No description available.

No description available.

No description available.

A computationally efficient analysis approach is developed to predict buckling load of geodesically stiffened composite panels under in-plane loads. The analysis procedure accounts for the contribution of the in-plane extensional and out-of-plane bending stiffnesses of the stiffeners through the use of Lagrange multipliers in an energy method solution. The analysis is used to isolate the effect of various stiffener deformation modes on the buckling load and skin deformation patterns of geodesically stiffened panels under various load combinations. The analysis routines are then coupled with the numerical optimizer ADS to create a package for the design of minimum-mass stiffened panels, subject to constraints on buckling of the panel assembly and material strength failure. Material failure in the skin and stiffeners are estimated using a maximum strain criterion. The design variables that can be used for optimization include thickness of the skin laminate, stiffener thickness and height, and positions of straight stiffeners. Applied loads are uniaxial compression, pure shear, and combined compression-shear. / Master of Science

The Rayleigh-Ritz method is used to determine both the buckling and postbuckling behavior of completely, enclosed prolate spheroidal shells under uniform external pressure. It is assumed that the shells are isotropic and elastic, and have a uniformly thin wall. It is further assumed that the prebuckling state can be described by membrane theory. In this analysis the buckling displacements are confined to a shallow cap located in the region of least curvature of the shell and the boundary of the buckled zone is considered to lie in a plane parallel to the axis of revolution of the shell. A coordinate transformation is performed so that one of the new coordinate curves coincides with the boundary or the buckled zone. The in-plane displacement component is then restricted to be normal to the family of curves which contain this boundary. In addition, both the in-plane and normal displacement components are considered to be functions of one variable only. Series expressions for the in-plane and normal displacement components, each involving M unknown parameters, are inserted into the total potential energy expression. The resulting functional is then minimized with respect to each displacement parameter and also the parameter characterizing the extent of the buckled zone to yield a system of 2M + 1 nonlinear algebraic equations. M + 1 of these equations are eliminated and the Newton-Raphson iterative procedure is employed to obtain the solutions to the remaining M equations. Results are presented for five shell geometries characterized by the ratio of the major diameter to the minor diameter: spherical shell is included as one of the cases. The numerical computations were performed on the Sperry-Rand LARC computer located at David Taylor Model Basin, Washington, D. C. Solutions for increasing values of M are compared in order to evaluate the convergence of the Rayleigh-Ritz method. For the determination of the buckling loads, the maximum value of M used is ten; the solutions to the postbuckling equations are limited to M=5. The numerical results represent a considerable improvement in scope and accuracy over previously published solutions to this problem. This is the first time that the postbuckling behavior associated with higher modes has been considered and it is demonstrated that there is a possibility of a mode shifting phenomenon occurring in the postbuckled state. A discussion is presented of a series of exploratory tests on spheroidal shell models made from an Epon-Versamid resin. Measurements and photographs of the buckled models give qualitative support to the theoretical work presented. / Ph. D.

A method is developed for analyzing the flexural-torsional and lateral-torsional buckling ("tripping") behavior of flanged stiffeners subjected to axial force, end moment, lateral pressure and any combination of these. The effects of cross-sectional distortion, postbuckling behavior of the plate (incorporated by considering the plate effective width), initial imperfections and plasticity are included. The method uses the Rayleigh-Ritz approach. Based on an assumed strain distribution, a displacement field is obtained for the tripping model, and the total potential energy functional is then derived. The strain distribution assumptions coincide with van der Neut's assumption. However, unlike the somewhat obscure differential equation approach given by van der Neut, this study provides a simple, clear, energy approach. Also the resulting method is applicable in the inelastic range, which is not possible with van der Neut's approach. Both the rigid web case and the flexible web case are studied. The effect of plate rotational restraint in the elastic range is accounted for. The method requires only four degrees of freedom and therefore the solution process is rapid. In order to verify the method in the elastic range, a number of sample stiffened panels are analyzed using the ABAQUS foote element program; the results are in quite good agreement. An inelastic tripping model is then developed based on the established elastic model, using deformation theory. Results obtained using the inelastic tripping method are shown to be in good agreement with experimental results, and to be more accurate than other methods. / Ph. D.

Unlike simple columns, rectangular plates which are supported on all edges may carry considerable load beyond their buckling load. Under some conditions it may be advantageous to utilize this additional load-carrying capacity. Von Karman has presented the basic nonlinear differential equations for a plate element undergoing large deflections. In this dissertation the nonlinear equations of von Karman are converted into a s~t of linear equations by expanding the displacements into a prn,er series in terms of an arbitrary parameter. The first few equations of the set can be identified as the usual (linear) small deflection equations. Solution of these and then some of the succeeding equations permits a study of the behavior of the plate at buckling and then beyond into the large deflection range. At present it seems that only plates without initial eccentricities subject to in-plane loading may be solved by the present method. The advantage of the present method is the simplicity of solution. The elastic postbuckling behavior of simply supported rectangular plates subjected to longitudinal compression and subjected to a unIform temperature rise is investigated in detail by solving the first few of the equations. Results are presented for these problems in the form of equations and curves. Load-shortening curves for the compression problem and similar curves for one of the temperature problems solved indicate that changes in buckle pattern will occur. Because of the incompleteness and the inconsistencies of the treatment of the phenomenon of change in buckle pattern in the literature, a study of this phenomenon is made. In order to analyze change in buckle pattern in a rigorous fashion the postbuckling behavior of a symmetric three element column on a nonlinear elastic foundation is determined. It is indicated how the principles learned from the column analysis may be applied qualitatively to plate problems. The results for the plate in compression are compared to previous theoretical results and to experiment. For a square plate the present results agree with previous exact results. For an infinitely long plate the present thesis gives more accurate (lower) loads than previous results. Experimental results which have not been reported previously are described in this thesis, and results from these and other experiment are compared with the present theory. Comparisons are made for total shortening and local strains and deflections which indicate good agreement between experimental results and theoretical results. / Doctor of Philosophy

This research presents an analytical investigation of the buckling of eccentrically stiffened orthotropic cylinders and includes the influence of prebuckling deformations. Nonlinear equilibrium equations and boundary conditions are derived by using energy principles. The stiffened cylinder consists of a cylindrical shell made of a homogeneous orthotropic material with eccentric stiffeners on its surface. The rings, or circumferential stiffeners, are considered to be located discretely on circumferential lines along the length of the cylinder and the stringers, or longitudinal stiffeners, are considered to be closely spaced so that their properties can be averaged (smeared out) over the stringer spacing. The stiffeners are considered to be beam elements, to be equally spaced, and to have the stiffener twisting accounted for in an approximate manner. Non-linear Donnell type strain-displacement relations for the shell and the stiffeners are defined and the strain energy of the stiffener-cylinder system is formulated. The governing nonlinear equilibrium equations and boundary conditions are then obtained by the principle of minimum potential energy and the fundamental lemma of calculus of variations. The discrete ring terms are included in the nonlinear equilibrium equations by use of a Dirac delta function. By a perturbation of the nonlinear equilibrium equations and boundary conditions, a set of nonlinear prebuckling equations and boundary conditions and linear buckling equations and boundary conditions are obtained which govern the prebuckling deformations and stresses and buckling of a stiffened orthotropic cylinder with discrete rings. Solutions of the prebuckling and buckling equations are obtained for classical simple support boundary conditions and for loadings of axial compression, lateral pressure, and combinations of axial compression and external or internal pressure. The solutions are obtained by the method of finite differences in which the governing equations and boundary conditions are changed to a system of second order differential equations which are then written in terms of finite differences at stations along the length of the cylinder. The difference equations are formulated in terms of a matrix equation which is solved by a modified G~ussian elimination technique. Solutions of the prebuckling and buckling equations for the case where the rings are considered to be smeared out are presented for comparison with the discrete case. A Galerkin solution of the buckling equations for discrete rings assuming classical prebuckling deformations is also presented in the Appendix. Computed results for two types of contemporary stiffened cylinders are presented in order to study and illustrate the importance of prebuckling deformations, discrete rings, and eccentrically applied compressive loads. The results show that the predicted buckling loads for stiffened cylinders may be substantially affected by using an analysis which takes into account prebuckling deformations. / Doctor of Philosophy

The local buckling, postbuckling, and crippling (failure) of channel, zee, and I- and J-section stiffeners made of AS4/3502 graphite-epoxy unidirectional tape are studied by experiment and analysis. Thirty-six specimens were loaded in axial compression as intermediate length columns. Examination of the experimental results indicates the existence of a number of damage initiation modes, all of which involve either delamination in some part of the specimen or local material strength failure in a comer of the specimen. The ratio of the flange width to thickness has a strong influence on the buckling stress and damage initiation mode. The inner corner radius strongly affects the buckling and crippling stresses for the I- and J-section specimens. Comparison of the numerical results from a computer code for shell analysis (STAGS) with experimental data shows good correlation prior to buckling and at the buckling load, but diminished agreement in the postbuckling region. This lack of postbuckling correlation is attributed to the neglecting of transverse shearing deformations in the structural theory, inaccuracies in the modeling of in-plane boundary conditions, and damage initiation in the experimental specimens. A plane stress failure analysis for five of the specimens shows the compressive fiber mode criterion of Hashin correlates reasonably well with the first detectable damage event. Equilibrium is used to develop interlaminar stress equations for classical laminated plate theory that require high order derivatives of the displacements. Derivatives computed from discrete displacement data using the Discrete Fourier Transform are inaccurate due to the Gibbs phenomenon. / Master of Science

The large deformation response of composite beams subjected to a dynamic axial load was studied. The beams were loaded with a moderate amount of eccentricity to promote bending. The study was primarily experimental but some finite element results were obtained. Both the deformation and the failure of the beams were of interest. The static response of the beams was also studied in order to determine the difference between the static and dynamic failure. Twelve different laminate types were tested. The beams tested were 23 in. by 2 in. and generally 30 plies thick. The beams were loaded dynamically with a gravity-driven impactor traveling at 19.6 ft./sec. and quasi-static tests were done on identical beams in a displacement controlled manner. For laminates of practical interest, the failure modes under static and dynamic loadings were identical. Failure in most of the laminate types occurred in a single event involving 40% to 50% of the plies. However, failure in laminates with 30° or 15° off axis plies occurred in several events. All laminates exhibited bimodular properties. The compressive flexural moduli in some laminates was measured to be 1/2 the tensile flexural modulus. No simple relationship could be found among the measured ultimate failure strains of the different laminate types. Using empirically determined flexural properties, a finite element analysis was reasonably accurate in predicting the static and dynamic deformation response. / Master of Science