of buckling coefficients calculated using the semi-analytical model and the finite element model show agreement within 5%, in general, and within 15% for the worst cases. In addition to the good agreement between the semi-analytical analysis and the finite element results, the finite element model also suggests that the simplified inplane force resultant distribution assumed in the analysis is valid. In order to verify both the finite element and semi-analytical analyses, buckling loads are measured for graphite/epoxy plates having a wide range of plate geometries and stacking sequences. Test fixtures, instrumentation system, and experimental technique are described. Experimental results for the buckling load, the buckled mode shape, and the prebuckling plate stiffness are presented and show good agreement with the analytical results regarding the buckling load and the prebuckling plate stiffness. However, the experimental results show that for some cases the analysis underpredicts the number of halfwaves in the buckled mode shape. In the context of the definitions of taper ratio and aspect ratio used in this study, it is concluded that the buckling load always increases as taper ratio increases for a given aspect ratio for plates having simply supported boundary conditions on the parallel ends. There are combinations of plate geometry and ply stacking sequences, however, that reverse this trend for plates having clamped boundary conditions on the parallel ends such that an increase in the taper ratio causes a decrease in the buckling load. The clamped boundary conditions on the parallel ends of the plate are shown to increase the buckling load compared to simply supported boundary conditions. Also, anisotropy (the D₁₆ and D₂₆ terms) is shown to decrease the buckling load and skew the buckled mode shape for both the simply supported and clamped boundary conditions. / Master of Science

The plane in-plane compression response for a symmetrically laminated composite panel with a spatially varying fiber orientation has been analyzed for four different boundary conditions. Variation of the fiber angle along the length of a composite laminate results in stiffness properties that change as a function of location. The laminates are therefore termed variable stiffness panels. This work presents an analysis of the stiffness variation and its effect on the in-plane and buckling response of the panel. The fiber orientation is assumed to vary only in one spatial direction, although the analysis can be extended to fibers that vary in two spatial directions. A system of coupled elliptic partial differential equations that govern the in-plane behavior of these panels has been derived. Solving these equations yields the displacement fields, from which the strains, stresses, and stress resultants can be subsequently calculated. A numerical solution has been obtained using an iterative collocation technique. Corresponding closed form solutions are presented for the in-plane problem for four different sets of boundary conditions. Three of the cases presented have exact solutions, and therefore serve to validate the numerical model. The Ritz Method has been used to find the buckling loads and buckling modes for the variable stiffness panels. Improvements in the buckling load of up to 80% over straight fiber configurations were found. Results for three different panel aspect ratios are presented. / Master of Science

This thesis is concerned with the buckling problem of a cantilever thin plate with its tree end subjected to uniform shear. The same problem waa originally solved by Prandtl in 1899, based on the equilibrium condition of a deep beam. The author baa used the energy method based on the thin plate theory to attack the problem. After the displacement is assumed, the potential energy can be formulated. From the condition that the potential energy assume a minimum value in an equilibrium configuration, results a system of n linear homogeneous algebraic equations ot n parameters which are introduced in the assumed displacement. For a non-trivial solution, the determinant of the coefficients must vanish. This gives a characteristic equation from which the buckling load is determined. The author has obtained a curve for maximum stress at buckling state, which shows that the result 1a better than that obtained by Prandtl in certain cases. The energy method has been generalized to a three dimensional problem to consider the displacement in all directions. / Master of Science

A circular thin plate under radial compressive forces resting on an elastic foundation not in attachment was studied with regard to it's behavior in the 2nd mode shape. Two regions of action are controlled by two differential equations of the fourth order which were solved in terms of the Bessel functions. The relations between the foundation modulus and the buckling load were found from two characteristic equations expressed in terms of Bessel functions. / Master of Science

A method is presented for the calculation of the crippling stress of structural sections as a function of material properties and the proportions of the section. The presence of formed or anisotropic material is accounted for by the use of an effective stress-strain curve in defining the material properties. The method applies to many sections for which a procedure for calculating crippling was not previously available.. / Master of Science

Structural optimization was performed by either mathematical programming methods or optimality criteria methods. Both types of methods are based on iterative resizing of structures in the expectation that it will lead to the satisfaction of optimality conditions. Recent developments in methods for solving nonlinear equations gave a way to an alternative approach in which the optimality conditions are treated as a set of nonlinear equations and solved directly. Two different formulations are presented; one is a conventional nested approach and the other is a simultaneous analysis and design approach. Two procedures are explored to solve the nonlinear optimality conditions; a Newton-type iteration method and a homotopy method. Here, the homotopy method is adapted to the optimal design so that we can trace a path of optimum solutions. The solution path has several branches due to changes in the active constraint set and transitions from unimodal to bimodal solutions. The Lagrange multipliers and second-order optimality conditions are used to detect branching points and to switch to the optimum solution path. This study specifically deals with buckling load maximization which requires highly nonlinear eigenvalue analysis and the procedure is applied to design of a column or laminated composite plate structures. A formulation to obtain multimodal solutions is given. Also, a special property in a laminate bending stiffness is found. That is, for a given stacking sequence of ply orientations, we showed an existence of a design with the same bending stiffness matrix and same total thickness even when the stacking sequence is changed. / Ph. D.

Scale model testing can improve the cost-effectiveness of composite structures by reducing the reliance on full size component testing. Use of scale models requires the relationship be known between the responses of the small scale model and full size component. This relationship may be predicted by dimensional analysis or through mechanics formulations. The presence of physical constraints may prevent the complete reproduction of all responses in small scale models. Scaling relationships may not be available at the level necessary to predict all scaled responses. Investigations of the scalability of composite structures are needed in order to evaluate the reliability of small scale model predictions of the responses of full size components. The scaling of the responses of graphite-epoxy laminated composite Z-section stiffeners subjected to uniaxial, compressive loading has been evaluated. The response regimes investigated are prebuckling, initial local buckling, postbuckling and crippling. A mechanistic approach to scaling has been used, in which the scalability of the responses has been judged relative to governing mechanics models. A linked-plate analytical model has been obtained which predicts the buckling loads, and from which two nondimensional load parameters have been obtained. The finite element method has been used for prediction of the buckling and postbuckling responses. The analytical and numerical analyses were used to define an experimental program involving fifty-two specimens of seventeen basic geometrical configurations and three stacking sequences. The buckling, postbuckling and crippling responses were largely determined by the flange-to-web width ratio and both the absolute and relative values of the bending stiffnesses. Buckling loads increased with decreasing flange width and the laminate orthotropy ratio, and increasing flange-to-web corner radii and laminate thickness. The postbuckling load range was the greatest for specimens having wider flanges, but the failure stresses were greatest among the narrower specimens. The crippling mechanisms included flange free edge delamination at both nodal and anti-nodal axial positions, material crushing in the flange-to-web corner at nodal axial positions, and ply splitting in the flange-to-web corner at anti-nodal axial locations. The constraint of the potted end supports of the experimental specimens was not scaled. The effect of displacements within the end supports was manifested by lower prebuckling axial stiffnesses than predicted based on the gage length properties alone. This phenomenon required a post-test adjustment to the data in order to permit comparisons of the experimental and finite element predictions of the response of the gage length on an equivalent basis. Once corrected, the prebuckling stiffnesses were generally observed to have scaled. One of the nondimensional load parameters normalized the buckling loads for specimens of various web widths only. The second parameter normalized the buckling loads for all of the geometric and material variables contained in the model. This parameter also normalized the postbuckling loads, and is, therefore, a general nondimensional parameter for the buckling and postbuckling responses of the Z-section stiffeners. No scale effects were observed in the buckling response. The quality of the postbuckling load predictions degraded with the width of the postbuckling load range. It was not determined whether genuine scale effects were present in the postbuckling response or whether the observed error was a result of inadequate modelling of structural and material nonlinearities or other effects such as damage development in the specimens. Good correlation between experimental and finite element predictions of the out-of-plane displacements and load-axis strains has been demonstrated. Predicted local material strain development has been related to the structural deformation characteristics. Consideration of individual strain values, however, could not predict which of several competing failure modes would determine the actual crippling response. Neither could the strain data provide any quantitative prediction of the crippling loads. Thus, the determination of strength scale effects is hindered by the complex structural-material interaction and the lack of a mechanics-based interactive failure model. / Ph. D.

The postbuckling failure of square composite plates with central holes is analyzed numerically and experimentally. The particular plates studied have stacking sequences of [± 45/0/90]₂₅, [± 45/02]₂₅, [± 45/06]₅. and [± 45]₄₅. A simple plate geometry, one with a hole diameter to plate width ratio of 0.3 , is considered. Failure load, failure mode, and failure locations are predicted numerically by using the finite-element method. The predictions are compared with experimental results. In the experiments in order to be accommodated by the test fixture it is necessary for the plates to be slightly larger than the analYSis region, extending somewhat beyond the supports. The region outside the supports is included in the numerical study. It is shown that not considering these regions can lead to erroneous numerical predictions. In numerical failure analysis the interlaminar shear stresses, as well as the inplane stresses, are taken into account. By comparing the interlaminar shear stress calculations from the finite-element method with analytical results for simple cases, a solid foundation for interlaminar shear stress calculation is established. / Ph. D.

The strength and behavior of gusset plates in buckling is evaluated herein based on data from the experimental investigations conducted by other researchers and the analytical work presented herein. A set of design guidelines has been recommended through the review of the current practice. Representative single and double brace gusset plates normally adopted for connections with compressive bracing/diagonal members in braced frames and trusses, were modeled and analyzed using linear and nonlinear finite element methods to determine the buckling loads. The buckling analysis data along with the test data indicated the occurrence of inelastic buckling of the gusset plates. Current design practice and a set of formulas for determination of gusset plate thickness have been reviewed. A set of guidelines has been recommended for the design and evaluating gusset plate buckling loads.

Natural frequencies of the double and triple-walled carbon nanotubes are determined exactly and approximately for both types. Approximate solutions are found by using Bubnov-Galerkin and Petrov-Galerkin methods. For the first time explicit expressions are obtained for the natural frequencies of double and triple-walled carbon nanotubes for different combinations of boundary conditions. Comparison of the results with recent studies shows that the above methods constitute quick and effective alternative techniques to exact solution for studying the vibration properties of carbon nanotubes. The natural frequencies of the clamped-clamped double-walled carbon nanotubes are obtained; exact solution is provided and compared with the solution reported in the literature. In contrast to earlier investigation, an analytical criterion is derived to establish the behavior of the roots of the characteristic equation. Approximate Bubnov-Galerkin solution is also obtained to compare natural frequencies at the lower end of the spectrum. Simplified version of the Bresse-Timoshenko theory that incorporates the shear deformation and the rotary inertia is proposed for free vibration study of double-walled carbon nanotubes. It is demonstrated that the suggested set yields extremely accurate results for the lower spectrum of double-walled carbon nanotube. The natural frequencies of double-walled carbon nanotubes based on simplified versions of Donnell shell theory are also obtained. The buckling behavior of the double-walled carbon nanotubes under various boundary conditions is studied. First, the case of the simply supported double-walled carbon nanotubes at both ends is considered which is amenable to exact solution. / Then, approximate methods of Bubnov-Galerkin and Petrov-Galerkin are utilized to check the efficacy of these approximations for the simply supported double-walled carbon nanotubes. Once the extreme accuracy is demonstrated for simply supported conditions, the approximate techniques are applied to two other cases of the boundary conditions, namely to clamped-clamped and simply supported-clamped double-walled carbon nanotubes. For the first time in the literature approximate expression for the buckling loads are reported for these boundary conditions. The dynamic deflection of a single-walled carbon nanotube under impact loading is analyzed by following a recently study reported on the energy absorption capacity of carbon nanotubes under ballistic impact. / by Demetris Pentaras. / Thesis (Ph.D.)--Florida Atlantic University, 2009. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.